Skew Pairs of Idempotents in Partial Transformation Semigroups

Panuwat Luangchaisri,Thawhat Changphas

doi:10.3390/math10010002

Abstract

Let S be a regular semigroup. A pair (e,f) of idempotents of S is said to be a skew pair of idempotents if fe is idempotent, but ef is not. T. S. Blyth and M. H. Almeida (T. S. Blyth and M. H. Almeida, skew pair of idempotents in transformation semigroups, Acta Math. Sin. (English Series), 22 (2006), 1705–1714) gave a characterization of four types of skew pairs—those that are strong, left regular, right regular, and discrete—existing in a full transformation semigroup T(X). In this paper, we do in this line for partial transformation semigroups.

Highlights

In [1], the authors defined a skew pair of idempotents in a regular semigroup as follows: Citation: Luangchaisri, P.; Changphas, T
Assume that P( X ) contains a discrete skew pair of Lemmas 4 and 5, it follows that | X | > 5
We considered the existence of skew pairs of idempotents on partial transformation semigroups using the cardinality of its based set

Summary

Introduction

In [1], the authors defined a skew pair of idempotents in a regular semigroup as follows: Citation: Luangchaisri, P.; Changphas, T. We let x ∈ ran γ, there exists y ∈ dom γ such that γ(y) = x. P( X ) contains a strong skew pair of idempotents if and only if | X | ≥ 2.

Objectives

Results

Conclusion