Semisymmetrizations of Abelian Group Isotopes

Abstract

This note begins a study of the structure of quasigroup semisymmetrizations. For the class of quasigroups isotopic to abelian groups, a fairly complete description is available. The multiplication group is the split extension of the cube of the abelian group by a cyclic group whose order is identified as the semisymmetric index of the quasigroup. For a finite abelian group isotope, the dual of the semisymmetrization is isomorphic to the opposite of the semisymmetrization. The character table of the semisymmetrization is readily computed. The simplicity question for semisymmetrizations is raised. It is shown that a simple, non-abelian quasigroup need not have a simple semisymmetrization.

Quasigroup Homotopies, Semisymmetrization, and Reversible Automata

“…Each quasigroup Q yields a reversible automaton Q at or (Q, Q, Q) with equal state spaces. From (10), it is then apparent that a quasigroup homotopy f = (f 1 , f 2 , f 3 ) : Q → Q yields a corresponding homomorphism f at : Q at → Q at of reversible automata. Thus a functor at : Qtp → RAt is defined, corestricting to an equivalence at : Qtp → QAt of the homotopy category Qtp with a category QAt of homomorphisms between reversible automata.…”

“…Let (A, +) be a non-trivial abelian group. Now the characteristic congruence ν on the semisymmetrization A∆ is defined by [10]. Then ν is a non-trivial congruence.…”

Semisymmetrization and Mendelsohn quasigroups

“…For a quasigroup Q that is an isotope of an abelian group A, Section 4 shows how the semisymmetrization Q ∆ may be recovered from the Mendelsohnization Q Γ , and an action of Q Γ on the abelian group A, by the short exact sequence (4.4). In particular, the characteristic congruence on Q ∆ , originally introduced in [8], is now recognized as the kernel of the surjection in the short exact sequence.…”

Modules over semisymmetric quasigroups

“…The trivial quasigroup). This example is introduced in[2]. We reproduce it here because of the nice connection it makes between representation theory and semisymmetrization.…”