# On determinability of idempotent medial commutative quasigroups by their endomorphism semigroups/Idempotentsete mediaalsete kommutatiivsete kvaasiruhmade maaratavusest oma endomorfismipoolruhmadega.

1. INTRODUCTION

In this paper we study the endomorphism semigroups of idempotent medial commutative quasigroups (IMC-quasigroups, for short). K. Toyoda established a connection between medial quasigroups and Abelian groups (Theorem 2.10 in [2]). Endomorphism rings of Abelian groups have been studied by several authors and the obtained results are presented in [4]. In [6] Puusemp proved that if G and G' are finite Abelian groups, then the isomorphism G [congruent to] G' follows from the isomorphism between their endomorphism semigroups End G [congruent to] End G' (more precisely, it was proved that if G is a group such that its endomorphism semigroup is isomorphic to the endomorphism semigroup of a finite Abelian group, then the groups G and G' are isomorphic). Motivated by Toyoda's result (Theorem 2.10 in [2]) and Puusemp's result, we study endomorphisms of magmas (groupoids) which are "very close" to Abelian groups. To be more precise, we replace the associativity by a weaker assumption--mediality. It is known that every Abelian group G has the zero-endomorphism corresponding to the maximal congruence G x G. There exist medial quasigroups with no proper subquasigroups, such that all their endomorphisms are invertible. Motivated by results of endomorphism semigroups of groups, we restrict ourselves to the medial quasigroups with zero-endomorphisms. For this purpose we consider idempotent quasigroups.

As a result we generalize Puusemp's result to finite IMC-quasigroups, that is, if the endomorphism semigroups of finite IMC-quasigroups Q and Q' are isomorphic, then the quasigroups Q and Q' are isomorphic too.

Idempotent medial commutative quasigroups arise in several examples of mid-point quasigroups. Let R and [R.sub.+] denote the set of real numbers and the set of positive reals, respectively. Define two binary operations x [??] y = (x + y)/2 and x [dot encircle] y = [square root of xy]. Then both (r, [??]) and ([R.sub.+], [dot encircle]) are IMC-quasigroups.

The paper is organized as follows. In Section 2 we present the necessary definitions and propositions needed for the main theorem. These results are elementary and can be found also in [3,5]. For the convenience of the reader, we recall them together with the proofs.

The connection between the given ICM-quasigroup and associated commutative Moufang loops will be studied in Section 3. The main theorem will be given and proved in Section 4.

2. CONNECTION BETWEEN IMC-QUASIGROUPS AND ABELIAN GROUPS

Let us start by recalling the classical definition of the quasigroup.

Definition 1. A magma <Q, x> is called a quasigroup if each of the equations ax = b andya = b has a unique solution for any a, b [member of] Q.

The solutions of these equations will be denoted by x = a\b and y = b/a, respectively. We also need the following definition of quasigroups.

Definition 2. A set Q with three binary operations x, \,/ is called a quasigroup if the following identities hold:

x\(x x y) = x x (x\y)= y, (y x x)/x = (y/x) x x = y.

Definitions 1 and 2 are equivalent (see [2]). Next it is assumed that Q is a quasigroup <Q, x,\, />.

It follows from the definitions that the mappings [L.sub.a], [R.sub.b]: Q [right arrow] Q, defined by [L.sub.a](x) = ax, [R.sub.b](x) = xb, are bijective. Hence, to each quasigroup Q one can associate the subgroup M(Q) = <{[L.sub.a], [R.sub.b] | a,b [member of] Q},o> of the group of all bijections Q [right arrow] Q. The group M(Q) is called a multiplication group or an associated group of the quasigroup Q.

The mapping [phi]: Q [right arrow] Q is called an endomorphism of Q if [phi] preserves the binary operation x, that is [phi](x x y) = [phi](x) x [phi](y) for all x, y [member of] Q. An invertible endomorphism of Q is called an automorphism of Q. The set of all endomorphisms (automorphisms) of Q will be denoted by End Q (resp. Aut Q). By abuse of notation, we let End Q stand for the endomorphism monoid of Q. Immediate computations show that if [phi][member of] End Q, then [phi] preserves also the binary operations \ and /.

Definition 3. A quasigroup Q is called medial (commutative) if it satisfies the identity (x x y) x (u x v) = (x x u) x (y x v) (resp. x x y = y x x).

Similarly to Abelian groups, all endomorphisms of a medial quasigroup Q are summable, i.e. if [phi], [psi] are endomorphisms of a medial quasigroup Q, then [phi] + [psi] defined by ([phi] + [psi])(x) = [phi](x) x [psi](x) is an endomorphism too.

Theorem 1 (Toyoda's theorem (Theorem 2.10 in [2])). If Q is a medial quasigroup, then there exist an Abelian group <Q, +, -, 0>, its commuting automorphisms [phi] and [psi], and an element c [member of] Q such that

x x y = [phi](x) + [psi](y)+ c. (1)

The Abelian group ( Q, + , - , 0) is called an underlying Abelian group of the medial quasigroup Q.

Definition 4. A quasigroup Q is called idempotent if in Q the identity x x x = x is fulfilled.

Proposition 1. If Q is an idempotent medial quasigroup, then

1. Q is a distributive quasigroup, that is, Q satisfies both x x (y x z) = (x x y) x (x x z) and (y x z) x x = (y x x) x (z x x);

2. M(Q) is a subgroup of Aut Q.

The proof is straightforward.

If e [member of] Q is an idempotent, then <{e}, x> is a subquasigroup in Q. From now on, we write [[pi].sub.e] for the endomorphism Q [right arrow] {e}. Clearly, [[pi].sub.e] is a left-zero in End Q, i.e. [[pi].sub.e] [omicron] [phi] = [[pi].sub.e] for each [phi] [member of] End Q.

Proposition 2. If Q is an idempotent quasigroup, then [phi] is a left-zero in End Q iff [phi] = [[pi].sub.e] for some e [member of] Q.

Proof. If [phi] [member of] End Q, then obviously [[pi].sub.e] [omicron] [[pi].sub.e] = [[pi].sub.e] for each e [member of] Q.

Conversely, suppose that an endomorphism [theta] is left-zero in End Q. For each e, x [member of] Q we have

[theta](x) = ([theta] [omicron] [[pi].sub.e])(x) = [theta](e) = [[pi].sub.[theta](e)](x).

Hence, [theta] = [[pi].sub.f], where f = [theta](e).

Proposition 3. Suppose that <Q, x> is a medial quasigroup, where x is in the form (1). Then the following hold:

1. Q is commutative iff [phi] = [psi];

2. Q is idempotent iff c = 0 and [phi] + [psi] = [1.sub.Q].

See also Theorem 4 in [7].

Let 2 denote the endomorphism x [??] x + x of an Abelian group <Q, +, -, 0>.

Proposition 4. A quasigroup Q is an IMC-quasigroup iff there exists an Abelian group (Q, +, -, 0) such that the mapping 2 is its automorphism and x x y = [2.sup.-1](x + y).

Proof. Suppose that <Q, x> is an IMC-quasigroup. It follows from Toyoda's theorem that x x y = [phi](x) + [psi](y) + c, where [phi] and [psi] are automorphisms of <Q, +, -, 0>. Since <Q, x> is commutative and idempotent, Proposition 3 shows that [phi] = [psi], c = 0, and [1.sub.Q] = [phi] + [phi] = 2 [omicron] [phi]. As [phi] is an automorphism, we have that [2.sup.-1] = [phi] and finally that x x y = [2.sup.-1](x + y).

Conversely, if x x y = [2.sup.-1] (x + y), then it is staightforward to check that <Q, x> is an IMC-quasigroup.

One should note that any IMC-quasigroup is uniquely determined by its underlying Abelian group. Next we assume that everywhere <Q, +> is an underlying Abelian group of the given IMC-quasigroup (Q, x).

Corollary 1. If the Abelian group <Q, +, -, 0> is finite, then 2 is an automorphism iff <Q, +, -, 0> is of odd order.

The next corollary is a special case of Proposition 3 in [8].

Corollary 2. If Q is an IMC-quasigroup, then End(Q, +) [??] End(Q, x) (embedding of monoids).

Indeed, if [eta] is an endomorphism of the Abelian group <Q, +, -, 0>, then 2 [omicron] [eta] = [eta] [omicron] 2, [2.sup.-1] [omicron][eta] = [eta][omicron][2.sup.-1] and for each x, y [member of] Q we have

[eta](x) x [eta](y) = [2.sup.-1]([eta](x) + [eta](y)) = [2.sup.-1]([eta](x)) + [2.sup.-1]([eta](y)) = [eta]([2.sup.-1](x + y)) = [eta](x x y),

i.e. [eta] [member of] End(Q, x).

Corollary 3. If Q is an IMC-quasigroup, then x\y = y/x = 2(y) - x.

Indeed, since y = ( y/ x) x x, we have y = [2.sup.-1] ( y/ x + x), i.e. y/x = 2(y) - x. In a commutative quasigroup always y/x = x\y. So we have x\y = 2(y) - x.

Definition 5. Two quasigroups <Q, x> and <Q', *> are called isomorphic, if there exist the bijective mapping [tau] : Q [right arrow] Q', such that [tau](x x y) = [tau](x) * [tau](y) for each x,y [member of] Q.

3. CONNECTION BETWEEN IMC-QUASIGROUPS AND COMMUTATIVE MOUFANG LOOPS

It was shown in Proposition 1 that any idempotent medial quasigroup is distributive. Therefore, any IMC-quasigroup is also a distributive quasigroup. Distributive quasigroups are connected to commutative Moufang loops (see [2] for more details).

Definition 6. A quasigroup <Q, x> is called a loop if there exists e [member of] Q such that e x x = x = x x e for each x [member of] Q.

We denote it by <Q, x, e>.

Definition 7. A loop <Q, x, e> is called a commutative Moufang loop if it satisfies the identity

(x x x) x ( y x z) = (x x y) x (x x z). (2)

From Definition 7 it follows that the binary operation x is commutative and there exists the [mapping.sup.-1] : Q [right arrow] Q such that [x.sup.-1] x (x x y) = y holds.

Let <Q, x> be an IMC-quasigroup and k [member of] Q. Let us define a new binary operation on Q as follows:

x [[direct sum].sub.k] y = [R.sup.-1.sub.k] x x [L.sup.-1.sub.k] y. (3)

The magma (Q, [[direct sum].sub.k]> is a commutative Moufang loop with the identity element k by Theorem 8.1 in [2]. Due to commutativity of <Q, x>, we have that

x [[direct sum].sub.k]y = [L.sup.-1.sub.k] x [L.sup.-1.sub.k] = (x x y).

Moreover, the unary operation [sup.-1] of <Q, [[direct sum].sub.k]> will be denoted by [[??].sub.k] and the element [[??].sub.k] x coincides with k/x.

Proposition 5. The commutative Moufang loop <Q, [[direct sum].sub.k], [[??].sub.k], k> is an Abelian group.

Proof (see also Theorem 9 in [5]). It is sufficient to prove only the associativity of [[direct sum].sub.k]. Let x, y, z [member of] Q. Then

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Corollary 4. If <Q, x> is an IMC-quasigroup with the underlying Abelian group <Q, +, -, 0> , that is x x y = [2.sup.-1](x + y), then x [[direct sum].sub.k] y = x + y - k.

Proof. Let x, y [member of] Q. Then

x [[direct sum].sub.k] y = [L.sup.-1.sub.k](x x y)= k\(x x y) = 2(x x y) - k = 2 ([2.sup.-1](x + y)) - k = x + y - k.

As a particular case we have

Corollary 5. Let the assumptions be as in Corollary 4. Then x [[direct sum].sub.0] y = x + y.

By the last corollary we have

x + y = x [[direct sum].sub.0] y = [L.sup.-1.sub.0] (x x y)= [L.sup.-1.sub.0] ([2.sup.-1] (x + y)).

Therefore, 2 = [L.sup.-1.sub.0], i.e. [L.sup.-1.sub.0] is an automorphism of the underlying Abelian group. We have a more general result.

Corollary 6. Let [2.sub.k], with [2.sub.k] (x) = x [[direct sum].sub.k] x, be an automorphism of the Abelian group (Q, [[direct sum].sub.k], [[??].sub.k], k>. Then [2.sub.k] = [L.sup.-1.sub.k].

Obviously,

[2.sub.k] (x) = x [[direct sum].sub.k] x = [L.sup.-1.sub.k] (x x x) = [L.sup.-1.sub.k] (x).

Proposition 6. For each k, l [member of] Q the Abelian groups <Q, [[direct sum].sub.k], [[??].sub.k], k> and (Q, [[direct sum].sub.l], [[??].sub.l], l) are isomorphic.

Proof (see also Theorem 10 in [5]). For each k [member of] Q, the mapping [[psi].sub.k]: Q [right arrow] Q, given by [[psi].sub.k] (x) = x + k, is an isomorphism from <Q, +, -, 0> to (Q, [[direct sum].sub.k], [[??].sub.k], k>. Now the proposition follows immediately.

We will write [0.sup.Q] for the set of all left-zero endomorphisms of an ICM-quasigroup Q. By Proposition 2,

[0.sup.Q] = {[[pi].sub.k]: Q [right arrow] {k} | k [member of] Q}.

Let k [member of] Q. The set of all submonoids M of End(Q, x) such that [1.sub.q], [[pi].sub.k] [member of] M and Mn [intersection] [0.sup.Q] = {[[pi].sub.k]} is non-empty and, by Zorn's lemma, has a maximal element. From now on, [M.sub.k] denotes a maximal submonoid in End(Q, x) such that [M.sub.k] [intersection] [0.sup.Q] = {[[pi].sub.k]}.

Proposition 7. The submonoid [M.sub.k] coincides with the endomorphism monoid of the Abelian group <Q, [[direct sum].sub.k], [[??].sub.k], k).

Proof. The proof is divided into three steps:

1. [phi](k) = k for each [phi] [member of] [M.sub.k];

2. [phi](x [[direct sum].sub.k] y) = [phi](x) [[direct sum].sub.k] [phi](y) for each [phi] [member of] [M.sub.k] and for each x, y [member of] Q;

3. the endomorphism monoid End(Q, [[direct sum].sub.k]) of the Abelian group <Q, [[direct sum].sub.k], [[??].sub.k], k> coincides with [M.sub.k].

It follows from the first two steps that [M.sub.k] [subset or equal to] End(Q, [[direct sum].sub.k]).

1. Let [phi] [member of] [M.sub.k]. By definition, [[pi].sub.k] [member of] [M.sub.k]. Therefore [phi] [omicron] [[pi].sub.k] [member of] [M.sub.k]. For each x [member of] Q we have ([phi] [omicron] [[pi].sub.k])(x) = [phi](k) i.e. [phi] [omicron] [[pi].sub.k] = [[pi].sub.[phi](k)]. By the definition of [M.sub.k] we conclude that [[pi].sub.k] = [[pi].sub.[phi](k)] and hence k = [phi](k).

2. Let [phi] [member of] [M.sub.k] and let x, y [member of] Q. Then

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3. By Corollary 6, [2.sub.k] = [L.sup.-1.sub.k]. Hence, for each [xi] [member of] End(Q, [[direct sum].sub.k]) we have [2.sub.k] [omicron] [xi] = [xi] [omicron] [2.sub.k], [L.sup.-1.sub.k] [omicron] [xi] = [xi] [omicron] [L.sup.-1.sub.k], [xi] [omicron] [L.sub.k] = [L.sub.k] [omicron] [xi] and, in view of x [[direct sum].sub.k] y = [L.sup.-1.sub.k](x x y), it follows that

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i.e. [xi] [member of] End(Q, x). Therefore, [M.sub.k] [subset or equal to] End(Q, [[direct sum].sub.k]) [subset or equal to] End(Q, x). On the other hand, it is easy to check that End(Q, [[direct sum].sub.k]) [intersection] = [0.sup.Q] = {[[pi].sub.k]} by the definition of [M.sub.k], it follows that End(Q, [[direct sum].sub.k]) = [M.sub.k]. The proposition is proved.

Corollary 7. Let [phi] [member of] End(Q, x). Then [phi] [member of] [M.sub.k] [??] [phi](k) = k.

Proof. Let

M' = {[phi] [member of] End(Q, x) | [phi](k)= k}.

Obviously [1.sub.Q] [member of] M' and [phi] [omicron] [psi] [member of] M' whenever [phi], [psi] [member of] M'. Hence, M' is a monoid and [[pi].sub.k] [member of] M' by the definition of [[pi].sub.k]. The first part of the proof of Proposition 7 implies [M.sub.k] [subset or equal to] M'. Clearly, M' is a submonoid of End(Q, x) such that M' [intersection] [0.sup.Q] = {[[pi].sub.k]}. By the definition of [M.sub.k] we have M' = [M.sub.k].

Corollary 8. If End(Q, x) is finite, then (Q, x) is also finite.

We give two proofs for this corollary. The first one is more elementary. The second proof uses results of the group theory and the analogue of its corollary for groups.

Elementary proof. If End(Q, x) is finite, then [0.sup.Q] is finite too. Hence the IMC-quasigroup Q is finite due to the one-to-one correspondence between [0.sup.Q] and Q, i.e. k [left and right arrow] [[pi].sub.k].

Group-theoretic proof. Let End(Q, x) be finite. Since End(Q, +) [subset or equal to] End(Q, x), the monoid End(Q, +) is finite, too. It is well known that if the endomorphism monoid of a group G is finite, then the group G is finite by Theorem 2 in [1]. Therefore, the group (Q, +) is finite and so is the IMC-quasigroup (Q, x).

4. MAIN THEOREM

Theorem 2. Let <Q, x> and <Q', *> be IMC-quasigroups and <Q, x> be finite. If the endomorphism monoids End(Q, x) and End(Q', *) are isomorphic, then the quasigroups Q and Q' are isomorphic.

Proof. By Corollary 8, Q' is finite, too. Let x,y [member of] Q and x', y' [member of] Q'. By Proposition 4 we have

x x y = [2.sup.-1](x + y) and x' * y' = [2'.sup.-1](x' +' y'),

where <Q, +> and <Q', +'> are the underlying Abelian groups of Q and Q', respectively. By Corollary 2, the monoids End(Q, +) and End(Q', +") are contained in the monoids End(Q, x) and End(Q', *), respectively.

Let M [less than or equal to] End(Q, x) be the maximal submonoid, such that M [intersection] [0.sup.Q] = {[[pi].sub.k]} for some k [member of] Q. By Propositions 6 and 7 and Corollary 5 we have an isomorphism M [congruent to] End(Q, +). Let [GAMMA]: End(Q, x) [right arrow] End(Q', *) be an isomorphism of monoids. Hence, the image of the restriction of [GAMMA] to End(Q, +) is the maximal submonoid M' [less than or equal to ] End(Q', *) such that M' [intersection] [0.sup.Q'] = {[[pi].sub.k']} for some k' [member of] Q'. From Propositions 6 and 7 we conclude that M' [congruent to] End(Q', +') and finally that End(Q, +) [congruent to] End(Q', +').

Since (Q, +) and (Q', +') are finite Abelian groups, their isomorphism follows from End(Q, +) [congruent to] End( Q', +') (see Theorem 4.2 in [6]). Let [xi] : Q [right arrow] Q' be the corresponding isomorphism. Clearly, 2' [omicron] [xi] = [xi] [omicron] 2 and [2'.sup.-1] [omicron] [xi] = [xi] [omicron] [2.sup.-1]. We finish the proof by showing that [xi] is also an isomorphism between the quasigroups Q and Q'. Let x, y [member of] Q. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 9. Let endomorphism monoids of two IMC-quasigroups Q and Q' be isomorphic. If Q is finite, then the underlying Abelian groups o Q and Q' are isomorphic.

On uuritud loplike kommutatiivsete idempotentsete mediaalsete kvaasiruhmade maaratavust oma endomorfismipoolriihmaga. Latitudes Toyoda teoreemist, mis seob omavahel mediaalsed kvaasiriihmad ja Abeli riihmad, ning loplike Abeli ruhmade maaratavusest oma endomorfismipoolruhmaga [6], on mainitud tulemust laiendatud looplikele kommutatiivsetele idempotentsetele mediaalsetele kvaasiruhmadele. On naidatud, et ka selliste kvaasiruhmade jaoks saab laiendada ruhmateooriast tuntud tulemust, et ruhma G endomorfismipoolruhma loplikkusest jareldub ruhma loplikkus (jareldus 8).

doi: 10.3176/proc.2011.2.02

REFERENCES

[1.] Alperin, J. L. Groups with finitely many automorphisms. Pacific J. Math., 1962, 12(1), 1-5.

[2.] Belousov, V. D. Foundations of the Theory of Quasigroups and Loops. Nauka, Moscow, 1967 (in Russian).

[3.] Havel, V. and Sedlarova, M. Golden section quasigroups as special idempotent medial quasigroups. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 1994, 33, 43-50.

[4.] Krylov, P. A., Mikhalev, A. V., and Tuganbaev, A. A. Endomorphism Rings of Abelian Groups. Kluwer Academic Publisher, Dordrecht-Boston-London, 2003.

[5.] Murdoch, D. C. Structure of Abelian quasi-groups. Trans. Amer. Math. Soc., 1941, 49, 392-409.

[6.] Puusemp, P. Idempotents of the endomorphism semigroups of groups. Acta Comment. Univ. Tartuensis, 1975, 366, 76- 104 (in Russian).

[7.] Shcherbacov, V. A. On the structure of finite medial semigroups. Bul. Acad. Sti. Rep. Moldova, Matematica, 2005, 47(1), 11-18.

[8.] Tabarov, A. Kh. Homomorphisms and endomorphisms of linear and alinear quasigroups. Discrete Math. Appl., 2007, 17(3), 253-260.

Alar Leibak * and Peeter Puusemp

Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia; puusemp@staff.ttu.ee

Received 8 April 2010, accepted 17 June 2010

* Corresponding author, alar@staff.ttu.ee

In this paper we study the endomorphism semigroups of idempotent medial commutative quasigroups (IMC-quasigroups, for short). K. Toyoda established a connection between medial quasigroups and Abelian groups (Theorem 2.10 in [2]). Endomorphism rings of Abelian groups have been studied by several authors and the obtained results are presented in [4]. In [6] Puusemp proved that if G and G' are finite Abelian groups, then the isomorphism G [congruent to] G' follows from the isomorphism between their endomorphism semigroups End G [congruent to] End G' (more precisely, it was proved that if G is a group such that its endomorphism semigroup is isomorphic to the endomorphism semigroup of a finite Abelian group, then the groups G and G' are isomorphic). Motivated by Toyoda's result (Theorem 2.10 in [2]) and Puusemp's result, we study endomorphisms of magmas (groupoids) which are "very close" to Abelian groups. To be more precise, we replace the associativity by a weaker assumption--mediality. It is known that every Abelian group G has the zero-endomorphism corresponding to the maximal congruence G x G. There exist medial quasigroups with no proper subquasigroups, such that all their endomorphisms are invertible. Motivated by results of endomorphism semigroups of groups, we restrict ourselves to the medial quasigroups with zero-endomorphisms. For this purpose we consider idempotent quasigroups.

As a result we generalize Puusemp's result to finite IMC-quasigroups, that is, if the endomorphism semigroups of finite IMC-quasigroups Q and Q' are isomorphic, then the quasigroups Q and Q' are isomorphic too.

Idempotent medial commutative quasigroups arise in several examples of mid-point quasigroups. Let R and [R.sub.+] denote the set of real numbers and the set of positive reals, respectively. Define two binary operations x [??] y = (x + y)/2 and x [dot encircle] y = [square root of xy]. Then both (r, [??]) and ([R.sub.+], [dot encircle]) are IMC-quasigroups.

The paper is organized as follows. In Section 2 we present the necessary definitions and propositions needed for the main theorem. These results are elementary and can be found also in [3,5]. For the convenience of the reader, we recall them together with the proofs.

The connection between the given ICM-quasigroup and associated commutative Moufang loops will be studied in Section 3. The main theorem will be given and proved in Section 4.

2. CONNECTION BETWEEN IMC-QUASIGROUPS AND ABELIAN GROUPS

Let us start by recalling the classical definition of the quasigroup.

Definition 1. A magma <Q, x> is called a quasigroup if each of the equations ax = b andya = b has a unique solution for any a, b [member of] Q.

The solutions of these equations will be denoted by x = a\b and y = b/a, respectively. We also need the following definition of quasigroups.

Definition 2. A set Q with three binary operations x, \,/ is called a quasigroup if the following identities hold:

x\(x x y) = x x (x\y)= y, (y x x)/x = (y/x) x x = y.

Definitions 1 and 2 are equivalent (see [2]). Next it is assumed that Q is a quasigroup <Q, x,\, />.

It follows from the definitions that the mappings [L.sub.a], [R.sub.b]: Q [right arrow] Q, defined by [L.sub.a](x) = ax, [R.sub.b](x) = xb, are bijective. Hence, to each quasigroup Q one can associate the subgroup M(Q) = <{[L.sub.a], [R.sub.b] | a,b [member of] Q},o> of the group of all bijections Q [right arrow] Q. The group M(Q) is called a multiplication group or an associated group of the quasigroup Q.

The mapping [phi]: Q [right arrow] Q is called an endomorphism of Q if [phi] preserves the binary operation x, that is [phi](x x y) = [phi](x) x [phi](y) for all x, y [member of] Q. An invertible endomorphism of Q is called an automorphism of Q. The set of all endomorphisms (automorphisms) of Q will be denoted by End Q (resp. Aut Q). By abuse of notation, we let End Q stand for the endomorphism monoid of Q. Immediate computations show that if [phi][member of] End Q, then [phi] preserves also the binary operations \ and /.

Definition 3. A quasigroup Q is called medial (commutative) if it satisfies the identity (x x y) x (u x v) = (x x u) x (y x v) (resp. x x y = y x x).

Similarly to Abelian groups, all endomorphisms of a medial quasigroup Q are summable, i.e. if [phi], [psi] are endomorphisms of a medial quasigroup Q, then [phi] + [psi] defined by ([phi] + [psi])(x) = [phi](x) x [psi](x) is an endomorphism too.

Theorem 1 (Toyoda's theorem (Theorem 2.10 in [2])). If Q is a medial quasigroup, then there exist an Abelian group <Q, +, -, 0>, its commuting automorphisms [phi] and [psi], and an element c [member of] Q such that

x x y = [phi](x) + [psi](y)+ c. (1)

The Abelian group ( Q, + , - , 0) is called an underlying Abelian group of the medial quasigroup Q.

Definition 4. A quasigroup Q is called idempotent if in Q the identity x x x = x is fulfilled.

Proposition 1. If Q is an idempotent medial quasigroup, then

1. Q is a distributive quasigroup, that is, Q satisfies both x x (y x z) = (x x y) x (x x z) and (y x z) x x = (y x x) x (z x x);

2. M(Q) is a subgroup of Aut Q.

The proof is straightforward.

If e [member of] Q is an idempotent, then <{e}, x> is a subquasigroup in Q. From now on, we write [[pi].sub.e] for the endomorphism Q [right arrow] {e}. Clearly, [[pi].sub.e] is a left-zero in End Q, i.e. [[pi].sub.e] [omicron] [phi] = [[pi].sub.e] for each [phi] [member of] End Q.

Proposition 2. If Q is an idempotent quasigroup, then [phi] is a left-zero in End Q iff [phi] = [[pi].sub.e] for some e [member of] Q.

Proof. If [phi] [member of] End Q, then obviously [[pi].sub.e] [omicron] [[pi].sub.e] = [[pi].sub.e] for each e [member of] Q.

Conversely, suppose that an endomorphism [theta] is left-zero in End Q. For each e, x [member of] Q we have

[theta](x) = ([theta] [omicron] [[pi].sub.e])(x) = [theta](e) = [[pi].sub.[theta](e)](x).

Hence, [theta] = [[pi].sub.f], where f = [theta](e).

Proposition 3. Suppose that <Q, x> is a medial quasigroup, where x is in the form (1). Then the following hold:

1. Q is commutative iff [phi] = [psi];

2. Q is idempotent iff c = 0 and [phi] + [psi] = [1.sub.Q].

See also Theorem 4 in [7].

Let 2 denote the endomorphism x [??] x + x of an Abelian group <Q, +, -, 0>.

Proposition 4. A quasigroup Q is an IMC-quasigroup iff there exists an Abelian group (Q, +, -, 0) such that the mapping 2 is its automorphism and x x y = [2.sup.-1](x + y).

Proof. Suppose that <Q, x> is an IMC-quasigroup. It follows from Toyoda's theorem that x x y = [phi](x) + [psi](y) + c, where [phi] and [psi] are automorphisms of <Q, +, -, 0>. Since <Q, x> is commutative and idempotent, Proposition 3 shows that [phi] = [psi], c = 0, and [1.sub.Q] = [phi] + [phi] = 2 [omicron] [phi]. As [phi] is an automorphism, we have that [2.sup.-1] = [phi] and finally that x x y = [2.sup.-1](x + y).

Conversely, if x x y = [2.sup.-1] (x + y), then it is staightforward to check that <Q, x> is an IMC-quasigroup.

One should note that any IMC-quasigroup is uniquely determined by its underlying Abelian group. Next we assume that everywhere <Q, +> is an underlying Abelian group of the given IMC-quasigroup (Q, x).

Corollary 1. If the Abelian group <Q, +, -, 0> is finite, then 2 is an automorphism iff <Q, +, -, 0> is of odd order.

The next corollary is a special case of Proposition 3 in [8].

Corollary 2. If Q is an IMC-quasigroup, then End(Q, +) [??] End(Q, x) (embedding of monoids).

Indeed, if [eta] is an endomorphism of the Abelian group <Q, +, -, 0>, then 2 [omicron] [eta] = [eta] [omicron] 2, [2.sup.-1] [omicron][eta] = [eta][omicron][2.sup.-1] and for each x, y [member of] Q we have

[eta](x) x [eta](y) = [2.sup.-1]([eta](x) + [eta](y)) = [2.sup.-1]([eta](x)) + [2.sup.-1]([eta](y)) = [eta]([2.sup.-1](x + y)) = [eta](x x y),

i.e. [eta] [member of] End(Q, x).

Corollary 3. If Q is an IMC-quasigroup, then x\y = y/x = 2(y) - x.

Indeed, since y = ( y/ x) x x, we have y = [2.sup.-1] ( y/ x + x), i.e. y/x = 2(y) - x. In a commutative quasigroup always y/x = x\y. So we have x\y = 2(y) - x.

Definition 5. Two quasigroups <Q, x> and <Q', *> are called isomorphic, if there exist the bijective mapping [tau] : Q [right arrow] Q', such that [tau](x x y) = [tau](x) * [tau](y) for each x,y [member of] Q.

3. CONNECTION BETWEEN IMC-QUASIGROUPS AND COMMUTATIVE MOUFANG LOOPS

It was shown in Proposition 1 that any idempotent medial quasigroup is distributive. Therefore, any IMC-quasigroup is also a distributive quasigroup. Distributive quasigroups are connected to commutative Moufang loops (see [2] for more details).

Definition 6. A quasigroup <Q, x> is called a loop if there exists e [member of] Q such that e x x = x = x x e for each x [member of] Q.

We denote it by <Q, x, e>.

Definition 7. A loop <Q, x, e> is called a commutative Moufang loop if it satisfies the identity

(x x x) x ( y x z) = (x x y) x (x x z). (2)

From Definition 7 it follows that the binary operation x is commutative and there exists the [mapping.sup.-1] : Q [right arrow] Q such that [x.sup.-1] x (x x y) = y holds.

Let <Q, x> be an IMC-quasigroup and k [member of] Q. Let us define a new binary operation on Q as follows:

x [[direct sum].sub.k] y = [R.sup.-1.sub.k] x x [L.sup.-1.sub.k] y. (3)

The magma (Q, [[direct sum].sub.k]> is a commutative Moufang loop with the identity element k by Theorem 8.1 in [2]. Due to commutativity of <Q, x>, we have that

x [[direct sum].sub.k]y = [L.sup.-1.sub.k] x [L.sup.-1.sub.k] = (x x y).

Moreover, the unary operation [sup.-1] of <Q, [[direct sum].sub.k]> will be denoted by [[??].sub.k] and the element [[??].sub.k] x coincides with k/x.

Proposition 5. The commutative Moufang loop <Q, [[direct sum].sub.k], [[??].sub.k], k> is an Abelian group.

Proof (see also Theorem 9 in [5]). It is sufficient to prove only the associativity of [[direct sum].sub.k]. Let x, y, z [member of] Q. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 4. If <Q, x> is an IMC-quasigroup with the underlying Abelian group <Q, +, -, 0> , that is x x y = [2.sup.-1](x + y), then x [[direct sum].sub.k] y = x + y - k.

Proof. Let x, y [member of] Q. Then

x [[direct sum].sub.k] y = [L.sup.-1.sub.k](x x y)= k\(x x y) = 2(x x y) - k = 2 ([2.sup.-1](x + y)) - k = x + y - k.

As a particular case we have

Corollary 5. Let the assumptions be as in Corollary 4. Then x [[direct sum].sub.0] y = x + y.

By the last corollary we have

x + y = x [[direct sum].sub.0] y = [L.sup.-1.sub.0] (x x y)= [L.sup.-1.sub.0] ([2.sup.-1] (x + y)).

Therefore, 2 = [L.sup.-1.sub.0], i.e. [L.sup.-1.sub.0] is an automorphism of the underlying Abelian group. We have a more general result.

Corollary 6. Let [2.sub.k], with [2.sub.k] (x) = x [[direct sum].sub.k] x, be an automorphism of the Abelian group (Q, [[direct sum].sub.k], [[??].sub.k], k>. Then [2.sub.k] = [L.sup.-1.sub.k].

Obviously,

[2.sub.k] (x) = x [[direct sum].sub.k] x = [L.sup.-1.sub.k] (x x x) = [L.sup.-1.sub.k] (x).

Proposition 6. For each k, l [member of] Q the Abelian groups <Q, [[direct sum].sub.k], [[??].sub.k], k> and (Q, [[direct sum].sub.l], [[??].sub.l], l) are isomorphic.

Proof (see also Theorem 10 in [5]). For each k [member of] Q, the mapping [[psi].sub.k]: Q [right arrow] Q, given by [[psi].sub.k] (x) = x + k, is an isomorphism from <Q, +, -, 0> to (Q, [[direct sum].sub.k], [[??].sub.k], k>. Now the proposition follows immediately.

We will write [0.sup.Q] for the set of all left-zero endomorphisms of an ICM-quasigroup Q. By Proposition 2,

[0.sup.Q] = {[[pi].sub.k]: Q [right arrow] {k} | k [member of] Q}.

Let k [member of] Q. The set of all submonoids M of End(Q, x) such that [1.sub.q], [[pi].sub.k] [member of] M and Mn [intersection] [0.sup.Q] = {[[pi].sub.k]} is non-empty and, by Zorn's lemma, has a maximal element. From now on, [M.sub.k] denotes a maximal submonoid in End(Q, x) such that [M.sub.k] [intersection] [0.sup.Q] = {[[pi].sub.k]}.

Proposition 7. The submonoid [M.sub.k] coincides with the endomorphism monoid of the Abelian group <Q, [[direct sum].sub.k], [[??].sub.k], k).

Proof. The proof is divided into three steps:

1. [phi](k) = k for each [phi] [member of] [M.sub.k];

2. [phi](x [[direct sum].sub.k] y) = [phi](x) [[direct sum].sub.k] [phi](y) for each [phi] [member of] [M.sub.k] and for each x, y [member of] Q;

3. the endomorphism monoid End(Q, [[direct sum].sub.k]) of the Abelian group <Q, [[direct sum].sub.k], [[??].sub.k], k> coincides with [M.sub.k].

It follows from the first two steps that [M.sub.k] [subset or equal to] End(Q, [[direct sum].sub.k]).

1. Let [phi] [member of] [M.sub.k]. By definition, [[pi].sub.k] [member of] [M.sub.k]. Therefore [phi] [omicron] [[pi].sub.k] [member of] [M.sub.k]. For each x [member of] Q we have ([phi] [omicron] [[pi].sub.k])(x) = [phi](k) i.e. [phi] [omicron] [[pi].sub.k] = [[pi].sub.[phi](k)]. By the definition of [M.sub.k] we conclude that [[pi].sub.k] = [[pi].sub.[phi](k)] and hence k = [phi](k).

2. Let [phi] [member of] [M.sub.k] and let x, y [member of] Q. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. By Corollary 6, [2.sub.k] = [L.sup.-1.sub.k]. Hence, for each [xi] [member of] End(Q, [[direct sum].sub.k]) we have [2.sub.k] [omicron] [xi] = [xi] [omicron] [2.sub.k], [L.sup.-1.sub.k] [omicron] [xi] = [xi] [omicron] [L.sup.-1.sub.k], [xi] [omicron] [L.sub.k] = [L.sub.k] [omicron] [xi] and, in view of x [[direct sum].sub.k] y = [L.sup.-1.sub.k](x x y), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e. [xi] [member of] End(Q, x). Therefore, [M.sub.k] [subset or equal to] End(Q, [[direct sum].sub.k]) [subset or equal to] End(Q, x). On the other hand, it is easy to check that End(Q, [[direct sum].sub.k]) [intersection] = [0.sup.Q] = {[[pi].sub.k]} by the definition of [M.sub.k], it follows that End(Q, [[direct sum].sub.k]) = [M.sub.k]. The proposition is proved.

Corollary 7. Let [phi] [member of] End(Q, x). Then [phi] [member of] [M.sub.k] [??] [phi](k) = k.

Proof. Let

M' = {[phi] [member of] End(Q, x) | [phi](k)= k}.

Obviously [1.sub.Q] [member of] M' and [phi] [omicron] [psi] [member of] M' whenever [phi], [psi] [member of] M'. Hence, M' is a monoid and [[pi].sub.k] [member of] M' by the definition of [[pi].sub.k]. The first part of the proof of Proposition 7 implies [M.sub.k] [subset or equal to] M'. Clearly, M' is a submonoid of End(Q, x) such that M' [intersection] [0.sup.Q] = {[[pi].sub.k]}. By the definition of [M.sub.k] we have M' = [M.sub.k].

Corollary 8. If End(Q, x) is finite, then (Q, x) is also finite.

We give two proofs for this corollary. The first one is more elementary. The second proof uses results of the group theory and the analogue of its corollary for groups.

Elementary proof. If End(Q, x) is finite, then [0.sup.Q] is finite too. Hence the IMC-quasigroup Q is finite due to the one-to-one correspondence between [0.sup.Q] and Q, i.e. k [left and right arrow] [[pi].sub.k].

Group-theoretic proof. Let End(Q, x) be finite. Since End(Q, +) [subset or equal to] End(Q, x), the monoid End(Q, +) is finite, too. It is well known that if the endomorphism monoid of a group G is finite, then the group G is finite by Theorem 2 in [1]. Therefore, the group (Q, +) is finite and so is the IMC-quasigroup (Q, x).

4. MAIN THEOREM

Theorem 2. Let <Q, x> and <Q', *> be IMC-quasigroups and <Q, x> be finite. If the endomorphism monoids End(Q, x) and End(Q', *) are isomorphic, then the quasigroups Q and Q' are isomorphic.

Proof. By Corollary 8, Q' is finite, too. Let x,y [member of] Q and x', y' [member of] Q'. By Proposition 4 we have

x x y = [2.sup.-1](x + y) and x' * y' = [2'.sup.-1](x' +' y'),

where <Q, +> and <Q', +'> are the underlying Abelian groups of Q and Q', respectively. By Corollary 2, the monoids End(Q, +) and End(Q', +") are contained in the monoids End(Q, x) and End(Q', *), respectively.

Let M [less than or equal to] End(Q, x) be the maximal submonoid, such that M [intersection] [0.sup.Q] = {[[pi].sub.k]} for some k [member of] Q. By Propositions 6 and 7 and Corollary 5 we have an isomorphism M [congruent to] End(Q, +). Let [GAMMA]: End(Q, x) [right arrow] End(Q', *) be an isomorphism of monoids. Hence, the image of the restriction of [GAMMA] to End(Q, +) is the maximal submonoid M' [less than or equal to ] End(Q', *) such that M' [intersection] [0.sup.Q'] = {[[pi].sub.k']} for some k' [member of] Q'. From Propositions 6 and 7 we conclude that M' [congruent to] End(Q', +') and finally that End(Q, +) [congruent to] End(Q', +').

Since (Q, +) and (Q', +') are finite Abelian groups, their isomorphism follows from End(Q, +) [congruent to] End( Q', +') (see Theorem 4.2 in [6]). Let [xi] : Q [right arrow] Q' be the corresponding isomorphism. Clearly, 2' [omicron] [xi] = [xi] [omicron] 2 and [2'.sup.-1] [omicron] [xi] = [xi] [omicron] [2.sup.-1]. We finish the proof by showing that [xi] is also an isomorphism between the quasigroups Q and Q'. Let x, y [member of] Q. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 9. Let endomorphism monoids of two IMC-quasigroups Q and Q' be isomorphic. If Q is finite, then the underlying Abelian groups o Q and Q' are isomorphic.

On uuritud loplike kommutatiivsete idempotentsete mediaalsete kvaasiruhmade maaratavust oma endomorfismipoolriihmaga. Latitudes Toyoda teoreemist, mis seob omavahel mediaalsed kvaasiriihmad ja Abeli riihmad, ning loplike Abeli ruhmade maaratavusest oma endomorfismipoolruhmaga [6], on mainitud tulemust laiendatud looplikele kommutatiivsetele idempotentsetele mediaalsetele kvaasiruhmadele. On naidatud, et ka selliste kvaasiruhmade jaoks saab laiendada ruhmateooriast tuntud tulemust, et ruhma G endomorfismipoolruhma loplikkusest jareldub ruhma loplikkus (jareldus 8).

doi: 10.3176/proc.2011.2.02

REFERENCES

[1.] Alperin, J. L. Groups with finitely many automorphisms. Pacific J. Math., 1962, 12(1), 1-5.

[2.] Belousov, V. D. Foundations of the Theory of Quasigroups and Loops. Nauka, Moscow, 1967 (in Russian).

[3.] Havel, V. and Sedlarova, M. Golden section quasigroups as special idempotent medial quasigroups. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 1994, 33, 43-50.

[4.] Krylov, P. A., Mikhalev, A. V., and Tuganbaev, A. A. Endomorphism Rings of Abelian Groups. Kluwer Academic Publisher, Dordrecht-Boston-London, 2003.

[5.] Murdoch, D. C. Structure of Abelian quasi-groups. Trans. Amer. Math. Soc., 1941, 49, 392-409.

[6.] Puusemp, P. Idempotents of the endomorphism semigroups of groups. Acta Comment. Univ. Tartuensis, 1975, 366, 76- 104 (in Russian).

[7.] Shcherbacov, V. A. On the structure of finite medial semigroups. Bul. Acad. Sti. Rep. Moldova, Matematica, 2005, 47(1), 11-18.

[8.] Tabarov, A. Kh. Homomorphisms and endomorphisms of linear and alinear quasigroups. Discrete Math. Appl., 2007, 17(3), 253-260.

Alar Leibak * and Peeter Puusemp

Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia; puusemp@staff.ttu.ee

Received 8 April 2010, accepted 17 June 2010

* Corresponding author, alar@staff.ttu.ee

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Author: | Leibak, Alar; Puusemp, Peeter |
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Publication: | Proceedings of the Estonian Academy of Sciences |

Article Type: | Report |

Geographic Code: | 4EXES |

Date: | Jun 1, 2011 |

Words: | 3508 |

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