Abelization of quasi-hypergroups, [H.sub.v]-rings and transposition [H.sub.v]-groups as a categorial reflection.

Abstract

The aim of this paper is to use the construction of the abelization of hyperstructures (quasi-hypergroups, weakly associative groups, weakly associative hyperrings and transposition [H.sub.v]-groups) for the construction of categorial reflections.

AMS Subject Classification: 20N20.

Keywords: quasi-hypergroups, weakly associative groups, weakly associative hyperrings, transposition [H.sub.v]-groups, join spaces.

1. Introduction

This paper deals with the constructions of commutative hyperstructures from non-commutative ones. The construction of this abelization is expressed in terms of category theory and as in the classical case of abelian groups it creates a functor called reflector. Various classes of hyperstructures have been investigated in many countries since 1934. One motivating example of such generalizations which can be found among others was given by Marty [16]. Let (G, *) be a group and H any subgroup of G. Then G/H = {x x H : x [member of] G} becomes a hypergroup, where the hyperoperation is defined in a usual manner: [bar.x] [dot encircle] [bar.y] = {[bar.z] : z [member of] [bar.x] x [bar.y], where [bar.x] = x x H.

Apart from the motivation for the investigation of hyperstructures coming from noncommutative algebra, geometrical structures and other mathematical fields, there exist such physical phenomenon as nuclear fission. Nuclear fission occurs when a heavy nucleus such as [U.sup.235] splits or fissions into two smaller nuclei. As a result of this fission process we can get several dozens of different combinations of two medium-mass elements and several neutrons. More precisely, the input of this reaction is always the same--heavy uranium is bombarded with neutrons but the result is in general different--there are about 90 different daughter nuclei that can be formed. So, the nuclear reaction can be considered to be an example of a hyperoperation.

There is another typical example of the situation when the result of interaction between two particles is the whole set of particles. It is the interaction between a photon with certain energy and an electron. The result of this interaction is not deterministic. A photo-electric effect or Coulomb repulsion effect or changeover of a photon onto a pair electron and positron can arise.

It is to be noted that a similar situation which occurs during uranium fission appears during several nuclear fissions as well. The result depends on conditions. Although the input 2 particles are the same, the output can be variant. It can differ both in the number of arising particles and in their kind.

In recent times the so called weak hyperstructures have been in the centre of scientific interest of many algebraists. Weak hyperstructures with one or two hyperoperations appearing in connection with the representation theory (T. Vougiouklis 1994) are the generalization of usual hyperstructures in such a way that hyperidentities, as associative and commutative law, are substituted by the requirement of the set incidence of left and right hand sides. Although the used construction of abelization does not preserve the associativity law (only the weakened form of this law) it can be applied onto various classes of weak hyperstructures. In this connection, let us note, that quasi-hyperstructures (hypergroupoids satisfying the reproduction axiom) have a deep motivation resulting from geometry.

Further, weak axioms are generalized by using proximity relations because two incident sets can be considered as sets being near one another with respect to the discrete proximity. Nevertheless, the proximity relation seems to be a very useful tool for the investigation of weak hyperstructures. Many classical constructions in the domain of mathematical structures have categorial character of reflection. So this paper ranks among the series of papers on reflective substructures of corresponding structures in algebra, topology, and category theory.

Transposition hypergroups, which are called also non-commutative join spaces, belong to important binary hyperstructures. In fact, they are hypergroups possessing certain property of transposition. Numerous hypergroups, well known in the literature, which form wide classes as join spaces, weak cogroups, double cosets spaces, polygroups and canonical hypergroups including usual groups and also some geometrically motivated non-commutative hyperstructures--belong to transposition hypergroups. The above mentioned classes of hypergroups are intensively investigated both as general structures and also in connection with their various applications. At the end of this paper the abelization of join space is given.

2. Reflections in Categories

Let A be a subcategory of B with embedding functor E: A [right arrow] B. If D is a class of B-morphisms (i.e.D [subset] Mor(B)), then A is called D-reflective in B provided that for each B-object B there exists an A-reflection ([r.sub.B], [A.sub.B]) such that each [r.sub.B] [member of] D. By an A-reflection ([r.sub.B], [A.sub.B]) we mean--as usual--an E-universal map ([r.sub.B],[A.sub.B]) for a B-object B, i.e., [r.sub.B] : B [right arrow] E([A.sub.B]) is a B-morphism for [A.sub.B] [member of] ObA, and for each A' [member of] ObA and each morphism f : B [right arrow] E(A') there exists a unique A-morphism [bar.f] : [A.sub.B] [right arrow] A' such that the following triangle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

commutes. By this construction a functor R: B [right arrow] A is defined, which is a left adjoint of E: A [right arrow] B, called a reflector for A.

In case D is the class of all epimorphisms (monomorphisms) of B we say that A is epireflective (monoreflective) in B.

3. Basic Notions from Hypergroup Theory

The definitions of hyperstructures are overtaken from [3] and [4]. The definitions of weak hyperstructures are overtaken from [17]. You can find them also in [6], [8], [10], [9] and [11]. Let us recall some of them.

A hypergroupoid is a pair (M; [omicron]), where M is a nonempty set and further [omicron]: M x M [right arrow] [P.sup.*](M) is a binary hyperoperation. ([P.sup.*](M) is the system of all nonempty subsets of M). A semihypergroup is an associative hypergroupoid, i.e., hypergroupoid satisfying the equality (a [omicron] b) [omicron] c = a [omicron] (b [omicron] c) for every triple a; b; c [member of] M. A quasi-hypergroup is a hypergroupoid (M; [omicron]) fulfilling the reproduction axiom, i.e., a [omicron] M = M = M [omicron] a for any a [member of] M. A hypergroup is an associative hypergroupoid (M; [omicron]), satisfying the reproduction axiom.

Let (H, *) and (H', *) be hypergroupoids. Then a mapping f : (H, *) [right arrow] (H', *) is called (inclusion) homomorphism if it satisfies the condition: f(x * y) [subset or equal to] f(x) * f(y) for all pairs x; y [member of] H.

Let M [not equal to] 0 be a set. The hyperoperation [omicron]: M x M [right arrow] [P.sup.*](M) is called a weakly associative if (a [omicron] (b [omicron] c) [intersection]((a [omicron] b) [omicron] c) [not equal to] 0 ; for any triple a; b; c [member of] H. A weak semihypergroup ([H.sub.v]-semigroup) is a set H (H [not equal to] 0) equipped with a weakly associative hyperoperation. A [H.sub.v]-semigroup is called a weak hypergroup if moreover the reproduction axiom is satisfied.

The [H.sub.v]-ring is a triple (R, +, x), where (R, +) is [H.sub.v]-group, (R, x) is a [H.sub.v]-semigroup and the hyperoperation "x" is weakly distributive with respect to the hyperoperation "+", which means for all elements x; y; z [member of] R that

x x (y + z) [intersection] (x x y + x x z) [not equal to] 0, (x + y) x z [intersection] (x x z + y x z) [not equal to] 0.

The [H.sub.v]-ring homomorphism or weak homomorphism of [H.sub.v]-ring (R, +, x) into another one (S; +, x) is mapping f : R [right arrow] S such that for any pair x; y [member of] R holds f(x + y) [intersection] (f(x) + f(y) [not equal to] 0 and f(x x y) [intersection] (f(x) x f(y)) [not equal to] 0.

The mapping of a [H.sub.v]-ring (R, +, x) into another one (S, +, x) is called an inclusion homomorphism or simply a homomorphism if for all elements x; y [member of] R holds f(x) + f(y) [subset] f(x + y) and f(x) x f(y) [subset] f(x x y).

4. Reflective Subcategories of Hyperstructures

By [[DELTA].sub.H] we mean the diagonal of the Cartesian product H x H, i.e., [[DELTA].sub.H] = {[x, x]; x [member of] H}. Let us define a mapping D: H [right arrow] H x H by D(x) = [x, x] for all x [member of] H, i.e., [[DELTA].sub.H] = D(H). For the proof of the next Lemma see [8].

Lemma 4.1. Let (H, x) be a hypergroupoid. Define a hyperoperation "*" on the diagonal [[DELTA].sub.H] as follows: [x; x] * [y; y] = D(x x y [union] y x x) = {[u; u]; u [member of] x x y [union] y x x} for any pair [x; x]; [y; y] [member of] [[DELTA].sub.H]. Then the following assertions hold:

1 For any hypergroupoid (H, x) we have that ([[DELTA].sub.H], *) is a commutative hypergroupoid.

2 If (H, x) is a weakly associative hypergroupoid, then the hypergroupoid ([[DELTA].sub.H], *) is weakly associative, as well.

3 If (H, x) is a quasi-hypergroup, the hypergroupoid ([[DELTA].sub.H], *) also satisfies the reproduction law, i.e., it is a quasi-hypergroup.

4 If (H, x) is associative, i.e., it is a semihypergroup, then the hypergroupoid ([[DELTA].sub.H]; *) is weakly associative (but not associative in general).

Let (H, x) be a quasi-hypergroup and [r.sub.H](x) = D(x) = [x, x]. Then the mapping [r.sub.H]: (H, x) [right arrow] (D(H), *) is a homomorphism of quasi-hypergroups.

Let quasi-hypergroups ([H.sub.1, x 1]), ([H.sub.2, x 2]) be given. Suppose f: ([H.sub.1, x 1]) [right arrow] ([H.sub.2, x 2]) is a homomorphism. For an arbitrary [x, x] [member of] D([H.sub.1]) we define

[bar.f]([x, x]) = [f(x), f(x)] [member of] D([H.sub.2]).

Consider the following diagram:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (D1)

In [8] the following lemma is proved.

Lemma 4.2. The following assertions hold:

1 The mapping [bar.f]: (D([H.sub.1]), [*.sub.1]) [right arrow] (D([H.sub.2]), [*.sub.2]) is a homomorphism.

2 The diagram (D1) is commutative for any homomorphism f:([H.sub.1, x 1]) [right arrow] ([H.sub.2, x 2]).

3 The homomorphism [bar.f] completes the diagram (D1) for any homomorphism f: ([H.sub.1, x 1]) [right arrow] ([H.sub.2, x 2]) uniquely.

4 The homomorphism [r.sub.H]: (H, x) [right arrow] ([[DELTA].sub.H], *) is a bimorphism, i.e., both a mono- and an epimorphism, for any quasi-hypergroup (H, x).

Let QHG be the category of all quasi-hypergroups and their homomorphisms, AQHG be its full subcategory of all commutative (i.e., abelian) quasi-hypergroups.

Define a functor F: QHG [right arrow] AQHG by F((H, x)) = (D(H), *) = ([[DELTA].sub.H], *) for any quasi-hypergroup (H, x) [member of] Ob(QHG), F(f) = [bar.f]: F(([H.sub.1, x 1])) [right arrow] F(([H.sub.2, x 2]) for any pair of quasi-hypergroups and any homomorphism f : ([H.sub.1, x 1]) [right arrow] ([H.sub.2, x 2]). By the above considerations (concentrated in Lemma 4.2) we can prove the next.

Theorem 4.3. ([8] or [11]) The following assertions hold:

1 The functor F : QHG [right arrow] AQHG is a reflector; more precisely the pair ([r.sub.H], ([[DELTA.sub.H], *)) is a reflection for (H, x) [member of] Ob(QHG), hence the category AQHG is a bireflective (i.e., mono- and epi-) full subcategory of the category QHG.

2 Let [H.sub.v] G be the full subcategory of all [H.sub.v]-groups of the category QHG, A[H.sub.v]G its full subcategory of all commutative [H.sub.v]-groups. Then the functor G: [H.sub.v]G [right arrow] A[H.sub.v]G is a reflector and A[H.sub.v]G is a bireflective full subcategory of the category [H.sub.v]G.

In the preceeding theorem the functor G: [H.sub.v]G [right arrow] A[H.sub.v]G was defined as a restriction of the functor F, i.e., G(H, x) = F(H, x) for any (H, x) [member of] Ob ([H.sub.v]G) and similarly for morphisms.

Let (R, +, x) [member of] ObA[H.sub.v]R whenever (R, +, x) is a [H.sub.v]-ring such that x + y = y + x and x x y = y x x for any pair x; y [member of] R. Similarly as above we define for an arbitrary [H.sub.v]-ring (R, +, x) the hyperoperations "[direct sum], "[dot encircle]" on the diagonal D(R) = [[DELTA].sub.R] for all pairs x; y [member of] R by

[x, x] [direct sum] [y, y] = {[u, u]; u [member of] (x + y) [union] (y + x)},

[x, x] [dot encircle] [y, y] = {[v, v]; v [member of] (x x y) [union] (y x x)}.

Theorem 4.4. Let [H.sub.v]R be the category of all [H.sub.v]-rings and their inclusion homomorphisms, A[H.sub.v]R be its full subcategory of all commutative [H.sub.v]-rings. Then the functor [PHI]: [H.sub.v]R [right arrow] A[H.sub.v]R defined by

[PHI](R, +, x) = (D(R), [direct sum], [dot encircle]), [PHI](f) = [bar.f] for any (R, +, x) [member of] Ob [H.sub.v]R

and any morphism f [member of] Mor [H.sub.v]R, f : (R, +, x) [right arrow] (S, +, x), is a reflector; more precisely the pair ([r.sub.R], ([[DELTA].sub.R], [direct sum], [dot encircle])) is an A[H.sub.v]R-reflection for any (R, +, x) [member of] Ob([H.sub.v]R). Thus A[H.sub.v]R is a reflective full subcategory of the category [H.sub.v]R.

5. Abelization of Join Spaces

The described construction of abelization can be used on join spaces too. We can get the reflection as well.

The concept of join space was introduced by Walter Prenowitz and used by him and afterwards together with James Jantosciak to rebuilt several branches of geometry. This concept was motivated by various kinds of "join" occurring in ordered and partially ordered linear geometry, spherical geometry and protective geometry. These branches of mathematics assign to two distinct points an appropriate connective: in linear geometry, segment; in spherical geometry, minor arc of great circle; in projective geometry, line.

In the classical notion of a join space commutativity of the corresponding binary hyperoperation is assumed. However, in papers of James Jantosciak [14, 15] noncommutative join spaces or--in other words--(noncommutative) transposition hypergroups are investigated and interesting results are presented concerning that topics. It is to be noted that many well-known hypergroups forming wide classes as join spaces, weak cogroups, double coset spaces, polygroups and canonical hypergroups, including ordinary groups and also some geometrically motivated noncommutative hyperstructures are all transposition hypergroups.

Recall first the basic notions.

A hypergroup (H; x) is called a transposition hypergroup if it satisfies the transposition axiom: For all a; b; c; d [member of] H the relation b\a [approximately equal to] c/d implies a x d [approximately equal to] b x c , where b\a = {x [member of] H; a [member of] b x x}, c/d = {x [member of] H; c [member of] x x d} and [approximately equal to] means that the sets are incident. A commutative transposition hypergroup (H, *) is called a join space. A commutative [H.sub.v]-group satisfying the transposition axiom is a weakly associative join space

Let (F, +, x [??]) be an ordered field. Denote A(F) = (F {[0.sub.F]}) x F and define a binary operation "x" on A(F) by [a, b] x [c, d] = [axc, axd + b]. Denote K = {[a, b]; a; b [member of] F; a [>.sub.P] [0.sub.F] [subset] A(F) and define an ordering "[??]" on K by

[a, b] [??] [c, d] whenever a = c and b [[??].sub.P] d.

Then (K, x) is an ordered group. Define *: K x K [right arrow] [P.sup.*](K) by

[a, b] * [c, d] = {[x, y]; [a, b] x [c, d] [??] [x, y]} = {[a x c, y]; a x d + b [[??].sub.P] y}.

The hyperstructure (K, *) is non-commutative.

Let us define the set [[DELTA].sub.K] = {[[a, b], [a, b]]; [a, b] [member of] K} and a hyperoperation * : [[DELTA].sub.K] x [[DELTA].sub.K] [right arrow] [P.sup.*]([[DELTA].sub.K]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

It is evident and it is proved in [8] that the following theorem holds.

Theorem 5.1. The hypergroupoid ([[DELTA].sub.K]; *) is a weakly associative and commutative transposition hypergroup, i.e., a weakly associative join space.

Proof. First we will show that the reproduction axiom is fulfilled. Due to Proposition 2.2 in [8] the structure (K, *) is the hypergroup, thus [a, b] * K = K = K * [a, b] for any [a, b] [member of] K. If we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where z = min{a x d + b; c x b + d}, evidently [a,b] [??] [c,d] [contains] [a,b] * [c,d]. Therefore K [contains] [a,b] [??] K [contains] [a,b] * K = K,

which implies that K (*) [a, b] = K and similarly [a, b] (*) K = K. From this we obtain that reproduction axiom holds in ([[DELTA].sub.K], *).

Second we will verify the transposition axiom. With respect to the definition of join space and (5.1) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have [[x, y], [x, y]] [member of] A [intersection] B, i.e., A [approximately equal to] B, if and only if x = [a.sub.1] x [b.sup.-1.sub.1] = [c.sub.1] x [d.sup.-1.sub.1] and

y [[??].sub.P] min{max{([a.sub.2] - [b.sub.2])x[b.sup.-1.sub.1], [a.sub.2] - [b.sub.2]x[a.sub.1]x[b.sup.-1.sub.1]}, max {([c.sub.2] - [d.sub.2])x[d.sup.-1.sub.1], [c.sub.2]-[d.sub.2]x[c.sub.1]x[d.sup.-1.sub.1]}}.

If A [approximately equal to] B, then necessarily [a.sub.1]x[d.sub.1] = [b.sub.1]x[c.sub.1]. Let us denote [u.sub.0] = [a.sub.1]x[d.sub.1]. For any [v.sub.0] such that [v.sub.0] [[??].sub.P] max{min{[a.sub.1]x[d.sub.2] + [a.sub.2], [a.sub.2]x[d.sub.1] + [d.sub.2], min{[b.sub.1]x[c.sub.2] + [b.sub.2]x[c.sub.1] + [c.sub.2]}} we obtain [[[u.sub.0], [v.sub.0]], [[u.sub.0], [v.sub.0]] [member of] C [intersection] D which proves the transposition axiom.

Remark 5.2. It is easy to verify that under the assumption of the previous theorem even the following equivalence holds: b\a [approximately equal to] c/d if and only if a x d [approximately equal to] b x c.

Further directions of the development of this topic will concern the reflectivity of the category JS of all join spaces and their inclusion homomorphisms.

Sarka Hoskova

University of Defence Brno, Department of Mathematics and Physics, Kounicova 65, 612 00 Brno, Czech Republic 

E-mail: [e-mail]

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