Free right type A semigroups
John Fountain · Glasgow Mathematical Journal · 1991
The relation ℒ* is defined on a semigroup S by the rule that a ℒ* b if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S . A semigroup S is an E-semigroup if its set E ( S ) of idempotents is a subsemilattice of S . A right adequate semigroup is an E -semigroup in which every ℒ*-class contains an idempotent. It is easy to see that, in fact, each ℒ*-class of a right adequate semigroup contains a unique idempotent [8]. We denote the idempotent in the ℒ*-class of a by a *. Then we may regard a right adequate semigroup as an algebra with a binary operation of multiplication and a unary operation *. We will refer to such algebras as *-semigroups. In [10], it is observed that viewed in this way the class of right adequate semigroups is a quasi-variety.