Open Problems in Semigroup Theory.

Question

NB: If this is a duplicate in any way or is too broad, I'm sorry.

What are some open problems in Semigroup Theory?

I'm reading Howie's "Fundamentals of Semigroup Theory" and I'm looking for problems accessible from that point of view.

Answer 1

Here's one by Peter J. Cameron:

Given a finite group $G$, let $\operatorname{End}(G)$ be the semigroup of endomorphisms of $G$, and $\operatorname{PIso}(G)$ the semigroup of partial isomorphisms of $G$ (isomorphisms between subgroups of $G$).

If $G$ is abelian, then $|\operatorname{End}(G)| = |\operatorname{PIso}(G)|$. Does the converse hold?

Answer 2

Warning. In this answer, I exclude open problems from group theory (with one exception, but the question really came from semigroup theory). I also only consider semigroups in algebra.

Word problem. Is the word problem for one-relator semigroups decidable?

Likely the most important open problem on semigroups. See [1] for related questions and [2] for a survey.
[1] A. J. Cain and V. Maltcev, Victor. Decision problems for finitely presented and one-relation semigroups and monoids. Internat. J. Algebra Comput. 19 (2009), no. 6, 747--770.
[2] Nik Ruskuc, One Relation Semigroups (slides, 2009).

Finite semigroups

Big lists of open problems on finite semigroups can be found in [1, 2, 3]:
[1] J. Almeida, Finite semigroups and universal algebra, Series in Algebra, 3. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. 511 pp. ISBN: 981-02-1895-8 (60 open problems at the time of publication)
[2] J. Rhodes, B. Steinberg, The $q$-theory of finite semigroups. Springer Monographs in Mathematics. Springer, New York, 2009. xxii+666 pp. ISBN: 978-0-387-09780-0 (74 open problems at the time of publication)
[3] M. V. Volkov, The Finite Basis Problem for Finite Semigroups, 2014 update, originally published in Sci. Math. Jap. 53, no.1, (2001) 171–199

Many questions on finite semigroups deal with varieties of finite semigroups, that are classes of finite semigroups closed under taking homomorphic images, subsemigroups and finite products, and operations defined on them. I just list a few of them below:

(a) Semidirect products
Given two varieties of finite monoids $\mathbf{V}$ and $\mathbf{W}$, let $\mathbf{V} * \mathbf{W}$ be the variety generated by the semidirect products of a member of $\mathbf{V}$ by a member of $\mathbf{W}$. Let $\mathbf{A}$ be the variety of finite aperiodic monoids and let $\mathbf{G}$ be the variety of finite groups. The following questions are open:

1. Is the variety $\mathbf{A} * \mathbf{G} * \mathbf{A}$ decidable?
2. More generally, for each $n$, is the variety $(\mathbf{A} * \mathbf{G})^n * \mathbf{A}$ decidable? This is the famous decidability of group complexity problem.

(b) Power semigroups
Let $\mathcal{P}(S)$ denote the semigroup of all subsets of a semigroup $S$.

1. Let $S$ and $T$ be finite semigroups. If $\mathcal{P}(S)$ is isomorphic to $\mathcal{P}(T)$, does it imply that $S$ is isomorphic to $T$?
2. What is the variety of finite semigroups generated by the power semigroups of finite solvable groups. See this question.
3. What is the variety of finite semigroups generated by the power semigroups of idempotent semigroups (also called bands in the literature)?
4. Complete the classification of power varieties.

(c) Profinite semigroups

1. Describe the free pro-aperiodic semigroup and the free pro-finite semigroup

Inverse semigroups. The slides by J. Meakin (2012) Inverse Semigroups: some open questions contain a list of six open problems on inverse semigroups.

Regular semigroups. See M.B. Szendrei, Some open problems in the structure theory of regular semigroups. Semigroup Forum 64 (2002), no. 2, 213--223.