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New algebras and new connections between/in the algebras of logic and the monoidal algebras.

1 Introduction

Pseudo-MV algebras, the non-commutative generalizations of Chang's MV algebras [4], were introduced in 1999 [14] and developed in [16]. Pseudo-MV algebras are intervals [0, u] in l-groups [8]. The pseudo-Wajsberg algebras are termwise equivalent to the pseudo-MV algebras [2], [3]. Hence, the pseudo-Wajsberg algebras must be connected to a notion that is termwise equivalent to the l-group.

Pseudo-product algebras, the non-commutative generalizations of Haajek's product algebras [18], were introduced in 2002 [6]. Pseudo-product algebras are particular cases of pseudo-BL algebras; they are obtained as bounded subsets -{[infinity]} U [G.sup.-] in l-groups. The pseudo-product algebras are equivalent to pseudo-Hajek(pP) algebras verifying some properties. Hence, that pseudo-Haajek(pP) algebras verifying some properties must be connected also to a notion that is termwise equivalent to the l-group.

That notion is the great piece which missed from the puzzle showing the connections between the algebras of logic and the monoidal algebras and we introduce it in this paper: the l-implicative-group.

The research concerning the implicative-groups, the po-implicative-groups and the l-implicative-groups was done in 2009's; it was presented in international conferences and in 2011 in the preprints [30] and [31].

In 2012, G. Dymek [9] introduced the notion of p-semisimple pseudo-BCI algebra and proved that it is equivalent to the group.

The core of this paper is most of the first preprint [30].

We introduce the implicative-groups, the po-implicative-groups and the l-implicative-groups and prove their term equivalence with the groups, the po-groups and the l-groups, respectively. Since G. Dymek made the connection between the pseudo-BCI algebras and the groups, by introducing the subclass of p-semisimple pseudo-BCI algebras and proving that they are equivalent with the groups, we conclude that the p-semisimple pseudo-BCI algebras are equivalent with the implicative-groups. We define also the p-semisimple po-groups (po-implicative-groups) and prove that they are equivalent with the groups (implicative-groups, respectively). We draw the "map" of some algebras of logic and the analogous "map" of the corresponding monoidal algebras. The announced connections between the l-implicative-groups and the pseudo-Wajsberg algebras and the pseudo-Haajek(pP) algebras verifying some properties are established.

Since the pseudo-MV algebras were initially defined as right-algebras, while the pseudo-product algebras were initially defined as left-algebras, we were obliged to work both with left-algebras and right-algebras. One can find more about left- and right- algebras in [27].

The paper is organized in seven sections, as follows.

In Section 2, we recall some algebras of logic and some monoidal algebras, needed in the paper.

In Section 3, we recall some basic things concerning the groups and we introduce and study the notion of implicative-group. We prove that the implicative-groups are term equivalent to the groups.

In Section 4, we recall some basic things concerning the po-groups and we introduce and study the notion of po-implicative-group. We prove that the po-implicative-groups are term equivalent to the po-groups.

In Section 5, we recall some basic things concerning the l-groups and we introduce and study the notion of I-implicative-group. We prove that the l-implicative-groups are term equivalent to the l-groups.

In Section 6, we study the p-semisimple: pseudo-BCI algebras, po-groups and po-implicative-groups. We draw the "map" of some algebras of logic and the analogous "map" of the corresponding monoidal algebras.

In Section 7, we present the announced connections between the l-implicative-groups and the pseudo-Wajsberg algebras/ the pseudo-Hajek(pP) algebras verifying some properties.

The list of the basic (left and right) properties used in the paper is the following: for all x, y, z,

([pBB.sub.L]) y [[right arrow].sup.L] z [less than or equal to] (z [[right arrow].sup.L] x) [[??].sup.L] (y [[right arrow].sup.L] x), y [[??].sup.L] z [less than or equal to] (z [[??].sup.L] x) [[right arrow].sup.L] (y [[??].sup.L] x) (L comes from "Left"),

([pB.sup.L]) z [[right arrow].sup.L] x [less than or equal to] (y [[right arrow].sup.L] z) [[right arrow].sup.L] (y [[right arrow].sup.L] x), z [[??].sup.L] x [less than or equal to] (y [[??].sup.L] z) [[??].sup.L] (y [[??].sup.L] x),

([pD.sup.L]) y [less than or equal to] (y [[right arrow].sup.L] x) [[??].sup.L] x, y [less than or equal to] (y [[??].sup.L] x) [[right arrow].sup.L] x,

([pM.sup.L]) 1 [[right arrow].sup.L] x = x = 1 [[??].sup.L] x,

([pEq.sup.[less than or equal to]]) x [less than or equal to] y [??]x [[right arrow].sup.L] y = 1 [??] x [[??].sup.L] y = 1,

([pI.sup.L]) x [[right arrow].sup.L] 1 = x [[??].sup.L] 1,

([p*.sup.L]) x [less than or equal to] y [??] z [[right arrow].sup.L] x [less than or equal to] z [[right arrow].sup.L] y, z [[??].sup.L] x [less than or equal to] z [[??].sup.L] y,

([p**.sup.L]) x [less than or equal to] y [??] y [[right arrow].sup.L] z [less than or equal to] x [[right arrow].sup.L] z, y [[??].sup.L] z [less than or equal to] x [[??].sup.L] z,

([pRe.sup.L]) (Reflexivity) x [less than or equal to] x, i.e. x [[right arrow].sup.L] x = 1 = x [[??].sup.L] x,

([pAn.sup.L]) (Antisymmetry) x [less than or equal to] y and y [less than or equal to] x [??] x = y,

([pTr.sup.L]) (Transitivity) x [less than or equal to] y and y [less than or equal to] z [??] x [less than or equal to] z,

(pL) (Last element) x [less than or equal to] 1,

([pWV.sup.L]) (x [conjunction] y) [[right arrow].sup.L] z [greater than or equal to] (x [[right arrow].sup.L] z) [disjunction] (y [[right arrow].sup.L] z), (x [conjunction] y) [[??].sup.L] z [greater than or equal to] (x [[??].sup.L] z) [disjunction] (y [[??].sup.L] z);

([pBB.sup.R])) y [[right arrow].sup.R] z [greater than or equal to] (z [[right arrow].sup.R] x) [[??].sup.R] (y [[right arrow].sup.R] x), y [[??] sup.R] z [greater than or equal to] (z [[??] sup.R] x) [[right arrow].sup.R] (y [[??] sup.R] x) (R comes from "Right"),

([pB.sup.R]) z [[right arrow].sup.R] x [greater than or equal to] (y [[right arrow].sup.R] z) [[right arrow].sup.R] (y [[right arrow].sup.R] x), z [[??] sup.R] x [greater than or equal to] (y [[??] sup.R] z) [[??] sup.R] (y [[??] sup.R] x),

([pD.sup.R]) y [greater than or equal to] (y [[right arrow].sup.R] x) [[??] sup.R] x, y [greater than or equal to] (y [[??] sup.R] x) [[right arrow].sup.R] x,

([pM.sup.R]) 0 [[right arrow].sup.R] x = x = 0 [[??] sup.R] x,

[mathematical expression not reproducible]

([pId.sup.R]) x [[right arrow].sup.R] 0 = x [[??] sup.R] 0,

([p*.sup.R]) x [less than or equal to] y [??] z [[right arrow].sup.R] x [less than or equal to] z [[right arrow].sup.R] y, z [[??] sup.R] x [less than or equal to] z [[??] sup.R] y,

([p**.sup.R]) x [less than or equal to] y [??] y [[right arrow].sup.R] z [less than or equal to] x [[right arrow].sup.R] z, y [[??] sup.R] z [less than or equal to] x [[??] sup.R] z,

([pRe.sup.R]) (Reflexivity) x [greater than or equal to] x, i.e. x [[right arrow].sup.R] x = 0 = x [[??] sup.R] x,

([pAn.sup.R]) (Antisymmetry) x [greater than or equal to] y and y [greater than or equal to] x [??] x = y,

([pTr.sup.R]) (Transitivity) x [greater than or equal to] y and y [greater than or equal to] z [??] x [greater than or equal to] z,

(pF) (First element) x [greater than or equal to] 0,

([pWV.sup.R]) (x [conjunction] y) [[right arrow].sup.R] z [less than or equal to] (x [[right arrow].sup.R] z) [disjunction] (y [[right arrow].sup.R] z), (x [conjunction]y) [[??] sup.R] z [less than or equal to] (x [[??] sup.R] z) [disjunction] (y [[??] sup.R] z);

[mathematical expression not reproducible]

[mathematical expression not reproducible]

[mathematical expression not reproducible]

(pcoR) (pseudo-coresiduum) [mathematical expression not reproducible] min{x | y [symmetry] x [greater than or equal to] z};

(pPR)=(pRP) x [??] y [less than or equal to] z [??] x [less than or equal to] y [[right arrow].sup.L] z [??] y [less than or equal to] x [[??].sup.L] z,

(pScoR)=(pcorRS) x [symmetry] y [greater than or equal to] z [??] x [greater than or equal to] y [[right arrow].sup.R] z [??] y [greater than or equal to] x [[??] sup.R] z;

([pDN.sup.L]) (pseudo-Double Negation) [(x[~.sup.L]).sup.-L] = x = [(x[-.sup.L]).sup.~L], where [x.sup.-L] = x [[right arrow].sup.L] 0 and [x.sup.~L] = x [[??].sup.L] 0,

([pDN.sup.R]) (pseudo-Double Negation) [(x[~.sup.R]).sup.-R] = x = [(x[-.sup.R]).sup.~R], where [x.sup.-R] = x [[right arrow].sup.R] 1 and [x.sup.~R] = x [[??] sup.R] 1;

([pC.sup.L]) x [disjunction] y = (x [[??].sup.L] y) [[right arrow].sup.L] y = (x [[right arrow].sup.L] y) [[??].sup.L] y,

([pC.sup.R]) x [conjunction] y = (x [[right arrow].sup.R] y) [[??] sup.R] y = (x [[??] sup.R] y) [[right arrow].sup.R] y;

([p@rel.sup.L]) x [[right arrow].sup.L] y [less than or equal to] (x [??] z) [[right arrow].sup.L] (y [??] z), x [[??].sup.L] y [less than or equal to] (z [??] x) [[??].sup.L] (z [??] y),

([p@rel.sup.R]) x [[right arrow].sup.R] y [greater than or equal to] (x [symmetry] z) [[right arrow].sup.R] (y [symmetry] z), x [[??] sup.R] y [greater than or equal to] (z [symmetry] x) [[??] sup.R] (z [symmetry] y);

([p@.sup.L]) x [[right arrow].sup.L] y = (x [??] z) [[right arrow].sup.L] (y [??] z), x [[??].sup.L] y = (z [??] x) [[??].sup.L] (z [??] y),

([p@.sup.R]) x [[right arrow].sup.R] y = (x [symmetry] z) [[right arrow].sup.R] (y [symmetry] z), x [[??] sup.R] y = (z [symmetry] x) [[??] sup.R] (z [symmetry] y);

([pprel.sup.L]) (pseudo-prelinearity) (x [[right arrow].sup.L] y) [disjunction] (y [[right arrow].sup.L] x) = 1 = (x [[??].sup.L] y) [disjunction] (y [[??].sup.L] x),

([pdiv.sup.L])(pseudo-divisibility) x [conjunction] y = (x [[right arrow].sup.L] y) [??] x = x [??] (x [[??].sup.L] y),

([pprel.sup.R]) (pseudo-prelinearity) (x [[right arrow].sup.R] y) [conjunction] (y [[right arrow].sup.R] x) = 0 = (x [[??] sup.R] y) [conjunction] (y [[??] sup.R] x),

([pdiv.sup.R]) (pseudo-divisibility) x [disjunction] y = (x [[right arrow].sup.R] y) [symmetry] x = x [symmetry] (x [[??] sup.R] y);

([pP1.sup.L]) x [conjunction] [x.sup.-L] = 0 = x [conjunction] [x.sup.~L],

(([pP2.sup.L]) ([[z.sup.-L]).sup.-L] [??] [(x [??] z) [[right arrow].sup.L] (y [??] z)] [less than or equal to] x [[right arrow].sup.L] y, ([[z.sup.~ (L]).sup.~ (L] [??] [[(z [??] x) [??].sup.L] (z [??] y)] [less than or equal to] x [[??].sup.L] y,

([pP1.sup.R]) x [disjunction] [x.sup.-R] = 1 = x [disjunction] [x.sup.~R],

([pP2.sup.R]) ([[z.sup.-R]).sup.-R] [symmetry] [(x [symmetry]z) [[right arrow].sup.R] (y [symmetry]z)] [greater than or equal to] x [[right arrow].sup.R] y, ([[z.sup.~R]).sup.~ (R] [symmetry] [(z [symmetry]x) [[??] sup.R] (z [symmetry]y)] [greater than or equal to] x [[??] sup.R] y;

(pG1) (Associativity) x + (y + z) = (x + y) + z,

(pG2) x + 0 = x = 0 + x,

(pG3) x + (-x) = 0 = (-x) + x,

(pG4) x [less than or equal to] y [??] a + x [less than or equal to] a + y and x + a [less than or equal to] y + a,

(pG4') x [less than or equal to] y [??] a + x [less than or equal to] a + y [??] x + a [less than or equal to] y + a,

(pG5) -(-x) = x,

(pG6) -0 = 0,

(pG7) x [less than or equal to] y [??] -y [less than or equal to] -x,

(pG8) a + (x [disjunction] y) + b = (a + x + b) [disjunction] (a + y + b) and dually

(pG9) a + (x [conjunction] y) + b = (a + x + b) [conjunction] (a + y + b),

(pG10) -(x [disjunction] y) = (-x) [conjunction] (-y) and dually

(pG11) -(x [conjunction] y) = (-x) [disjunction] (-y),

(pG12) x [disjunction] y = x - (x [conjunction] y) + y = [(x [conjunction] y) - x] + y = x + [(x [conjunction] y) [??] y] and dually

(pG13) x [conjunction] y = x - (x [disjunction] y) + y = [(x [disjunction] y) - x] + y = x + [(x [disjunction] y) [??] y],

(pG14) The lattice (G, [disjunction], [conjunction]) is distributive;

(#) x + y = z [??] x = y - z [??] y = x [??] z,

([alpha]) (x [right arrow] y) + x = y = x + (x [??] y),

([beta]) x - (y + x) = y = x [??] (x + y),

([#.sup.[less than or equal to]])x + y [less than or equal to] z [??] x [less than or equal to] y - z [??] y [less than or equal to] x [??] z,

([#.sup.[greater than or equal to]])x + y [greater than or equal to] z [??] x [greater than or equal to] y [right arrow] z [??] y [greater than or equal to] x [??] z;

(pBB) y [right arrow] z = (z [right arrow] x) [??] (y [right arrow] x), y [??] z = (z [??] x) [right arrow] (y [??] x),

(pB) z [right arrow] x = (y [right arrow] z) [right arrow] (y [right arrow] x), z [??] x = (y [??] z) [??] (y [??] x),

(pD) y = (y [right arrow] x) [??] x, y = (y [??] x) - x,

(pM) 0 [right arrow] x = x = 0 [??] x,

(pEq=) x = y [??] x - y = 0 [??] x [??] y = 0,

(Id) x [right arrow] 0 = x [??] 0,

(p*) x [less than or equal to] y z [right arrow] x [less than or equal to] z [right arrow] y, z [??] x [less than or equal to] z [??] y,

(p*Eq) x [less than or equal to] y [??] z [right arrow] x [less than or equal to] z [right arrow] y z [??] x [less than or equal to] z [??] y,

(p**) x [less than or equal to] y [??] y [right arrow] z [less than or equal to] x [right arrow] z, y [??] z [less than or equal to] x [??] z,

(pWV) (x [conjunction] y) [right arrow] z = (x [right arrow] z) [disjunction] (y [right arrow] z), (x [conjunction] y) [??] z = (x [??] z) [disjunction] (y [??] z),

(p@) x [right arrow] y = (x + z) [right arrow] (y + z), x [??] y = (x + z) [??] (y + z),

(pEx) z [??] (y [right arrow] x) = y [right arrow] (z [??] x),

(pRe) x [right arrow] x = 0 = x [??] x,

(pNeg1) (-x) [??] y = (-y) [right arrow] x,

(pNeg2) x [??] (-y) = y [right arrow] (-x).

We believe that the new notions and the new connections presented in this paper open new perspectives in the study of both algebras of logic and monoidal algebras.

2 Preliminaries

In this Section, we recall some of the notions and results needed in the sequel concerning monoidal algebras and algebras of logic.

Most of the surveyed algebras have essentially an order relation (i.e. a binary relation that is reflexive, antisymmetrique and transitive), denoted by [less than or equal to], which usually does not appear explicitely in the definitions. The pair (A, [less than or equal to]) is called a partially-ordered set or poset (po-set) for short.

The presence of the order relation [less than or equal to] implies the presence of the duality principle. Thus, each such algebra has a dual algebra, where the dual order relation is denoted by [greater than or equal to]. We have given names to the dual algebras: left-algebra and right-algebra [10], [21], [22], [27]. Hence, the notions of "left-algebra" and "right-algebra" are dual; they are connected with the left-continuity of a pseudo-t-norm and with the right-continuity of a pseudo-t-conorm, respectively, or with the negative (left) cone and the positive (right) cone of a partially-ordered group, respectively.

Note also that an algebra belongs either to "the world of algebras of logic", i.e. "the world of ([right arrow],[??]), 1", or to "the world of monoidal algebras", i.e. "the world of [??], 1".

Note that the logicians work usually with the logic of truth, where the truth is represented by 1, and there is essentially one implication (or two implications, in the non-commutative case); we could name this logic "left-logic". One can imagine also a "right-logic", as a logic of false, where the false is represented by 0. Hence, the logicians usually work with the left-algebras of logic (or the algebras of left-logic).

By contrast, note that the algebraists work usually with the positive (right) cone of a partially-ordered group (G, [less than or equal to], +, -, 0), where there are essentially a sum [symmetry] = + and an element 0. Sometimes, the negative (left) cone is needed also, where there are essentially a product [??] = + and an element 1 = 0. Hence, the algebraists usually work with the right-monoidal algebras (or monoidal right-algebras).

Consequently, for logicians, the appropriate algebras are the algebras of logic, not the monoidal algebras, and among the algebras of logic, the appropriate algebras are the left-algebras. For algebraists, by contrast, the appropriate algebras are the monoidal algebras, not the algebras of logic, and among the monoidal algebras, the appropriate algebras are the right-algebras. This explains why, for examples, the MV algebras (and hence the pseudo-MV algebras), were introduced as right-monoidal algebras, while the Wajsberg algebras (and hence the pseudo-Wajsberg algebras) were introduced as left-algebras of logic. Consequently, this is the motivation for us to work in this paper both with left-algebras and with right-algebras. Many connections were established in time between algebras of logic and monoidal algebras.

We shall recall in this section the definitions of some left notions from the worlds of monoidal algebras and of algebras of logic and some connections between them. The definitions of the corresponding dual (right) notions and the dual (right) connections are omitted.

2.1 The world of monoidal algebras

In this subsection, we recall the definitions of the monoid and of the left po-ims, po-rims, l-ims, l-rims, pseudo-residuated lattices, pseudo-BL algebras, pseudo-product algebras, pseudo-MV algebras and Boolean algebras.

* A monoid is an algebra of type (2,0), denoted additively by (A, +, 0), or (A, [symmetry], 0), or multiplicatively by (A, ., 1), or (A, [??], 1), verifying (pG1) and (pG2). The monoid is commutative if its binary operation is commutative.

* A partially-ordered, integral left-monoid, or a left-po-im for short, is a structure ([A.sup.L], [less than or equal to], [??], 1) such that ([A.sup.L], [less than or equal to], 1) is a poset with greatest element 1 (i.e. verifying the properties ([pRe.sup.L]), ([pAn.sup.L]), ([pTr.sup.L]) and (pL)), ([A.sup.L], [??], 1) is a monoid and [less than or equal to] is compatible with [??] (i.e. (pG4) holds); integral means that the greatest element of the poset coincides with the neutral element of the monoid. Denote by [po-im.sup.L] their class.

* A commutative left-po-im is named left-po-cim.

* A partially-ordered, residuated, integral left-monoid, or a left-po-rim for short, is a left-po-im ([A.sup.L], [less than or equal to], [??], 1) verifying additionally the property (pR). Denote by [po-rim.sup.L] their class.

* Note that in a left-po-im, (pR) [??] (pPR)=(pRP), where (pPR)=(pRP) is a double Galois connection.

* A commutative left-po-rim is named left-po-crim.

* If the order relation [less than or equal to] is a lattice, we denote by [conjunction] and [disjunction] the Dedekind lattice operations (x [less than or equal to] y [??] x [conjunction] y = x [??] x [disjunction] y = y, for all x, y). Note that the lattice operations are self-dual. We shall say that (A, [conjunction], [disjunction]) is the left-lattice, while (A, [conjunction], [disjunction]) is the right-lattice.

* A lattice-ordered, integral left-monoid, or a left-l-im for short, is a left-po-im with [less than or equal to] a lattice order. A left-l-im will be denoted by ([A.sup.L], [disjunction], [conjunction], 0, 1). Denote by [l-im.sup.L] their class.

* A lattice-ordered, residuated, integral left-monoid, or a left-l-rim for short, is a left-po-rim with [less than or equal to] a lattice order. A left-l-rim will be denoted by ([A.sup.L], [conjunction], [disjunction], [??], 1). Denote by [l-im.sup.L] their class.

* A left-algebra is bounded if, besides 1, there exists a smallest (first) element 0 too (i.e. verifying the property (pF)). Dually, a right-algebra is bounded if, besides 0, there exists a greatest (last) element 1 too (i.e. verifying the property (pL)). If we bound the algebra A, then the bounded algebra will be denoted by [A.sup.b].

* In a bounded left-po-rim ([A.sup.L], [less than or equal to], [??], 0, 1), we can define two implications: for all x [member of] [A.sup.L],

[x.sup.-L] = max{y | y [??] x = 0} and [x.sup.~L] = max{y | x [??] y = 0}.

* A bounded left-algebra is said to be with property ([pDN.sup.L]) if it verifies ([pDN.sup.L]). Denote by [mathematical expression not reproducible] their class.

* A left-non-commutative residuated lattice, or a left-pseudo-residuated lattice, is an algebra

[A.sup.L] = ([A.sup.L], [conjunction], [disjunction], [??], [[right arrow].sup.L], [[??].sup.L], 1)

of type (2, 2, 2, 2, 2, 0) such that ([A.sup.L], [conjunction], [disjunction], 1) is a lattice with last element 1 (under lattice order [less than or equal to]), ([A.sup.L], [??], 1) is a monoid and the property (pRP) holds. Denote by [pR-L.sup.L] their class.

* A left-pseudo-BL algebra is a bounded left-pseudo-residuated lattice

[A.sup.L] = ([A.sup.L], [conjunction], [disjunction], [??], [[right arrow].sup.L],[[??].sup.L], 0, 1)

verifying ([pprel.sup.L]) and ([pdiv.sup.L]). Denote by [pBL.sup.L] their class.

* A left-pseudo-product algebra is a left-pseudo-BL algebra verifying additionally (pP1 (L)) and ([pP2.sup.L]). Denote by [pproduct.sup.L] their class.

* A left-pseudo-MV algebra [10] is an algebra [A.sup.L] = ([A.sup.L], [??], (-), ~, 0, 1) of type (2, 1, 1, 0, 0), where [mathematical expression not reproducible], such that the following axioms are satisfied: for all x, y, z G [A.sup.L],

(pMVL1) x [??] (y [??] z) = (x [??] y) [??] z,

(pMVL2) x [??] 1 = 1 [??] x = x,

(pMVL3) x [??] 0 = [??] 0 x = 0,

(pMVL4) 0 (-) = 1, 0~ = 1,

(pMVL5) [([x.sup.-] [??] [y.sup.-]).sup.~] = [([x.sup.~] [??] [y.sup.~]).sup.-],

(pMVL6) x [??] ([x.sup.~] [symmetry] y) = y [??] ([y.sup.~] [symmetry] x) = (x [symmetry] [y.sup.-]) [??] y = (y [symmetry] [x.sup.-]) [??] x,

(pMVL7) x [symmetry] ([x.sup.-] [??] y) = (x [??] [y.sup.~]) [symmetry] y,

(pMVL8) ([x.sup.~])~ = x,

where [mathematical expression not reproducible]. Denote by [pMV.sup.L] their class.

* Boolean algebras were introduced in 1854 by George Boole. They are only commutative. The most used definition is as a complemented, distributive, bounded lattice. It is self-dual.

A left-Boolean algebra is denoted ([A.sup.L], [conjunction], [disjunction], [sup.-L], 0, 1). Denote by [Boole.sup.L] their class.

2.2 The world of algebras of logic. Connections with the world of monoidal algebras

In this subsection we recall the definitions of left pseudo-BCI algebras, (bounded) pseudo-BCK algebras, (bounded) pseudo-BCK(pP) algebras (lattices), of bounded pseudo-BCK algebras with property (pDN) (pseudo-Double-Negation), pseudo-Hajek(pP) algebras, pseudo-Wajsberg algebras and the implicative-Boolean algebras and their connections with the corresponding monoidal algebras.

2.2.1 The pseudo-BCI algebras and the pseudo-BCK algebras

* The pseudo-BCK algebras were introduced by G. Georgescu and A. Iorgulescu in 2001 [15], as a non-commutative generalization of Iseki's BCK algebras introduced in 1966 [32]. As the BCK algebras, the pseudo-BCK algebras also were introduced as right-algebras with 0: [A.sup.R] = ([A.sup.R], [less than or equal to],*, [omicron], 0). The reversed right-pseudo-BCK algebra is obtained by reversing the operations * and [omicron] i.e. by replacing x * y by y [[right arrow].sup.R] x and x [omicron] y by y [[??] sup.R] x, for all x, y [member of] [A.sup.R] [23]: [A.sup.R] = (A, [less than or equal to], [[right arrow].sup.R], [[??] sup.R], 0).

Dually, the left-pseudo-BCK algebra is obtained from the right-pseudo-BCK algebra by replacing the relation [less than or equal to] by the dual relation, [greater than or equal to], * by [??], [omicron] by # and 0 by 1: [A.sup.L] = ([A.sup.L], [greater than or equal to], [??], #, 1). The reversed left-pseudo-BCK algebra is obtained by reversing the operations [??] and #, i.e. by replacing x[??]y by y [[right arrow].sup.L] x and x#y by y [[??].sup.L] x, for all x,y G [A.sup.L]: [A.sup.L] = ([A.sup.L], [greater than or equal to], [[right arrow].sup.L], 1).

A reversed left-pseudo-BCK algebra, or a left-pBCK algebra for short, is a structure [23], [27]

[A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], 1),

where [less than or equal to] is a binary relation on [A.sup.L], [[right arrow].sup.L] and [[??].sup.L] are binary operations on [A.sup.L] and 1 is an element of [A.sup.L] verifying the axioms: for all x, y, z G [A.sup.L], ([pBB.sup.L]), ([pD.sup.L]), ([pRe.sup.L]), (pL), ([pAn.sup.L]), ([pEq.sup.[less than or equal to]]).

Note that we can define alternatively a left-pBCK algebra as an algebra [A.sup.L] = ([A.sup.L], [[right arrow].sup.L],[[??].sup.L], 1), where the binary relation is then obtained by (pEq[less than or equal to]).

Denote by [pBCK.sup.L] the class of left-pBCK algebras.

* The pseudo-BCI algebras were introduced by W.A. Dudek and Y.B. Jun in 2008 [7], as a non-commutative generalization of Iseki's BCI algebras introduced in 1966 [32]. As the pseudo-BCK algebras, the pseudo-BCI algebras were also introduced as right-algebras with 0.

A reversed left-pseudo-BCI algebra, or a left-pBCI algebra for short, is a structure

[A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L],[[??].sup.L], 1),

verifying the axioms: for all x, y, z G [A.sup.L], ([pBB.sup.L]), ([pD.sup.L]), ([pRe.sup.L]), ([pAn.sup.L]) and (pEq[less than or equal to]).

Hence, a left-pBCK algebra is a left-pBCI algebra verifying (pL) (x [less than or equal to] 1, for all x).

Denote by [pBCI.sup.L] the class of left-pBCI algebras.

* A left-pBCI (left-pBCK) algebra is commutative if [[right arrow].sup.L]=[[??].sup.L]. A commutative left-pBCI (left-pBCK) algebra is a left-BCI (left-BCK) algebra.

Proposition 2.1. The following property holds in a left-pBCK algebra and does not hold in a left-pBCI algebra: for all x, y,

([pK.sup.L]) x [less than or equal to] y [[right arrow].sup.L] x and x [less than or equal to] y [[??].sup.L] x.

Proposition 2.2. (see [27], Proposition 9.5.4)

The following properties hold in a left-pBCI algebra (hence in a left-pBCK algebra): for all x, y, z, ([pM.sup.L]) 1 [[right arrow].sup.L] x = x = 1 [[??].sup.L] x;

([pEx.sup.L]) (Exchange) z [[??].sup.L] (y [[right arrow].sup.L] x) = y [[right arrow].sup.L] (z [[??].sup.L] x);

([pB.sup.L]) z [[right arrow].sup.L] x [less than or equal to] (y [[right arrow].sup.L] z) [[right arrow].sup.L] (y [[right arrow].sup.L] x), z [[??].sup.L] x [less than or equal to] (y [[??].sup.L] z) [[??].sup.L] (y [[??].sup.L] x);

([p*.sup.L]) x [less than or equal to] y z [[right arrow].sup.L] x [less than or equal to] z [[right arrow].sup.L] y and z [[??].sup.L] x [less than or equal to] z [[??].sup.L] y;

([p**.sup.L]) x [less than or equal to] y y [[right arrow].sup.L] z [less than or equal to] x [[right arrow].sup.L] z and y [[??].sup.L] z [less than or equal to] x [[??].sup.L] z;

([pWV.sup.L]) (x conjunction y) [[right arrow].sup.L] z [greater than or equal to] (x [[right arrow].sup.L] z) [disjunction] (y [[right arrow].sup.L] z), (x [conjunction] y) [[??].sup.L] z [greater than or equal to] (x [[??].sup.L] z) [disjunction] (y [[??].sup.L] z).

The following theorem gives us an equivalent definition of a left-pBCI algebra, and hence of a left-pBCK algebra.

Theorem 2.3. ([27], Theorem 9.5.5)

i) Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L],[[??].sup.L], 1) be a structure such that ([A.sup.L], [less than or equal to]) is a poset and the following properties hold:

([pBB.sup.L]), ([pM.sup.L]), (pEq[less than or equal to]). Then, [A.sup.L] is a left-pBCI algebra.

ii) Conversely, let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) be a left-pBCI algebra. Then ([A.sup.L], [less than or equal to]) is a poset and the properties ([pBB.sup.L]), ([pM.sup.L]), (pEq[less than or equal to]) hold.

Hence, we obtain the following equivalent definitions, that will be used in the sequel:

- a left-pBCI algebra is a structure [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) such that ([A.sup.L], [less than or equal to]) is a poset and the properties ([pBB.sup.L]), ([pM.sup.L]), (pEq[less than or equal to]) hold;

- a left-pBCK algebra is a structure [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) such that ([A.sup.L], [less than or equal to], 1) is a poset with last element 1 and the properties ([pBB.sup.L]), ([pM.sup.L]), (pEq[less than or equal to]), (pL) hold.

Proposition 2.4. The following properties hold in a left-pBCI algebra: for all x, y,

([pIdM.sup.L]) x [[right arrow].sup.L] 1 = x [[??].sup.L] 1 [9],

([1.sup.L]) x [[right arrow].sup.L] y [less than or equal to] (y [[right arrow].sup.L] x) [[??].sup.L] 1, x [[??].sup.L] y [less than or equal to] (y [[??].sup.L] x) [[right arrow].sup.L] 1 [9],

([2.sup.L]) (y [[right arrow].sup.L] x) [[right arrow].sup.L] 1 = (y [[right arrow].sup.L] 1) (x [[??].sup.1] 1), (y [[??].sup.1] x) [[??].sup.1] 1 = (y [[??].sup.1] 1) [[right arrow].sup.L] (x [[right arrow].sup.L] 1) [9],

([3.sup.L]) x [less than or equal to] (x [[right arrow].sup.L] 1) x [less than or equal to] (x [[??].sup.L] 1) [[right arrow].sup.L] 1,

([4.sup.L]) y [[right arrow].sup.L] x [less than or equal to] (x [[right arrow].sup.L] 1) [[??].sup.L] (y [[right arrow].sup.L] 1), y [[??].sup.L] x [less than or equal to] (x [[??].sup.L] 1) [[right arrow].sup.L] (y [[??].sup.L] 1).

Remark 2.5.

(1) Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) be a left-pBCI algebra. We can define a negation [sup.-] = [sup.-L] by: for all x G [A.sup.L],

[mathematical expression not reproducible].

Then we have, by Proposition 2.4:

([1'.sup.L]) x [[right arrow].sup.L] y [less than or equal to] (y [[right arrow].sup.L] x)_, x [[??].sup.L] y [less than or equal to] (y [[??].sup.L] x)-,

([2'.sup.L]) (y [[right arrow].sup.L] x)_ = y_ [[??].sup.L] x_, (y [[??].sup.L] x)_ = y_ [[right arrow].sup.L] [x.sup.-],

([3'.sup.L]) x [less than or equal to] (x_)_,

([4'.sup.L]) y [[right arrow].sup.L] x [less than or equal to] x_ [[??].sup.L] y_, y [[??].sup.L] x [less than or equal to] x_ [[right arrow].sup.L] [y.sup.-];

([5'.sup.L]) 1 (_) = 1.

(2) We could define alternatively a left-pBCI algebra as a structure ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], [sup.-], 1).

(3) Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) be a left-pBCK algebra. Then, for all x [member of] [A.sup.L], [x.sup.-] = 1. It follows that the negation does not work here.

Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) be a left-pBCI algebra. Define the "negative (left) cone" of [A.sup.L] as follows:

[mathematical expression not reproducible].

Note that [A.sup.L.sub.-] verifies the property (pL) (x [less than or equal to] 1, for all x).

The following proposition shows that the negative cone of [A.sup.L] is closed under [[right arrow].sup.L], [[??].sup.L] and how to obtain left-pBCK algebras from left-pBCI algebras.

Proposition 2.6. Let [A.sup.L] be a left-pBCI algebra. Then ([mathematical expression not reproducible]) is a left-pBCK algebra.

Proof. It is sufficient to prove that [A.sup.L.sub.-] is closed under [[right arrow].sup.L], [[??].sup.L]. Indeed, if x [less than or equal to] 1 and y [less than or equal to] 1, then x [[right arrow].sup.L] y [less than or equal to] x [[right arrow].sup.L] 1 = 1 and x [[??].sup.L] y [less than or equal to] x [[??].sup.L] 1 = 1, by ([p.sup.* (L]) and by ([pEq.sup.[less than or equal to]]).

2.2.2 Other algebras of logic

* Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) be a left-pBCI algebra (left-pBCK algebra). If the order relation [less than or equal to] is a lattice order relation (x [less than or equal to] y [??] x [conjunction] y = x [??] x [disjunction] y = y, for all x, y G [A.sup.L]), then we say that [A.sup.L] is a left-pBCI lattice (left-pBCK lattice, respectively) and it is denoted by [A.sup.L] = ([A.sup.L], [conjunction], [disjunction], [[right arrow].sup.L], [[??].sup.L], 1). Denote by [pBCI-L.sup.L] ([pBCK-L.sup.L]) the class of left-pBCI lattices (left-pBCK lattices, respectively).

* A left-pBCK algebra with property (pP) (i.e. with pseudo-product), or a left-pBCK(pP) algebra for short, is a left-pBCK algebra such that property (pP) holds. Denote by [pBCK(pP).sup.L] their class.

* If [A.sup.L] is a left-pBCK algebra, then (pP) [??] (pRP).

* We have the following basic equivalence:

[pBCK(pP).sup.L] [??] po-[rim.sup.L],

namely we have:

Theorem 2.7.

(1) Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) be a left-pBCK(pP) algebra, where for all x, y [member of] [A.sup.L]:

[mathematical expression not reproducible].

Define [mathematical expression not reproducible]. Then, [PHI]([A.sup.L]) is a left-po-rim, where for all y, z [member of] [A.sup.L]:

max{x | x [??] y [less than or equal to] z} = y [[right arrow].sup.L] z, max{x | y [??] x [less than or equal to] z} = y [[??].sup.L] z.

(1') Conversely, let [A.sup.L] = ([A.sup.L], [less than or equal to], [??], 1) be a left-po-rim, where for any y, z [member of] [A.sup.L]:

[mathematical expression not reproducible].

Define [mathematical expression not reproducible]. Then, [PSI]([A.sup.L]) is a left-pBCK(pP) algebra, where for all x, y [member of] [A.sup.L]:

min{z | x [less than or equal to] y [[right arrow].sup.L] z} = min{z | y [less than or equal to] x [[??].sup.L] z} = x [??] y.

(2) The above defined mappings [PHI] and *[PSI] are mutually inverse.

* A left-pBCK(pP) lattice is an algebra that is simultaneously a left-pBCK(pP) algebra and a left-pBCK lattice. Denote by pBCK(pP)-L (L) their class.

* We have the equivalence:

[pBCK(pP)-L.sup.L] [??] [l-rim.sup.L].

* If there is an element 0 of a left-pBCK algebra [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) satisfying (pF) (0 [less than or equal to] x, for all x [member of] [A.sup.L]), then 0 is called the first element of [A.sup.L]. A left-pBCK algebra with 0 is called to be bounded and it is denoted by: ([mathematical expression not reproducible]). Denote by [pBCK.sup.bL] their class.

* We have the equivalences:

pBCK[(pP).sup.bL] [??] po-[rim.sup.bL], pBCK(pP)-[L.sup.bL] [??] l-[rim.sup.bL].

* A left-pseudo-Hajek(pP) algebra, or a left-pHajek(pP) algebra for short, is a bounded left-pBCK(pP) lattice verifying ([pprel.sup.L]) and ([pdiv.sup.L]). Denote by pHa[(pP).sup.L] their class.

* We have the equivalences:

pHA[(pP).sup.L] [??] [pBL.sup.L], pHA[(pP).sup.L] + ([pP1.sup.L]) + ([pP2.sup.L]) [??] [pproduct.sup.L].

* Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L],[[??].sup.L], 0,1) be a bounded left-pBCK algebra. Define, for all x [member of] [A.sup.L], two negations, [sup.-] = [sup.-L] and [sup.~] = [sup.~L], by: for all x [member of] [A.sup.L], [mathematical expression not reproducible].

A bounded left-pBCK algebra is said to be with property ([pDN.sup.L]), or involutive, if it verifies ([pDN.sup.L]). Denote by [pBCK.sup.L.sub.(pDN)] their class.

Theorem 2.8. Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 0, 1) be a bounded left-pBCK algebra with property ([pDN.sup.L]) (involutive). Then [A.sup.L] is with property (pP) and we have

[mathematical expression not reproducible], (2.1)

[mathematical expression not reproducible]. (2.2)

Let [A.sup.L] be a bounded left-pBCK algebra with property ([pDN.sup.L]) (involutive). Then we have also:

[mathematical expression not reproducible]. (2.3)

Remark 2.9. [26]

In a bounded left-pBCK(pP) algebra (with the pseudo-product [??]) with property ([pDN.sup.L]) [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 0, 1), we can define the following additional "right" operations (a pseudo-sum [symmetry] and two additional implications, [[??].sup.L], [approximately equal to][>.sup.L]):

[mathematical expression not reproducible], (2.4)

[mathematical expression not reproducible]. (2.5)

Then,

- [x.sup.-] = x [[right arrow].sup.L] 0 = x [[??].sup.L] 1 and [x.sup.~] = x [[??].sup.L] 0 = x [approximately equal to] [>.sup.L] 1;

- the connections between the "right" operations [[symmetry].sup.L], [[??].sup.L], [approximately equal to][>.sup.L] are:

[mathematical expression not reproducible], (2.6)

[mathematical expression not reproducible]; (2.7)

- the "left" operations expressed in terms of "right" operations are:

[mathematical expression not reproducible], (2.8)

[mathematical expression not reproducible]. (2.9)

Consequently, the algebra [A.sup.LR] = ([A.sup.L], [less than or equal to], [[??].sup.L], [approximately equal to][>.sup.L], 0, 1) is a bounded right-pBCK(pS) algebra (with the pseudo-sum [[symmetry].sup.L]) with property ([pDN.sup.R]) (involutive), that is termwise equivalent with [A.sup.L]. We say that [A.sup.L] is selfdual.

Namely, we have, for example, the equivalence from Figure 1.

* We have hence the equivalence:

[mathematical expression not reproducible].

* We say that a left-pBCK lattice [A.sup.L] = ([A.sup.L], [conjunction], [disjunction], [[right arrow].sup.L], [[??].sup.L], 1) is with property ([pC.sup.L]) if property ([pC.sup.L])holds.

Theorem 2.10. Let [A.sup.L] = ([A.sup.L], [conjunction], [disjunction], [[right arrow].sup.L],[[??].sup.L], 1) be a left-pBCK(pP) lattice. Then,

([pC.sup.L]) == ([pprel.sup.L]) + ([pdiv.sup.L]).

Corollary 2.11. Let [A.sup.L] = ([A.sup.L], [conjunction], [disjunction], [[right arrow].sup.L], [[??].sup.L], 0, 1) be a bounded left-pBCK(pP) lattice with ([pC.sup.L])property. Then [A.sup.L] is with ([pDN.sup.L]) property.

* A left-pseudo-Wajsberg algebra is an algebra [A.sup.L] = ([A.sup.L], [[right arrow].sup.L], [[??].sup.L], [sup.-], [sup.~], 1) of type (2, 2, 1, 1, 0), where [sup.-] = [sup.-L] and [sup.~] = [sup.~L], verifying: for all x, y, z [member of] [A.sup.L],

[mathematical expression not reproducible].

Denote by [pW.sup.L] their class.

* We have the equivalences:

[mathematical expression not reproducible].

Theorem 2.12. The bounded left-pBCK(pP) lattice with ([pC.sup.L]) is an equivalent definition of the left-pseudo-Wajsberg algebra.

* A left-implicative-Boolean algebra is an algebra [A.sup.L] = ([A.sup.L], [[right arrow].sup.L], [sup.-], 1) of type (2, 1, 0), where [sup.-] = [sup.-L], verifying the following axioms [28] [17]: for all x, y, z [member of] [A.sup.L],

(G1-L) x [[right arrow].sup.L] (y [[right arrow].sup.L] x) = 1,

(G2-L) [x [[right arrow].sup.L] (y [[right arrow].sup.L] z)] [[right arrow].sup.L] [(x [[right arrow].sup.L] y) [[right arrow].sup.L] (x [[right arrow].sup.L] z)] = 1,

(G3-L) ([y.sup.-] [[right arrow].sup.L] [x.sup.-]) [[right arrow].sup.L] (x [[right arrow].sup.L] y) = 1,

(G4-L)=([An.sup.L]) x [[right arrow].sup.L] y = 1 and y [[right arrow].sup.L] x = 1 implies x = y.

Denote by implicative-[Boole.sup.L] their class.

* We have the equivalence:

implicative-[Boole.sup.L] [??] [Boole.sup.L],

namely we have the following theorem.

Theorem 2.13. [17]

(1) Let [B.sup.L] = ([B.sup.L], [[right arrow].sup.L], [sup.-], 1) be a left-implicative-Boolean algebra.

Define [mathematical expression not reproducible] as follows: for each x, y [member of] [B.sup.L], x [conjunction] y = [(x [[right arrow].sup.L] [y.sup.-]).sup.-], x [disjunction] y = [([x.sup.-] [conjunction] [y.sup.-]).sup.-] = [x.sup.-] [[right arrow].sup.L] y, 0 = [1.sup.-].

Then, [PHI]([B.sub.L]) is a left-Boolean algebra.

(1') Conversely, let [B.sub.L] = ([B.sub.L], [conjunction], [disjunction], [sup.-], 0, 1) be a left-Boolean algebra.

Define [mathematical expression not reproducible] as follows: for every x, y [member of] [B.sup.L], [mathematical expression not reproducible].

Then, [PSI]([B.sup.L]) is a left-implicative-Boolean algebra.

(2) The mappings [PHI] and [PSI] are mutual inverse.

3 Groups and implicative-groups

The results from this section are taken from [30].

3.1 Groups

We consider the group as an algebra G = (G, +, -, 0) of type (2, 1, 0) verifying (pG1), (pG2), (pG3). Denote by group their class.

The group is said to be commutative, or abelian, if x + y = y + x, for all x, y [member of] G.

Proposition 3.1. Let G be a group. Then the properties (pG5), (pG6) hold.

If (G, +, -, 0) is a group, then (G, +, 0) is a monoid and for all x, y [member of] G, x + y = 0 [??] x = -y [??] y = -x and x + y = - [-y + (-x)].

3.1.1 New operations in groups: [right arrow] and [??]. Their Properties

Since the group is, by (pG5), an involutive structure and since in the involutive algebras of logic we have (2.2), we introduce the new operations [right arrow] and [??] on G, called "implications", defined by: for all x, y [member of] G,

[mathematical expression not reproducible]. (3.1)

Note that:

(i) in multiplicative notation, the group is (G, ., [sup.-1], 1) and hence (3.1) becomes:

[mathematical expression not reproducible];

(ii) in ([13], pag. 160), the implication [??] is denoted by \ (x\y = x [??] y) and the implication [right arrow] is replaced by its inverse, denoted by / (i.e. x/y = y [right arrow] x);

(iii) if the group is commutative, then the two implications coincide: [right arrow]=[??].

Remark 3.2. (See Remarks 2.9)

Note that, for all x, y [member of] G, we have:

[mathematical expression not reproducible],

i.e. the addition + is selfdual, the dual of [??] is [right arrow] and the dual of [right arrow] is [??] (the implication [right arrow] can be expressed in terms of [??] and viceversa). Consequently, one can better understand the results from papers [12] and [19] concerning algebras (G, [omicron]) of type (2), with two (one respectively) equations, that are termwise equivalent to groups.

Let (G, +, -, 0) be a group. Then, we have the special property: for all x, y, z [member of] G,

[mathematical expression not reproducible]. (3.2)

Indeed, [mathematical expression not reproducible].

Proposition 3.3. Let (G, +, -, 0) be a group. Then, for all x, y, z [member of] G, we have:

y [right arrow] z = (z [right arrow] x) [??] (y [right arrow] x), y [??] z = (z [??] x) [right arrow] (y [??] x), i.e. (pBB) holds (3.3)

(y [right arrow] x) [??] x = y = (y [??] x) [right arrow] x, i.e. (pD) holds (3.4)

-x = x [right arrow] 0 = x [??] 0, i.e. (Id) holds, (3.5)

x = y [??] x [right arrow] y = 0 [??] x [??] y = 0, i.e. (pEq =) holds. (3.6)

Proof. (3.3): [mathematical expression not reproducible].

(3.4): [mathematical expression not reproducible].

(3.5): -x = 0 -x = -x + 0, by (pG2).

(3.6): Obviously.

Proposition 3.4. In a group (G, +, -, 0), the following properties hold: for all x, y, z [member of] G,

x + y = z [??] x = y [right arrow] z [??] y = x [??] z (see [13], page 160), (3.7)

x = y [??] -y = -x (3.8)

0 [right arrow] x = x = 0 [??] x, (3.9)

z [??] (y [right arrow] x) = y [right arrow] (z [??] x), (3.10)

z [right arrow] x = (y [right arrow] z) [right arrow] (y [right arrow] x), z [??] x = (y [??] z) [??] (y [??] x), (3.11)

x [right arrow] x = 0 = x [??] x, (3.12)

x [??] (-y) = y [right arrow] (-x), (3.13)

-(x [right arrow] 0) = x = -(x [??] 0), (3.14)

[(y [right arrow] x) [??] x] [right arrow] x = y [right arrow] x, [(y [??] x) [right arrow] x] [??] x = y [??] x, (3.15)

x [right arrow] (y [right arrow] z) = (x + y) [right arrow] z, x [??] (y [??] z) = (y + x) [??] z, (3.16)

x [??] y = (-y) [right arrow] (-x), x [right arrow] y = (-y) [??] (-x), (3.17)

(-x) [??] y = (-y) [right arrow] x, (3.18)

(x [right arrow] y) + x = y = x + (x [??] y), (3.19)

x [right arrow] (y + x) = y = x [??] (x + y), (3.20)

x [right arrow] y = (x + z) [right arrow] (y + z), x [??] y = (z + x) [??] (z + y), (3.21)

(y + x) [right arrow] x = -y = (x + y) [??] x, (3.22)

y [right arrow] (x [right arrow] (y + x)) = 0 = y [??] (x [??] (x + y). (3.23)

Proof. (3.7): x + y = z implies x = z - y = y [right arrow] z and y = -x + z = x [??] z, by (pG3); conversely, x = y [right arrow] z, i.e. x = z - y, implies x + y = z and, similarly, y = x [??] z implies x + y = z too, by (pG3).

(3.8): (i) If x = y, then, obviously, -y = -x; (ii) if -y = -x, then, by (i), -(-x) = -(-y), i.e. x = y, by (pG5).

(3.9): [mathematical expression not reproducible].

(3.10): z [??] (y [right arrow] x) = z [??] (x - y) = -z + (x - y); [mathematical expression not reproducible]; thus (3.10) holds.

(3.11): (y [right arrow] z) [right arrow] (y [right arrow] x) = (x - y)-(z - y) = (x - y)+(y - z) = x - z = z [right arrow] x and (y [??] z) [??] (y [??] x) = -(-y+z)+(-y+x) = (-z+y)+(-y+x) = -z+x = z [??] x, by (pG1).

(3.12): [mathematical expression not reproducible].

(3.13): [mathematical expression not reproducible].

(3.14): [mathematical expression not reproducible].

(3.15): [mathematical expression not reproducible].

(3.16): [mathematical expression not reproducible].

(3.17): [mathematical expression not reproducible].

(3.18): (-x) [??] y = -(-x) + y = x + y = x - (-y) = (-y) [right arrow] x; or by (3.13) and (pG5).

(3.19): (x [right arrow] y) + x = (y - x) + x = y; x + (x [??] y) = x + (-x + y) = y.

(3.20): x [right arrow] (y + x) = (y + x) - x = y and x [??] (x + y) = -x + (x + y) = y.

(3.21): [mathematical expression not reproducible].

(3.22): (y + x) [right arrow] x = x - (y + x) = x + (-x - y) = -y and (x + y) [??] x = -(x + y) + x = (-y - x) + x = -y.

(3.23): y [right arrow] (x [right arrow] (y + x)) = y [right arrow] y = 0 and y [??] (x [??] (x + y)) = y [??] y = 0, by (3.20) and (3.12).

Proposition 3.5. In a group (G, +, -, 0), the following properties hold: for all x, y, z, [x.sub.1], [x.sub.2], ..., [x.sub.n] [member of] G,

(a) (x [??] y) + (y [??] z) = x [??] z,

(a') (y [right arrow] z) + (x [right arrow] y) = x [right arrow] z,

(b) ([x.sub.1] [??] [x.sub.2]) + ([x.sub.2] [??] [x.sub.3]) + ... + ([x.sub.n-1] [??] [x.sub.n]) = [x.sub.1] [??] [x.sub.n],

(b') ([x.sub.n-1] [right arrow] [x.sub.n]) + ... + ([x.sub.2] [right arrow] [x.sub.3]) + ([x.sub.1] [right arrow] [x.sub.2]) = [x.sub.1] [right arrow] [x.sub.n].

Proof. (a) (x [??] y) + (y [??] z) = (-x + y) + (-y + z) = -x + (y - y)+ z = -x + 0 + z = -x + z = x [??] z.

(a'), (b), (b') have similar proofs.

Let us introduce the new property: for all x, y, z [member of] G,

(#) x + y = z [??] x = y [right arrow] z [??] y = x [??] z, (it is (3.7)).

Lemma 3.6.

(#) [??] ([alpha]) + ([beta]),

where: for all x, y [member of] A,

([alpha]) (x [right arrow] y) + x = y = x + (x [??] y) (it is (3.19)),

([beta]) x [right arrow] (y + x) = y = x [??] (x + y) (it is (3.20)).

Proof. * (#) [??] ([alpha]) + ([beta]):

([alpha]): [mathematical expression not reproducible] and [mathematical expression not reproducible], which are true.

([beta]): [mathematical expression not reproducible] and [mathematical expression not reproducible], which are true.

* ([alpha]) + ([beta]) [??] (#):

Suppose x + y = z; then [mathematical expression not reproducible] and [mathematical expression not reproducible]. Conversely, suppose x = y [right arrow] z; then [mathematical expression not reproducible]; suppose y = x [??] z; then [mathematical expression not reproducible].

3.2 Implicative-groups

Definition 3.7. An implicative-group is an algebra G = (G, [right arrow], [??] 0) of type (2,2,0), such that the following axioms hold: for all x, y, z [member of] G,

(pBB) y [right arrow] z = (z [right arrow] x) [??] (y [right arrow] x), y [??] z = (z [??] x) [right arrow] (y [??] x),

(pD) y = (y [right arrow] x) [??] x, y = (y [??] x) [right arrow] x,

(pEq=) x = y [??] x [right arrow] y = 0 [??] x [??] y = 0.

Denote by implicative-group their class.

The implicative-group is said to be commutative, or abelian, if x [right arrow] y = x [??] y, for all x, y [member of] G.

Proposition 3.8. Let (G, [right arrow], [??], 0) be an implicative-group. Then, we have, for all x, y, z [member of] G:

(pM) 0 [right arrow] x = x = 0 [??] x,

(pEx) z [??] (y [right arrow] x) = y [right arrow] (z [??] x),

(pRe) x [right arrow] x = 0 = x [??] x,

(Id) x [right arrow] 0 = x [??] 0,

(pB) z [right arrow] x = (y [right arrow] z) [right arrow] (y [right arrow] x), z [??] x = (y [??] z) [??] (y [??] x).

Proof. [mathematical expression not reproducible], hence by (pEq=) we obtain [mathematical expression not reproducible], hence 0 [??] x = x.

(pEx): [mathematical expression not reproducible].

(pRe): [mathematical expression not reproducible].

(Id): [mathematical expression not reproducible].

(pB): [mathematical expression not reproducible].

Remark 3.9. An equivalent definition of the implicative-group is the following: an implicative-group is an algebra G = (G, [right arrow], [??], 0) of type (2, 2, 0), such that (pBB), (pM), (pEq=) hold. Indeed, (pD), (pEq=) imply (pM), while (pM), (pBB) imply (pD).

3.2.1 New operations in implicative-groups: - and +. Their properties

Let G = (G, [right arrow], [??], 0) be an implicative-group. Define an unary operation - by: for all x [member of] G,

[mathematical expression not reproducible]. (3.24)

Proposition 3.10. Let G be an implicative-group. Then, the following properties hold: for all x, y [member of] G,

(pG5) -(-x) = x,

(pG6) -0 = 0,

(pNeg1) (-x) [??] y = (-y) [right arrow] x,

(pNeg2) x [??] (-y) = y [right arrow] (-x).

Proof. (pG5): [mathematical expression not reproducible].

(pG6): [mathematical expression not reproducible].

(pNeg1): [mathematical expression not reproducible].

(pNeg2): [mathematical expression not reproducible].

Note that by (pNeg2), we have, for all x, y:

- [x [right arrow] (-y)] = -[y [??] (-x)]. (3.25)

Since the implicative-group is a special involutive structure, by (pG5) (there is only one negation), and since in the involutive algebras of logic we have (2.1), we introduce the new binary operation + on G by: for all x, y [member of] G,

[mathematical expression not reproducible]. (3.26)

Remark 3.11. By property (3.2) of a group, we could define equivalently x + y by:

[mathematical expression not reproducible].

Proposition 3.12. Let (G, [right arrow], [??], 0) be an implicative-group. Then, (pG1), (pG2), (pG3), ([alpha]) and ([beta]) hold.

Proof. (pG1): [mathematical expression not reproducible].

Corollary 3.13. Let (G, [right arrow], [??], 0) be an implicative-group. Then,

(i) (G, +, -, 0) is a group;

(ii) the property (#) holds.

Proof. (i) follows by Proposition 3.12 and (ii) follows by (i) or by Lemma 3.6.

3.2.2 The groups and the implicative-groups are termwise equivalent

The following theorem establishes the announced result:

Theorem 3.14.

(1) Let G = (G, [right arrow], [??], 0) be an implicative-group.

Define [mathematical expression not reproducible] by: for all x, y [member of] G,

[mathematical expression not reproducible].

Then [PHI](G) is a group.

(1') Conversely, let G = (G, +, -, 0) be a group.

Define [mathematical expression not reproducible] by: for all x, y [member of] G,

[mathematical expression not reproducible].

Then [PSI](G) is an implicative-group.

(2) The maps [PSI] and [PSI] are mutually inverse.

Proof. (1) follows by Proposition 3.3.

(1') follows by Corollary 3.13.

(2): Let [mathematical expression not reproducible]. Then, for all x, y [member of] G, [mathematical expression not reproducible] and [mathematical expression not reproducible], by (pG5).

Let now [mathematical expression not reproducible]. Then, for all x, y [member of] G, x [??] y = -(x + (-y)) = -(-(x [right arrow] (-(-y)))) = x [right arrow] y and x [??] y = -((-y) + x) = -(-(x [??] (-(-y)))) = x [??] y.

Hence we have the situation from Figure 2.

Remark 3.15. (See Remark 3.2) For all x [member of] G, we have:

0 = x [right arrow] x = x [??] x, -x = x [right arrow] 0 = x [??] 0 and -(-x) = x (there is only one involutive negation).

Proposition 3.16. The implicative-group is commutative iff the termwise equivalent group is commutative, i.e. x [right arrow] y = x [??] y for all x, y if and only if x + y = y + x for all x, y.

Proof. x [right arrow] y = x [??] y for all x, y implies -x [right arrow] y = -x [??] y [??] y - (-x) = - (-x) + y, i.e. y + x = x + y, by (pG5). Conversely, x + y = y + x for all x, y implies -x + y = y - x, i.e. x [??] y = x [right arrow] y.

4 Po-groups and po-implicative-groups

The results from this section are taken from [30].

4.1 Po-groups

Recall that a partially-ordered group, or a po-group for short, is a structure G = (G, [less than or equal to], +, -, 0) where (G, +, -, 0) is a group and [less than or equal to] is a partial order on G compatible with +, i.e. (pG4) holds.

Denote by po-group their class.

Corollary 4.1. Let G be a po-group. If x [less than or equal to] y and a [less than or equal to] b, then x + a [less than or equal to] y + b and a + x [less than or equal to] b + y.

If G is a po-group, then [G.sup.-] = {x [member of] G | x [less than or equal to] 0} will be called the negative cone and [G.sup.+] = {x [member of] G | x [greater than or equal to] 0} will be called the positive cone of G.

Corollary 4.2. [G.sup.-] and [G.sup.+] are closed under +.

Proposition 4.3. Let G be a po-group. Then, for all x, y, z [member of] G, we have:

(pG7) x [less than or equal to] y [??] -y [less than or equal to] -x,

(pG4') x [less than or equal to] y [??] a + x [less than or equal to] a + y [??] x + a [less than or equal to] y + a,

([#.sup.[less than or equal to]]) x + y [less than or equal to] z [??] x [less than or equal to] y [right arrow] z [??] y [less than or equal to] x [??] z and dually

([#.sup.[greater than or equal to]]) x + y [greater than or equal to] z [??] x [greater than or equal to] y [right arrow] z [??] y [greater than or equal to] x [??] z.

Proof. ([#.sup.[less than or equal to]]) x [less than or equal to] y [right arrow] z [??] x [less than or equal to] z - y [??] x + y [less than or equal to] z, by (pG4), and y [less than or equal to] x [??] z [??] y [less than or equal to] - x + z [??] x + y [less than or equal to] z, by (pG4);

([#.sup.[greater than or equal to]]) has a similar proof.

Corollary 4.4. Let G be a po-group. Then, for all x, y [member of] G, we have:

(i) y [less than or equal to] 0 [??] x [less than or equal to] y [right arrow] x [??] x [less than or equal to] y [??] x and dually

(ii) y [greater than or equal to] 0 [??] [greater than or equal to] y [right arrow] x [??] x [greater than or equal to] y [??] x.

Proof. Take z = x in ([#.sup.[less than or equal to]]) and then take z=y in ([#.sup.[greater than or equal to]]).

Note that the important properties (pG5) and (pG7) make the operation--be an involution.

Proposition 4.5. Let G be a po-group. Then, the following properties hold: for all x, y, z [member of] G,

(p*) x [less than or equal to] y [??] z [right arrow] x [less than or equal to] z [right arrow] y and z [??] x [less than or equal to] z [??] y,

(p**) x [less than or equal to] y [??] y [right arrow] z [less than or equal to] x [right arrow] z and y [??] z [less than or equal to] x [??] z.

Proof. (p*): Let x [less than or equal to] y; then [mathematical expression not reproducible] and [mathematical expression not reproducible].

(p**): Let x [less than or equal to] y; then by (pG7), -y [less than or equal to] -x; hence, [mathematical expression not reproducible] and [mathematical expression not reproducible].

Corollary 4.6. Let G be a po-group. For all x, y [member of] G:

if x [less than or equal to] y then x [right arrow] y [greater than or equal to] 0, x [??] y [greater than or equal to] 0 and y [right arrow] x [less than or equal to] 0, y [??] x [less than or equal to] 0.

Proof. Let x [less than or equal to] y; then,

- by (p*), x [right arrow] x [less than or equal to] x [right arrow] y, i.e., by (3.12), 0 [less than or equal to] x [right arrow] y; similarly, 0 [less than or equal to] x [??] y;

- by (p**), y [right arrow] x [less than or equal to] x [right arrow] x = 0, i.e. y [right arrow] x [less than or equal to] 0; similarly, y [??] x [less than or equal to] 0.

Lemma 4.7. (see Lemma 3.6)

(i) The properties ([#.sup.[less than or equal to]]) and ([#.sup.[greater than or equal to]]) imply the following properties: for all x, y, z [member of] A,

(#) x + y = z [??] x = y [right arrow] z [??] y = x [??] z,

(pG4') x [less than or equal to] y [??] z + x [less than or equal to] z + y [??] x + z [less than or equal to] y + z,

(p*) x [less than or equal to] y implies z [right arrow] x [less than or equal to] z [right arrow] y, z [??] x [less than or equal to] z [??] y,

(p**) x [less than or equal to] y implies y [right arrow] z [less than or equal to] x [right arrow] z, y [??] z [less than or equal to] x [??] z.

(ii) The following equivalence holds

([#.sup.[less than or equal to]]) + ([#.sup.[greater than or equal to]]) [??] (pG4) + (p*) + (#).

Proof. (i):

(#): properties ([#.sup.[less than or equal to]]) and ([#.sup.[greater than or equal to]]) imply property (#) since (p [left and right arrow] q and r [left and right arrow] s) imply (p and r)[left and right arrow](q and s) and since the order relation [less than or equal to] is antisymmetrique. Recall that (#) [??] ([alpha]) + ([beta]); we shall use this equivalence in the rest of the proof.

(pG4'): Suppose x [less than or equal to] y; by ([beta]), y = z [??] (z + y), hence x [less than or equal to] z [??] (z + y) and [mathematical expression not reproducible], i.e. first part of (pG4) holds. Conversely, [mathematical expression not reproducible].

Suppose again x [less than or equal to] y; by ([beta]), y = z [right arrow] (y + z), hence x [less than or equal to] z [right arrow] (y + z) and [mathematical expression not reproducible]; thus (pG4) holds. Conversely, [mathematical expression not reproducible].

(p*): Suppose x [less than or equal to] y; by ([alpha]), (z [right arrow] x) + z = x, hence (z [right arrow] x) + z [less than or equal to] y and [mathematical expression not reproducible]; by ([alpha]) also, z + (z [??] x) = x, hence z + (z [??] x) [less than or equal to] y and [mathematical expression not reproducible].

(p**): Suppose that x [less than or equal to] y; by (pG4'), [mathematical expression not reproducible] and [mathematical expression not reproducible]; by (pG4') also, [mathematical expression not reproducible] and [mathematical expression not reproducible].

(ii): By (i), it remains to prove that (pG4) + (p*) + (#) imply ([#.sup.[less than or equal to]]) + ([#.sup.[greater than or equal to]]). Indeed, (pG4) + (p*) + (#) imply ([#.sup.[less than or equal to]]):

- Suppose x + y [less than or equal to] z; then by (p*), y [right arrow] (x + y) [less than or equal to] y [right arrow] z and x [??] (x + y) [less than or equal to] x [??] z; hence, by ([beta]), we obtain that x [less than or equal to] y [right arrow] z and y [less than or equal to] x [??] z.

- Conversely, suppose x [less than or equal to] y [right arrow] z; then by (pG4), [mathematical expression not reproducible]; suppose y [less than or equal to] x [??] z; then by (pG4), [mathematical expression not reproducible].

Similarly, (pG4) + (p*) + (#) imply ([#.sup.[greater than or equal to]]).

If the partial order [less than or equal to] is linear (total), then G is a linearly-ordered group or a totally-ordered group. Note that in a linearly-ordered group G we have either x [less than or equal to] 0 or x [greater than or equal to] 0, for any x [member of] G, hence G = [G.sup.-] [union] [G.sup.+].

Note that the presence of the order relation implies the presence of the duality principle. It follows that there are two dual po-groups. If their support sets differ ([G.sub.1] [not equal to] [G.sub.2]), then their unit elements differ and suppose that [0.sub.1] [less than or equal to] [0.sub.2] in the union set [G.sub.1] [union] [G.sub.2]; we then shall call [G.sub.1] as left-po-group and [G.sub.2] as right-po-group. If their support sets coincide (G = [G.sub.1] = [G.sub.2]), we shall say that G is self-dual, i.e. (G, [less than or equal to], +, -, 0) is in the same time left-po-group and right-po-group.

4.2 Po-implicative-groups

Definition 4.8. A partially-ordered implicative-group, or a po-implicative-group for short, is a structure G = (G, [less than or equal to], [right arrow], [??], 0), where (G, [right arrow], [??], 0) is an implicative-group and [less than or equal to] is a partial order on G compatible with [right arrow], [??], i.e. we have: for all x, y, z [member of] G, (p*) x [less than or equal to] y implies z [right arrow] x [less than or equal to] z [right arrow] y and z [??] x [less than or equal to] z [??] y.

Denote by po-implicative-group their class.

Proposition 4.9. Let (G, [less than or equal to], [right arrow], [??], 0) be a po-implicative-group. Then, the following properties hold: for all x, y, a [member of] G,

(pG7) x [less than or equal to] y implies -y [less than or equal to] -x,

(pG4) x [less than or equal to] y implies a + x [less than or equal to] a + y and x + a [less than or equal to] y + a,

(p*Eq) x [less than or equal to] y [??] z [right arrow] x [less than or equal to] z [right arrow] y [??] z [??] x [less than or equal to] z [??] y,

(#) x + y = z [??] x = y [right arrow] z [??] y = x [??] z.

Proof. (pG7): Let x [less than or equal to] y; by (p*), 0 = x [right arrow] x [less than or equal to] x [right arrow] y, then -y = y [??] 0 [less than or equal to] y [??] (x [right arrow] y) = x [right arrow] (y [??] y) = x [right arrow] 0 = -x..

(pG4): Let x [less than or equal to] y; then -y [less than or equal to] -x, by (pG7); by (p*), a [right arrow] (-y) [less than or equal to] a [right arrow] (-x), hence by (pG7) again, we obtain a + x = -(a [right arrow] (-x)) [less than or equal to] -(a [right arrow] (-y)) = a + y and by (p*), a [??] (-y) [less than or equal to] a [??] (-x), hence by (pG7) again, we obtain x + a = -(a [??] (-x)) [less than or equal to] -(a [??] (-y)) = y + a.

(p*Eq): By (p*), it is sufficient to prove that z [right arrow] x [less than or equal to] z [right arrow] y implies x [less than or equal to] y and that z [??] x [less than or equal to] z [??] y implies x [less than or equal to] y. Indeed, z [right arrow] x [less than or equal to] z [right arrow] y implies, by above (pG4), that [mathematical expression not reproducible] and z [??] x [less than or equal to] z [??] y implies, by above (pG4), that [mathematical expression not reproducible].

(#): follows by Corollary 3.13.

Corollary 4.10. Let (G, [less than or equal to], [right arrow], [??], 0) be a po-implicative-group. Then,

(i) (G, [less than or equal to], +, -, 0) is a po-group;

(ii) the properties ([#.sup.[less than or equal to]]) and ([#.sup.[greater than or equal to]]) hold.

Proof. (i): By Corollary 3.13 and Proposition 4.9.

(ii): By Lemma 4.7 (ii).

4.2.1 The po-groups and the po-implicative-groups are termwise equivalent

The announced result follows by the next theorem.

Theorem 4.11. (See Theorem 3.14)

(1) Let G = (G, [less than or equal to], [right arrow], [??], 0) be a po-implicative-group.

Define [mathematical expression not reproducible], where (G, +, -, 0) = [PHI](G, [right arrow], [??], 0), with [PHI] from Theorem 3.14(1).

Then [PHI]'(G) is a po-group.

(1') Conversely, let G = (G, [less than or equal to], +, -, 0) be a po-group.

Define [mathematical expression not reproducible], where (G, [right arrow], [??], 0) = [PSI](G, +, -, 0), with [PSI] from Theorem 3.14(1').

Then [PSI]'(G) is a po-implicative-group.

(2) The maps [PHI]' and [PSI]' are mutually inverse.

Proof. (1) follows by Corollary 4.10.

To prove (1'), by Theorem 3.14 (1'), it remains to prove (p*), which holds by Proposition 4.5.

(2) follows by Theorem 3.14 (2).

Remark 4.12.

(1) There is a strong analogy between the po-groups and the involutive left- and right- po-rims and between the po-implicative-groups and the involutive left- and right- pBCK algebras.

(2) Since pseudo-Wajsberg algebras are termwise equivalent to pseudo-MV algebras [2], [3] and pseudo-MV algebras are intervals in l-groups with strong unit [8] and l-groups are termwise equivalent to l-implicative-groups, it follows that pseudo-Wajsberg algebras are intervals in l-implicative-groups with strong unit; find a direct proof of the last statement is an open problem (see [35]).

5 l-groups, l-implicative-groups

The results from this section are taken from [30].

5.1 l-groups

If the partial order [less than or equal to] is a lattice order, then the po-group G is called lattice-ordered group, or l-group for short, denoted additively by G = (G, [disjunction], [conjunction], +, -, 0). Denote by l-group their class. An introduction in l-groups is [1], see also [5].

Note that an l-group may be linearly-ordered or not, while a linearly-ordered group is an l-group.

Proposition 5.1. Let G be an l-group. Then the properties (pG8), (pG9), (pG10), (pG11), (pG12), (pG13), (pG14) hold.

Corollary 5.2. Let G be an l-group. Then we have, for all x, y [member of] G:

(i) x, y [member of] [G.sup.-] imply x + y [less than or equal to] x [conjunction] y,

(i') x, y [member of] [G.sup.+] imply x + y [greater than or equal to] x [disjunction] y.

Proposition 5.3. In an l-group (G, [disjunction], [conjunction], +, -, 0), the following properties hold, for all x, y, z [member of] G:

(x[disjunction]z) [right arrow] y = (x [right arrow] y)[conjunction](z [right arrow] y), (x[disjunction]z) [??] y = (x [??] y)[conjunction](z [??] y) and dually (5.1)

(x [conjunction] z) [right arrow] y = (x [right arrow] y) [disjunction] (z [right arrow] y), (x [conjunction] z) [??] y = (x [??] y) [disjunction] (z [??] y); (5.2)

y [right arrow] (x[disjunction]z) = (y [right arrow] x)[disjunction](y [right arrow] z), y [??] (x[disjunction]z) = (y [??] x)[disjunction](y [??] z) and dually (5.3)

y [right arrow] (x [conjunction] z) = (y [right arrow] x) [conjunction] (y [right arrow] z), y [??] (x [conjunction] z) = (y [??] x) [conjunction] (y [??] z); (5.4)

[(x [conjunction] 0) [??] 0] [conjunction] 0 = 0, [(x [conjunction] 0) [right arrow] 0] [conjunction] 0 = 0 and dually (5.5)

[(x [disjunction] 0) [??] 0] [disjunction] 0 = 0, [(x [disjunction] 0) [right arrow] 0] [disjunction] 0 = 0. (5.6)

5.1.1 Examples

(0) Z = (Z, [disjunction] = max, [conjunction] = min, +, -, 0) is a self-dual, linearly ordered, commutative l-group and R = (R, [less than or equal to], +, -, 0) is a self-dual commutative linearly-ordered l-group, where x [right arrow] y = y + (-x).

(1) D = (D = (0, +[infinity]) = {x [member of] R | x > 0}, [less than or equal to], ., [sup.-1], 1) is an commutative linearly-ordered l-group, where x [right arrow] y = y . ([1/x]) = [y/x].

(2) [G.sub.R] = ([G.sub.R] = (0, [infinity]) x R, [less than or equal to], +, -, [0.sub.[G.sub.R]]) [10] with:

[mathematical expression not reproducible], [0.sub.[G.sub.R]] = (1,0), -(a, b) = ([1/a], -[b/[a.sup.2]]) and with the lexicographic order [less than or equal to] is a linearly ordered commutative l-group, where: (a, b) [right arrow] (c, d) = ([c/a], [[ab-bc]/[a.sup.2]]).

(2') Its dual is [29] [G.sub.L] = ([G.sub.L] = (-[infinity], 0) x R, [less than or equal to], +, -, [0.sub.[G.sub.L]]) with: [mathematical expression not reproducible] and with the lexicographic order [less than or equal to].

[G.sub.R] is the linearly ordered commutative right-l-group and [G.sub.L] is the linearly ordered commutative left-l-group, since (-1, 0) < (1,0) in the set [G.sub.L] [union] [G.sub.R].

(3) Let G = [G.sub.R] = (0, [infinity]) x R and define a binary operation "+" on G by:

[mathematical expression not reproducible].

The operation "+" is associative, non-commutative and

[mathematical expression not reproducible].

The order relation is the lexicographic order: (a, b) < (c, d) iff a < c or (a = c and b < d). It makes G a lattice and the structure [G.sub.R] = (G, [disjunction], [conjunction], +, -, [0.sub.G]) a linearly-ordered, non-commutative, l-group [10], where:

[mathematical expression not reproducible].

(3') Dually, let G = [G.sub.L] = (-[infinity], 0) x R and define a binary operation "+" on G by:

[mathematical expression not reproducible].

The operation "+" is associative, non-commutative and

[mathematical expression not reproducible],

since [mathematical expression not reproducible].

The order relation is the lexicographic order; it makes G a lattice and the structure [G.sub.L] = (G, [conjunction], [disjunction], +, -, [0.sub.G]) a linearly-ordered, non-commutative l-group [29], where:

[mathematical expression not reproducible].

[G.sub.R] is the right-l-group and [G.sub.L] is the left-l-group, since (-1,0) < (1, 0) in the set [G.sub.L] [union] [G.sub.R].

5.2 l-implicative-groups

Let G = (G, [less than or equal to], [right arrow], 0) be a po-implicative-group. If the partial order relation [less than or equal to] is a lattice order relation, with the lattice operations [conjunction] and [disjunction] defined by: x [less than or equal to] y [??] x[conjunction]y = x [??] x [disjunction] y = y, then G is a lattice-ordered implicative-group, or an l-implicative-group for short, denoted G = (G, [disjunction], [conjunction], [right arrow], 0). Denote by l-implicative-group their class. Note that an l-implicative-group may be linearly-ordered or not, while a linearly-ordered implicative-group is an l-implicative-group.

Corollary 5.4. Let (G, [disjunction], [conjunction]) be the reduct ofan l-group or ofan l-implicative-group. Then, [G.sup.-] and [G.sup.+] are closed under [disjunction] and [conjunction].

Proof. Let x,y [member of] [G.sup.+], i.e. x [greater than or equal to] 0 and y [greater than or equal to] 0. Then, 0 is a lower bound of {x, y}; hence, 0 [less than or equal to] x [conjunction] y, i.e. x [conjunction] y [member of] [G.sup.+]; since 0 [less than or equal to] x [conjunction] y [less than or equal to] x [disjunction] y, it follows that 0 [less than or equal to] x [disjunction] y, i.e. x [disjunction] y [member of] [G.sup.+] too.

Let now x, y [member of] [G.sup.-], i.e. x [less than or equal to] 0 and y [less than or equal to] 0. Then, 0 is an upper bound of {x, y}; hence, 0 [greater than or equal to] x [disjunction] y, i.e. x [disjunction] y [member of] [G.sup.-]; since 0 [greater than or equal to] x [disjunction] y [greater than or equal to] x [conjunction] y, it follows that 0 [greater than or equal to] x [conjunction] y, i.e. x [conjunction] y [member of] [G.sup.-] too.

5.2.1 The l-groups and the l-implicative-groups are termwise equivalent

The announced result follows immediately (by Theorem 4.11):

Corollary 5.5. l-implicative-groups are termwise equivalent to l-groups.

6 p-semisimple algebras

6.1 p-semisimple pBCI algebras

G. Dymek [9] made the connection between the pseudo-BCI algebras (defined as algebras (A, [less than or equal to], *, [omicron], 0)) and the groups, by introducing, as in the commutative case, the subclass of p-semisimple pseudo-BCI algebras and by proving that these are equivalent with the groups.

Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], 1) be a left-pBCI algebra. We say that [A.sup.L] is p-semisimple if x [less than or equal to] 1 implies x = 1, for any x [member of] [A.sup.L]. [9]

Proposition 6.1. [9] Let [A.sup.L] be a left-pBCI algebra. Then the following are equivalent:

(i) [A.sup.L] is p-semisimple;

(ii) for all x, y [member of] [A.sup.L], if x [less than or equal to] y, then x = y.

Proof. (i) [??] (ii): Suppose that x [less than or equal to] y; then y [[right arrow].sup.L] x [less than or equal to] y [[right arrow].sup.L] y, by ([p*.sup.L]); hence y [[right arrow].sup.L] x [less than or equal to] 1, by ([pRe.sup.L]). Hence, y [[right arrow].sup.L] x = 1, by (i), i.e. y [less than or equal to] x, by ([pEq.sup.[less than or equal to]]). Since x [less than or equal to] y and y [less than or equal to] x, it follows by ([pAn.sup.L]) that x = y.

(ii) [??] (i): Obviously. □

Theorem 6.2. [9]

(1) Let [A.sup.L] = ([A.sup.L], [less than or equal to], [[right arrow].sup.L], [[??].sup.L]1) be a p-semisimple left-pBCI algebra.

Define [[PHI].sub.p-s]([A.sup.L]) = ([A.sup.L], +, -, 0) by: for any x, y [member of] [A.sup.L],

[mathematical expression not reproducible].

Then, [[PHI].sub.p-s]([A.sup.L]) is a group.

(1') Conversely, let A = (A, +, -, 0) be a group.

Define [[PSI].sub.p-s]([A.sup.L]) = (A, [less than or equal to], [[right arrow].sup.L], [[??].sup.L], 1) by: for any x, y [member of] A,

[mathematical expression not reproducible].

Then, [[PSI].sub.p-s]([A.sup.L]) is a p-semisimple left-pBCI algebra.

(2) The above defined mappings [[PHI].sub.p-s] and [[PSI].sub.p-s] are mutually inverse.

By Theorems 6.2 and 3.14, we obtain immediately the following:

Corollary 6.3. The p-semisimple-left-pBCI algebras "coincide" with the implicative-groups and, dually, the p-semisimple-right-pBCI algebras "coincide" with the implicative-groups.

6.2 The "map" of the hierarchies of algebras of logic

Looking for the hierarchies of the involved left-algebras of logic, we obtain the hierarchies from Figure 3; the four algebras which have no correspondent in the monoidal algebras are represented by *. The hierarchies of the involved right-algebras of logic can be obtained in mirror with those from Figure 3.

6.3 p-semisimple po-groups

Looking also at the connections presented in Section 2 between the involved algebras of logic and the corresponding monoidal algebras, we note that the place of the left-pBCI algebras in the hierarchies from Figure 3 is taken by the left-po-group. Indeed, the following proposition shows how to obtain left-po-ims from left-po-groups.

Let [G.sup.L] = ([G.sup.L], [less than or equal to], [??], -, 1) be a left -po-group (in an appropriate multiplicative notation). Define the "negative (left) cone" of [G.sup.L] as follows:

[mathematical expression not reproducible].

Note that [G.sup.L.sub.-] verifies the property (pL) (x [less than or equal to] 1, for all x).

Proposition 6.4. (See Proposition 2.6)

Let [G.sup.L] = ([G.sup.L], [less than or equal to], [??], -, 1) be a left-po-group. Then ([G.sup.L.sub.-], [less than or equal to], [??], 1) is a left-po-im.

Proof. First, note that if x [less than or equal to] 1, then -x [greater than or equal to] - 1 = 1, by (pG6), (pG7); hence [G.sup.L.sub.-] is not closed under the negation. Then note that 1 [less than or equal to] 1, by the reflexivity of [less than or equal to] (property (pRe)); hence 1 [member of] [G.sup.L.sub.-]. It remains to prove that [G.sup.L.sub.-] is closed under [??], which is true by Corollary 4.2.

Let [G.sup.L] = ([G.sup.L], [less than or equal to], [??], -, 1) be a left-po-group. We say that [G.sup.L] is p-semisimple if x [less than or equal to] 1 implies x = 1, for any x [member of] [G.sup.L].

Then, the following results are obtained.

Proposition 6.5. (See Proposition 6.1)

Let [G.sup.L] be a left-po-group. Then the following are equivalent:

(i) [G.sup.L] is p-semisimple;

(ii) for all x, y [member of] [G.sup.L], if x [less than or equal to] y, then x = y.

Proof. (i) [??] (ii): Suppose that x [less than or equal to] y; then [mathematical expression not reproducible], by (pG4); hence -y [??] x = 1, by (i); then, y [??] x = 1, by (3.1), and hence y = x, by (pEq=).

(ii) [??] (i): Obviously.

Theorem 6.6. (See Theorem 6.2)

(1) Let [G.sup.L] = ([G.sup.L], [less than or equal to], +, -, 0 be a p-semisimple left-po-group.

Define [[alpha].sub.m]([G.sup.L]) = ([G.sup.L], +, -, 0). Then, [[alpha].sub.m]([G.sup.L]) is a group.

(1') Conversely, let G = (G, +, -, 0) be a group.

Define [[beta].sub.m]([G.sup.L]) = (G, [less than or equal to], +, -, 0) by: for any x, y [member of] G, x [less than or equal to] y [??] y - x = 0 [??] -x + y = 0.

Then, [[beta].sub.m]([G.sup.L]) is a p-semisimple left-po-group.

(2) The above defined mappings [[alpha].sub.m] and [[beta].sub.m] are nutually inverse.

Proof. (1): Obviously.

(1'): It is sufficient to prove that [less than or equal to] is an order relation verifying (pG4) and that x [less than or equal to] 1 implies x = 1. Indeed, first note that x [less than or equal to] y [??] x [right arrow] y = 0 [??] x [??] y = 0, by (3.1); then we obtain that x [less than or equal to] y [??] x = y, by (3.6). Hence, [less than or equal to] is an order relation, verifies obviously (pG4) and, if x [less than or equal to] 1, we obviously have x = 1.

6.4 The "map" of the hierarchies of monoidal algebras

We are now able to draw in Figure 4 the hierarchies existing between the involved monoidal left-algebras.

The hierarchies between the involved monoidal right-algebras are in the mirror with those from Figure 4.

6.5 p-semisimple po-implicative-groups

A po-implicative-group (G, [less than or equal to], [right arrow], [??], 0) is called p-semisimple if x [less than or equal to] 0 implies x = 0.

By Theorem 4.11, the po-implicative groups are term equivalent with the po-groups, hence the p-semisimple po-implicative-groups are term equivalent with the p-semisimple po-groups. But the p-semisimple po-groups are equivalent ("coincide") with the groups, by Theorem 6.6 and its dual. And the groups are term equivalent with the implicative-groups, by Theorem 3.14. It follows that we have the following result: the p-semisimple po-implicative-groups are equivalent ("coincide") with the implicative-groups, namely:

Theorem 6.7. (See Theorem 6.6)

(1) G = (G, [less than or equal to], [right arrow], [??], 0) be a p-semisimple po-implicative-group.

Define [[alpha].sub.ig](G) = (G, [right arrow], [??], 0). Then, [[alpha].sub.ig](G) is an implicative-group.

(1') Conversely, let G = (G, [right arrow], [??], 0) be an implicative-group.

Define [[beta].sub.ig](G) = (G, [less than or equal to], [right arrow], [??], 0) by: for any x,y [member of] G, x [less than or equal to] y [??] x [right arrow] y = 0 [??] x [??] y = 0.

Then, [[beta].sub.ig](G) is a p-semisimple po-implicative-group.

(2) The above defined mappings [[alpha].sub.ig] and [[beta].sub.ig] are mutually inverse.

Remark 6.8. Let (G, [less than or equal to], [right arrow], [??], 0) be a po-implicative-group. Define the negative cone [G.sup.-] = {x [member of] G | x [less than or equal to] 0} and the positive cone [G.sup.+] = {x [member of] G | x [greater than or equal to] 0}. Then note that, while the negative cone [A.sup.L.sub.-] of a left-pBCI algebra [L.sup.L] is closed under [[right arrow].sup.L], [[??].sup.L] (Proposition 2.6) and the negative cone [G.sup.L.sub.-] of a left-po-group [G.sup.L] is closed under [??] = + (Corollary 4.2), [G.sup.-] and [G.sup.+] are not closed under [right arrow], [??].

Open problem Find the properties verified by the left-po-implicative-groups, and by their subclasses (see Section 2), and draw the "map" of hierarchies analogous to that from Figure 3. The research is already begun in the next section.

7 Connections between the l-implicative-groups and the pseudo-BCK lattices

The results from this section are taken from [30]. The following theorem is the main result of this section.

Theorem 7.1. Let G = (G, [disjunction], [conjunction], [right arrow], [??], 0) be an l-implicative-group.

(1) Define, for all x, y [member of] [G.sup.-]:

[mathematical expression not reproducible], (7.1)

[mathematical expression not reproducible]. (7.2)

Then, [G.sup.L] = ([G.sup.-], [disjunction], [conjunction], [[right arrow].sup.L], [[??].sup.L], 1 = 0) is a distributive left-pBCK(pP) lattice (with the pseudo-product [??] = +) verifying properties ([pC.sup.L]) and ([p@.sup.L]) (see property ([pP2.sup.L]) from the definition of a left-pseudo-product algebra and (3.21)).

(1') Define, for all x, y [member of] [G.sup.+]:

[mathematical expression not reproducible], (7.3)

[mathematical expression not reproducible]. (7.4)

Then, [G.sup.R] = ([G.sup.+], [disjunction], [conjunction], [[right arrow].sup.R], [[??] sup.R], 0 = 0) is a distributive right-pBCK(pS) lattice (with the pseudo-sum [direct sum] = +) verifying the dual properties ([pC.sup.R])and ([p@.sup.R]).

Proof. (1):

- [G.sup.-] is closed under the lattice operations [conjunction] and [disjunction] of G, by Corollary 5.4, and ([G.sup.-], [conjunction], [disjunction]) is a distributive lattice, since (G, [disjunction], [conjunction]) is a distributive lattice. Then [G.sup.-] is closed under [[right arrow].sup.L] and [[??].sup.L]

- We prove now that [G.sup.L] is a left-pBCK algebra (we use the equivalent definition), i.e. ([G.sup.-], [less than or equal to], 1 = 0) is a poset with greatest element and the properties ([pBB.sup.L]), ([pM.sup.L]), ([pEq.sup.[less than or equal to]]) hold.

* Obviously, ([G.sup.-], [less than or equal to], 1 = 0) is a poset with greatest element, where x [less than or equal to] y iff x [conjunction] y = x iff x [disjunction] y = y.

* ([pBB.sup.L]): We must prove that for all x, y, z [member of] [G.sup.-],

(z [[right arrow].sup.L] x) [[??].sup.L] (y [[right arrow].sup.L] x) [greater than or equal to] y [[right arrow].sup.L] z, (z [[??].sup.L] x) [[right arrow].sup.L] (y [[??].sup.L] x) [greater than or equal to] y [[??].sup.L] z. (7.5)

First, we shall prove that

(z [[right arrow].sup.L] x) [[??].sup.L] (y [[right arrow].sup.L] x) = (y [[right arrow].sup.L] z) [disjunction] (y [[right arrow].sup.L] x), (z [[??].sup.L] x) [[right arrow].sup.L] (y [[??].sup.L] x) = (y [[??].sup.L] z) [disjunction] (y [[??].sup.L] x). (7.6)

Indeed, denote A = (z [[right arrow].sup.L] x) [[??].sup.L] (y [[right arrow].sup.L] x). Then,

[mathematical expression not reproducible].

Similarly, denote B = (z [[??].sup.L] x) [[right arrow].sup.L] (y [[??].sup.L] x). Then

[mathematical expression not reproducible].

Thus, (7.6) holds and therefore (7.5) holds.

* ([pM.sup.L]): We must prove that for all x [member of] [G.sup.-] we have

1 [[right arrow].sup.L] x = x = 1 [[??].sup.L] x. (7.7)

Indeed, [mathematical expression not reproducible] and [mathematical expression not reproducible].

Thus, (7.7) holds.

* ([pEq.sup.[less than or equal to]]): We must prove that for all x, y [member of] [G.sup.-] we have

[mathematical expression not reproducible]. (7.8)

which follows immediately by the definitions of [[right arrow].sup.L] and [[??].sup.L].

Hence, ([G.sup.-], [conjunction], [conjunction], [[right arrow].sup.L], [[??].sup.L], 1 = 0) is a left-pBCK lattice.

- To prove that [G.sup.L] is with the pseudo-product [??] = +, it is equivalent to prove that for every x, y, z [member of] [G.sup.-], properties (pRP) hold, i.e.

[mathematical expression not reproducible], (7.9)

[mathematical expression not reproducible]. (7.10)

(7.9): By ([#.sup.[less than or equal to]]), we have x [??] y = x + y [less than or equal to] z [??] x [less than or equal to] y [right arrow] z; if x [less than or equal to] y [right arrow] z, then, since x [less than or equal to] 0, we obtain x [less than or equal to] (y [right arrow] z) [conjunction] 0 = y [[right arrow].sup.L] z, i.e. x [less than or equal to] y [[right arrow].sup.L] z; conversely, if x [less than or equal to] y [[right arrow].sup.L] z, then x [less than or equal to] y [right arrow] z, since y [[right arrow].sup.L] z = (y [right arrow] z) A 0 [less than or equal to] y [right arrow] z.

(7.10): similarly.

- To prove that the left-pBCK(pP) lattice [G.sup.L] satisfies property ([pC.sup.L]) means to prove that for all x, y [member of] [G.sup.-] we have

(y [[right arrow].sup.L] x) [[??].sup.L] x = y [disjunction] x = (y [[??].sup.L] x) [[right arrow].sup.L] x. (7.11)

Indeed, [mathematical expression not reproducible].

And [mathematical expression not reproducible]; thus, (7.11) holds.

- The left-pBCK(pP) lattice [G.sup.L] satisfies property ([p@.sup.L]) by (3.21). Thus, (1) holds.

(1') has a similar proof, where the analogous of (7.9), (7.10) are respectively:

[mathematical expression not reproducible], (7.12)

[mathematical expression not reproducible]. (7.13)

Remark 7.2. By Theorem 2.10, the properties ([pprel.sup.L]) and ([pdiv.sup.L]) are verified by [G.sup.L]. Dually, the properties ([pprel.sup.R])and ([pdiv.sup.R]) are verified by [G.sup.R].

We obtain obviously the following corollary:

Corollary 7.3. If G is a linearly-ordered l-implicative-group, then:

(1) the two implications from Theorem 7.1 (1) become, for all x, y [member of] [G.sup.-]:

[mathematical expression not reproducible].

(1') the two implications from Theorem 7.1 (1') become, for all x, y [member of] [G.sup.+]:

[mathematical expression not reproducible].

Lemma 7.4. Let (G, [disjunction], [conjunction], [right arrow], [??], 0) be an l-implicative-group. Let u' < 0 and u > 0 from G. Then,

(1) the interval [u', 0] = {x [member of] [G.sup.-] | u' [less than or equal to] x [less than or equal to] 0} [subset] [G.sup.-] is closed under [[right arrow].sup.L] and [[??].sup.L] in [G.sup.L] from Theorem 7.1 (1),

(1') the interval [0, u] [subset] [G.sup.+] is closed under [[right arrow].sup.R] and [[??] sup.R] in [G.sup.R] from Theorem 7.1 (1').

Proof. (1): Let u' [member of] [G.sup.-] and let u' [less than or equal to] x, y [less than or equal to] 0. By the properties of the left-pseudo-BCK algebra [G.sup.L], we have u' [less than or equal to] y [less than or equal to] x [[right arrow].sup.L] y = (x [right arrow] y) [conjunction] 0 [less than or equal to] 0 and u' [less than or equal to] y [less than or equal to] x [[??].sup.L] y = (x [??] y) [conjunction] 0 [less than or equal to] 0. Hence, u' [less than or equal to] x [[right arrow].sup.L] y [less than or equal to] 0 and u' [less than or equal to] x [[??].sup.L] y [less than or equal to] 0.

(1'): Let u [member of] [G.sup.+] and let 0 [less than or equal to] x, y [less than or equal to] u. By the properties of the right-pseudo-BCK algebra [G.sup.R], we have u [greater than or equal to] y [greater than or equal to] x [[right arrow].sup.R] y = (x [right arrow] y) [disjunction] 0 [greater than or equal to] 0 and u [greater than or equal to] y [greater than or equal to] x [[??] sup.R] y = (x [??] y) [disjunction] 0 [greater than or equal to] 0. Hence, u [greater than or equal to] x [[right arrow].sup.R] y [greater than or equal to] 0 and u [greater than or equal to] x [[??] sup.R] y [greater than or equal to] 0.

Let us "bound" the algebras [G.sup.L] and [G.sup.R] from Theorem 7.1 in two different ways: first, with an "internal" element, then, with an "external" element. We obtain the equivalent of known results, by Theorem 7.1, Remark 7.2 and Lemma 7.4:

Corollary 7.5.

* (i) Let [G.sup.L] = ([G.sup.-], [conjunction], [disjunction], [[right arrow].sup.L], [[??].sup.L], 1 = 0) from, Theorem, 7.1 (1). Let us "bound" this algebra in two different ways:

1) Let us take u' < 0 from [G.sup.-] and consider the interval [u', 0]. Then the algebra

[mathematical expression not reproducible]

is a bounded left-pBCK(pP) lattice (with the pseudo-product [mathematical expression not reproducible] with property ([pC.sup.L]), i.e. is an equivalent definition of the left-pseudo-Wajsberg algebra (see [24] and [34], [11] for the commutative case)

[mathematical expression not reproducible].

2) Let us consider a symbol -[infinity] distinct from, the elements of G. Define [mathematical expression not reproducible] and extend the operations [mathematical expression not reproducible] as follows:

[mathematical expression not reproducible].

We extend the lattice order relation [less than or equal to] as follows: we put -[infinity] [less than or equal to] x, for any [mathematical expression not reproducible]. Then, the algebra

[mathematical expression not reproducible]

is a left-pseudo-Hajek(pP) algebra (with the pseudo-product [[??].sub.2]) verifying properties ([pP1.sup.L]) and ([pP2.sup.L]) (see [24]).

* (i') Let [G.sup.R] = ([G.sup.+], [disjunction], [conjunction], [[right arrow].sup.R], [[??] sup.R], 0 = 0) from Theorem 7.1 (1'). Let us "bound" this algebra in two different ways:

1') Let us take u > 0 from [G.sup.+] and consider the interval [0, u] = {x [member of] [G.sup.+] | 0 [less than or equal to] x [less than or equal to] u}. Then the algebra

[mathematical expression not reproducible]

is a bounded right-pBCK(pS) lattice (with the pseudo-sum [mathematical expression not reproducible] with property ([pC.sup.R]), i.e. is an equivalent definition of the right-pseudo-Wajsberg algebra

[mathematical expression not reproducible].

2') Let us consider a symbol +[infinity] distinct from the elements of G. Define [mathematical expression not reproducible] and extend the operations [mathematical expression not reproducible] as follows:

[mathematical expression not reproducible].

We extend the lattice order relation [greater than or equal to] as follows: we put + [infinity] [greater than or equal to] x, for any x [member of] [G.sup.+.sub.+[infinity]].

Then, the algebra

[mathematical expression not reproducible]

is a right-pseudo-Hajek(pS) algebra (with the pseudo-sum [[direct sum].sub.2]) verifying the dual properties (pP[1.sup.R]) and (pP[2.sup.R]).

Acknowledgement. All my gratitude to Sergiu Rudeanu and Paul Flondor for their valuable and useful remarks and suggestions that helped me to improve the presentation. I am specially grateful to George Georgescu for his encouragements in doing this research.

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Afrodita Iorgulescu

Academy of Economic Studies, Bucharest, Romania

E-mail: afrodita.iorgulescu@ase.ro
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