Transposition Regular TA-Groupoids and Their Structures

Abstract

Tarski associative groupoid (TA-groupoid) is a kind of non-associative groupoid satisfying Tarski associative law. In this paper, the new notions of transposition regular TA-groupoid are proposed and their properties and structural characteristics are studied by using band and quasi-separativity. In particular, the following conclusions are strictly proved: (1) every left transposition regular TA-groupoid is a semigroup; (2) every left transposition regular TA-groupoid is the disjoint union of sub Abelian groups; and (3) a finite TA-groupoid with quasi-separativity and a finite left transposition regular TA-groupoid are equivalent.