On some problems of conjugacy in semigroups

Abstract

The conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019].

KEYWORDS:

• Cancellative semigroup
• o-conjugacy
• p-conjugacy
• r-conjugacy
• restriction semigroup
• universal relation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M10
• 20M18

Acknowledgments

The authors express their profound gratitude to the referee for the valuable comments and suggestions, which improve greatly the content and presentation of this article. In particular, according to the referee’s advices, we rewrite Problem 2.4 and add Problem 3.10 and Example 4.8.

Disclosure statement

The authors state no conflict of interest.

Additional information

Funding

The paper is supported by Nature Science Foundations of China (12271442, 11661082) and a Nature Science Foundation of Shandong Province of China (ZR2020MA002).