On determinability of idempotent medial commutative quasigroups by their endomorphism semigroups

Abstract

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.

Medial and isospectral algebras

“…This implies that Idm(A) is a commutative idempotent medial (CIM, for short) quasigroup with respect to the algebra multiplication. Then the classical Bruck-Murdoch-Toyoda theorem characterizes any medial quasigroup as an isotope of a certain abelian group G. The structure of a CIM quasigroups and their endomorphisms has been recently studied by A. Leibak and P. Puusemp [38], [37]. In [38] a refinement of the Bruck-Murdoch-Toyoda theorem for CIM quasigroups has also been established.…”

“…This implies (36). Setting c 2 = c 1 we arrive at (37). Finally, since x 2 is a λ 2 -eigenvector of c 2 ,…”

“…Note that the product in the left hand side maybe zero such that the inclusion in (37) is trivial in that case. Proposition 4.6.…”

“…Medial algebra extensions. Let (Q, •) be an idempotent medial commutative quasigroup, IMC-quasigroup, for short, following [37]. Note that the order |Q| of an IMC-quasigroup is always an odd number, see, for example, Corollary 2.5 in [34].…”

“…The latter choice is essentially the only possible one as a straightforward verification by virtue of the Bruck-Murdoch-Toyoda theorem shows; see also Proposition 4.1 in [38] for a general abelian group. See some further discussion in [38], [37].…”