On variants of a semigroup

Abstract

IfSis a (multiplicative) Semigroup anda∈S, the binary operation ∘ defined on the setSbyx∘y=x a yis associative and the resulting semigroup is called a variant ofS. We study the congruenceadefined onSby saying that two elements are α-related if and only if they determine the same variant ofS. Certain quotients of variants are used to provide an arbitrary semigroup with a generalised local structure. The variant formulation of Nambooripad's partial order on a regular semigroup is used to show that the order possesses a certain property (involvingD-equivalence).