Green's-Like Relations on Algebras and Varieties

Abstract

There are five equivalence relations known as Green's relations definable on any semigroup or monoid, that is, on any algebra with a binary operation which is associative. In this paper, we examine whether Green's relations can be defined on algebras of any type τ. Some sort of (super-)associativity is needed for such definitions to work, and we consider algebras which are clones of terms of type τ, where the clone axioms including superassociativity hold. This allows us to define for any variety V of type τ two Green's-like relations ℒV and ℛV on the term clone of type τ. We prove a number of properties of these two relations, and describe their behaviour when V is a variety of semigroups.