Regular elements of order-preserving transformation semigroups

Abstract

An element x of a semigroup S is called regular if there is an element y [is an element of a set] S such that x = xyx and S is said to be a regular semigroup if every element of S is regular. A mapping [alpha] from a partially ordered X into a partially ordered set Y is said to be order-preserving if for any x, x' [is an element of a set] X , x [is less than or equal to] x' n X-> x[alpha] [is less than or equal to] x'[alpha] in Y The semigroup, under composition, of all order-preserving transformations of a partially ordered set X is denoted by OT(X). Let Z and R be the chain of integers and the chain of real numbers, respectively, under the natural order. It is known that OT(X) is regular for every nonempty subset X of Z and for an interval X in R, OT(X) is regular if and only if X is closed and bounded. Moreover, for an interval X in a subfield F of , OT(X) is regular if and only if F =R and X is closed and bounded. In this research, we provide a necessary and sufficient condition for the elements of OT(X) to be regular when X is any chain. It is then applied to prove the above known results. For a chain X, the dictionary partially ordered set of X is the chain ( X x X , [is less than or equal to][subscript d] ) where [is less than or equal to][subscript d] defined by (a[subscript 1], b[subscript 1]) [is less than or equal to][subscript d] (a[subscript 2], b[subscript 2]) <-> a[subscript 1] < a[subscript 2] or a[subscript 1]=a[subscript 2] and b[subscript 1] [is less than or equal to] b[subscript 2] The characterization of regular elements is applied to determine when OT( X x X , [is less than or equal to][subscript d]) is a regular semigroup where X is a nonempty subset of Z, an interval in R or an interval in a subfield F of R.