ON QUASIGROUPS WITH SOME MINIMAL IDENTITIES

Abstract

Quasigroups with two identities (of typesT1 and T2) from Belousov-Bennett classification are considered. It is proved that a π-quasigroup of type T2 is also of type only if it satisfies the identity yx∙x=y (the “right keys law”), so π -quasigroups that are of both types T1and T2 are RIP - quasigroups. Also, it is proved that π-quasigroups of type T2 are isotopic to idempotent quasigroups. Necessary and sufficient conditions when a π -quasigroup of typeT2 is isotopic to a group (an abelian group) are found. It is shown that the set of all π-quasigroups of type T2 isotopic to abelian groups is a subvariety in the variety of all π -quasigroups of type T2 and that π- T-quasigroups of type T2 are medial quasigroups. Using the symmetric group on Q∙Q, some considerations for the spectrum of finite π-quasigroups (Q1) of type T1 are discussed.Sunt considerate cvasigrupuri cu doua identitati (de tipurile T1 si T2) din clasificarea Belousov-Bennett. Se demonstreaza ca un π-cvasigrup de tipul T2 este un π-cvasigrup de tipul T1 dacasi numai daca el verifica identitatea yx∙x=y (legea „cheilor la dreapta”), deci π-cvasigrupurile care sunt simultan de tipul T1 si T2 sunt RIPМесто для формулы.-cvasigrupuri. De asemenea, se arata ca π-cvasigrupurile de tipul T2 sunt izotope unor cvasigrupuri idempotente. Sunt determinate conditiile necesare si suficiente ca un π-cvasigrup de tipul T2 sa fie izotop unui grup (grup abelian). Astfel, se obtine ca π-cvasigrupurile de tipul izotope unor grupuri abeliene formeaza o subvarietate in varietatea tuturor π-cvasigrupurilor de tipul T2 si ca π T-cvasigrupurile de tipul T2 sunt mediale. Este data o caracterizare a spectrului π-cvasigrupurilor finite(Q1) de tipul T1 in limbajul substitutiilor multimii Q∙Q.