Eventually regular regressive transformation semigroups

Abstract

Les S be a semigroup. An element a of S is said to be regular if a = aba some b E S. An element a of S is said to be eventually regular if an is regular for some positive integer n. We call S an eventually regular semigroup if every element of S is eventually regular. A partial transformation alpha of a set is said to be almost identical if xa is not equal x for at most a finite number of elements X in the domain of alpha. Let X be a partially ordered set. A partial transformation alpha of X is said to be regressive xa is less than or equal to x for all x in the domain of alpha. Let PT RE(X), T RE(X), I RE(X), U RE(X), V RE(X) and W RE(X) denote the regressive partial transformation semigroup on X, the full regressive transformation semigroup on X, the regressive 1-1 partial transformation semigroup on X, the semigroup of all regressive almost identical partial transformations of X, the semigroup of all regressive almost identical transformations of X and the semigroup of all regressive almost identical 1-1 partial transformations of X, respectively. If S is a transformation semigroup on a set and theta E S, let (S,theta) denote the semigroup S with the product * defined by alpha * beta = alpha theta beta for all alpha, beta E S. The main results of this research are as follows: Theorem 1. Let X be a partially ordered set and let S be PT RE(X), T RE(X) or I RE(X). Then S is eventually regular if and only if there exists a positive integer n such that for every chain C of X, /C/ is less than or equal to n. Theorem 2. If X is a partially ordered set, then U RE(X), V RE(X) and W RE(X) are all eventually regular. Theorem 3. Let X be a partially ordered set and let S be PT RE(X), T RE(X) or I RE(X) and theta E S. Then (S,theta) is an eventually regular semigroup if and only if there exists a positive integer n such that /C/ is less than or equal to n for every chain C of the domain of theta having the property that for x,y E C, x < y implies x is less than or equal to y theta is less than or equal to y. Corollary 4. Let X be a partially ordered set and let S be PT RE(X), T RE(X) or I RE(X) and theta E S. If the range of theta is finite, then (S,theta) is eventually regular. Theorem 5. Let X be a partially ordered set. If S is U RE(X), V RE(X) or W RE(X), then for any theta E S, (S,theta) is eventually regular.