On determinability of idempotent medial commutative quasigroups by their endomorphism semigroups

We extend the result of P. Puusemp (Idempotents of the endomorphism semigroups of groups. Acta Comment. Univ. Tartuensis, 1975, 366, 76–104) about determinability of finite Abelian groups by their endomorphism semigroups to finite idempotent medial commutative quasigroups.


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 ∼ = G follows from the isomorphism between their endomorphism semigroups End G ∼ = 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 × 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 + denote the set of real numbers and the set of positive reals, respectively.Define two binary operations x y = (x + y)/2 and x y = √ xy.Then both (R, ) and (R + , ) 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.

CONNECTION BETWEEN IMC-QUASIGROUPS AND ABELIAN GROUPS
Let us start by recalling the classical definition of the quasigroup.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 •, \, / is called a quasigroup if the following identities hold: Definitions 1 and 2 are equivalent (see [2]).Next it is assumed that Q is a quasigroup Q, •, \, / .It follows from the definitions that the mappings L a , R b : The mapping ϕ : 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 ϕ ∈ End Q, then ϕ preserves also the binary operations \ and /.
Similarly to Abelian groups, all endomorphisms of a medial quasigroup Q are summable, i.e. if ϕ, ψ are endomorphisms of a medial quasigroup Q, then ϕ + ψ defined by 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 φ and ψ, and an element c The Abelian group Q, +, −, 0 is called an underlying Abelian group of the medial quasigroup Q.
From now on, we write π e for the endomorphism Q → {e}.Clearly, π e is a left-zero in End Q, i.e. π e • ϕ = π e for each ϕ ∈ End Q.
Conversely, suppose that an endomorphism θ is left-zero in End Q.For each e, x ∈ Q we have Hence, θ = π f , where f = θ (e).
Proposition 3. Suppose that Q, • is a medial quasigroup, where • is in the form (1). Then the following hold: 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 • y = 2 −1 (x + y).
Proof.Suppose that Q, • is an IMC-quasigroup.It follows from Toyoda's theorem that x • y = φ (x) + ψ(y) + c, where φ and ψ are automorphisms of Q, +, −, 0 .Since Q, • is commutative and idempotent, Proposition 3 shows that φ = ψ, c = 0, and 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, • .
The next corollary is a special case of Proposition 3 in [8].

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).
We denote it by Q, •, e .
Definition 7. A loop Q, •, e is called a commutative Moufang loop if it satisfies the identity From Definition 7 it follows that the binary operation • is commutative and there exists the mapping Let Q, • be an IMC-quasigroup and k ∈ Q.Let us define a new binary operation on Q as follows: The magma Q, ⊕ k is a commutative Moufang loop with the identity element k by Theorem 8.1 in [2].Due to commutativity of Q, • , we have that Moreover, the unary operation −1 of Q, ⊕ k will be denoted by k and the element k x coincides with k/x.
Proposition 5.The commutative Moufang loop Q, ⊕ k , k , k is an Abelian group.
Proof (see also Theorem 9 in [5]).It is sufficient to prove only the associativity of ⊕ k .Let x, y, z ∈ Q.Then As a particular case we have Corollary 5. Let the assumptions be as in Corollary 4. Then x ⊕ 0 y = x + y.
By the last corollary we have Therefore, 2 = L −1 0 , i.e.L −1 0 is an automorphism of the underlying Abelian group.We have a more general result.
Proof (see also Theorem 10 in [5]).For each k ∈ Q, the mapping We will write 0 Q for the set of all left-zero endomorphisms of an ICM-quasigroup Q.By Proposition 2, and, by Zorn's lemma, has a maximal element.From now on, M k denotes a maximal submonoid in End(Q, •) Proposition 7. The submonoid M k coincides with the endomorphism monoid of the Abelian group Proof.The proof is divided into three steps: . By the definition of M k we conclude that π k = π ϕ(k) and hence k = ϕ(k).
The proposition is proved.
Obviously 1 Q ∈ M and ϕ •ψ ∈ M whenever ϕ, ψ ∈ M .Hence, M is a monoid and π k ∈ M by the definition of π k .The first part of the proof of Proposition 7 implies 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.
Group-theoretic proof.Let End(Q, •) be finite.Since End(Q, +) ⊆ End(Q, •), 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, •).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, •) and End(Q , * ), respectively.Let M End(Q, •) be the maximal submonoid, such that M ∩ 0 Q = {π k } for some k ∈ Q.By Propositions 6 and 7 and Corollary 5 we have an isomorphism M ∼ = End(Q, +).Let Γ : End(Q, •) → End(Q , * ) be an isomorphism of monoids.Hence, the image of the restriction of Γ to End(Q, +) is the maximal submonoid M End(Q , * ) such that M ∩ 0 Q = {π k } for some k ∈ Q .From Propositions 6 and 7 we conclude that M ∼ = End(Q , + ) and finally that End(Q, +) ∼ = End(Q , + ).

Definition 1 .
A magma Q, • is called a quasigroup if each of the equations ax = b and ya = b has a unique solution for any a, b ∈ Q.
are bijective.Hence, to each quasigroup Q one can associate the subgroup M(Q) = {L a , R b | a, b ∈ Q}, • of the group of all bijections Q → Q.The group M(Q) is called a multiplication group or an associated group of the quasigroup Q.