The Regular Part of a Semigroup of Full Transformations with Restricted Range: Maximal Inverse Subsemigroups and Maximal Regular Subsemigroups of Its Ideals

Worachead Sommanee

doi:10.1155/2018/2154745

Abstract

Let TX be the full transformation semigroup on a set X. For a fixed nonempty subset Y of a set X, let TX,Y be the semigroup consisting of all full transformations from X into Y. In a paper published in 2008, Sanwong and Sommanee proved that the set FX,Y=α∈TX,Y:Xα=Yα is the largest regular subsemigroup of TX,Y. In this paper, we describe the maximal inverse subsemigroups of FX,Y and completely determine all the maximal regular subsemigroups of its ideals.

Highlights

Let T(X) be the set of all full transformations from a nonempty set X into itself
We describe the maximal inverse subsemigroups of F(X, Y) and completely determine all the maximal regular subsemigroups of its ideals
In 1999, Yang [6] described all of the maximal inverse subsemigroups of the finite symmetric inverse semigroup

Summary

Introduction

Let T(X) be the set of all full transformations from a nonempty set X into itself. It is well-known that T(X) is a regular semigroup under composition of functions; see [1, p. 63]. Later in 2001, Yang [7] obtained the maximal subsemigroups of the finite singular transformation semigroups. In 2002, You [8] determined all the maximal regular subsemigroups of all ideals of the finite full transformation semigroup. L. Yang [9] completely described the maximal subsemigroups of ideals of the finite full transformation semigroup. In 2014, Zhao et al [10] showed that any maximal regular subsemigroup of ideals of the finite full transformation semigroup is idempotent generated. Sanwong [17] described Green’s relations and ideals and all maximal regular subsemigroups of F(X, Y). In 2015, Sommanee and Sanwong [23] investigated the regularity and Green’s relations of the order-preserving transformation semigroup. We describe the maximal inverse subsemigroups of F(X, Y) and completely determine all the maximal regular subsemigroups of its ideals

Preliminaries and Notations

Maximal Inverse Subsemigroups