On Semigroups of Transformations Acting Transitively on a Set

Abstract

We call a semigroup S transitive if S is isomorphic to a semigroup T of transformations of some set M into itself, where T acts on M transitively, that is in such a manner that for all x, y ∊ M we have Xπ = y for some transformation π∊T. In [4] the author showed that S is transitive if and only if there exists a right congruence σ (i.e., an equivalence relation for which a σ b always implies ac σ bc for all c ∊ S) on S, satisfying:(1)There exists a left identity modulo σ, that is an element e such that ea σ a for all a ∊ S .(2)Each σ-class meets each right ideal, or, equivalently, for all a, b ∊ S we have ac σ b for some c ∊ S .(3)The relation σ contains ( i. e. , is less fine than) no left congruence except the identity relation (in which each class consists of a single element).