Four notions of conjugacy for abstract semigroups

Abstract

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations, and transformation semigroups). Here we study them in full generality. The paper ends with a large list of open problems.

Conjugacy in inverse semigroups

“…Finally, we characterize one of the two extreme cases for i-conjugacy on an inverse semigroup S, namely where ∼ i is the universal relation S × S. In Theorem 6.9 we will consider the opposite extreme, where ∼ i is the identity relation (equality). Similar discussions for other notions of conjugacy can be found in [1].…”

“…e m u m is canonical with the first idempotent piece xx −1 e 1 . By Lemma 3.3, e 1 = xx −1 e 1 , and so w = xx −1 w. Similarly, if x −1 / ∈ A 2 (w), then wxx −1 = w. We have proved (1). Statement (2) follows from (1) and Proposition 1.3.…”

“…where for a = 0, P(a) = {g ∈ S 1 : ∀ m∈S 1 (ma = 0 ⇒ (ma)g = 0)}, P(0) = {0}, and P 1 (a) = P(a) ∪ {1}. [1].…”

On the complexity of inverse semigroup conjugacy

“…Relatedly, this paper will give several hardness results left open in [9]. Finally, this paper will answer Open Problem 6.10 from [3].…”

“…It turns out there are manys ways to define such an equivalence relation. In [3], the authors analyze four ways of defining conjugacy for semigroups and this paper will begin with a brief review of these types of conjugacy. This paper will then build upon results from [2] related to conjugacy in inverse semigroups, with a particular focus on determining the computational complexity of decision problems involving conjugacy in finite inverse semigroups.…”