We study conjugacy relations on semigroups and monoids, focusing on the relation a similar to(n) b, defined by the existence of g,h E S1 such that ag = gb, bh = ha, hag = b, and gbh = a. This notion emerged as one that yields particularly elegant results. The interplay between similar to(n) and other standard conjugacy relations is analyzed, and some results on special classes of abstract semigroups are established. We then specialize to the case of transformation semigroups. A complete classification of similar to(n)-classes is obtained for the full transformation monoid similar to(n), the symmetric inverse monoid T-n, and the endomorphism monoid of G-sets, among others. We also investigate the natural conjugacy in diagram semigroups, including the partition monoid, the Brauer monoid, and the partial Brauer monoid. Finally, we investigate the conjugacy growth function in polycyclic monoids and obtain a precise asymptotic estimate. The paper concludes with some open problems.(c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

We introduce the inverse monoid of inner partial automorphisms of a semigroup - a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner automorphisms with a zero adjoined. We then describe this structure for completely simple semi-groups, the full transformation monoid, and the endomorphism monoid of a finite G-set, when G is a finite abelian group. The paper ends with some open problems. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

A complete mapping of a semigroup S is a bijection α S→ S such that the map θ S→ S defined by xθ=x· xα is also a bijection. Equivalently, it determines a transversal of the multiplication table of S. Complete mappings connect group theory, Latin squares, and cryptography, and their existence for finite groups was characterized by the resolution of the Hall–Paige conjecture. In this paper, we develop the corresponding theory for finite semigroups. We prove that every finite semigroup admitting a complete mapping is regular and that the problem reduces to principal factors. We classify the existence of a complete mapping in Rees matrix semigroups without zero, give a Hall-type criterion for Rees 0-matrix semigroups over groups with complete mappings, and prove sufficient conditions for Rees 0-matrix semigroups whose maximal subgroups do not have complete mappings. As the main application of the Rees 0-matrix analysis, we show that T_n has a complete mapping if and only if n=1 or n≥ 4. Equivalently, T_n has a complete mapping if and only if the same holds for S_n. We prove that the full linear monoid of a finite-dimensional vector space has a complete mapping except in dimension 1 over a field of odd order and in dimension 2 over 𝔽_2. We also prove that the partition monoid 𝒫_n has a complete mapping if and only if n=1 or n≥4, and that every finite aperiodic regular *-semigroup has a complete mapping. As a consequence, the planar partition, Motzkin and Jones monoids have complete mappings. The paper concludes with open problems.

We study conjugacy relations on semigroups and monoids, focusing on the relation a b, defined by the existence of g,h ∈ S^1 such that ag = gb, bh = ha, hag = b, and gbh = a. This notion emerged as one that yields particularly elegant results. The interplay between and other standard conjugacy relations is analyzed, and some results on special classes of abstract semigroups are established. We then specialize to the case of transformation semigroups. A complete classification of -classes is obtained for the full transformation monoid 𝒯_n, the symmetric inverse monoid ℐ_n, and the endomorphism monoid of G-sets, among others. We also investigate the natural conjugacy in diagram semigroups, including the partition monoid, the Brauer monoid, and the partial Brauer monoid. Finally, we investigate the conjugacy growth function in polycyclic monoids and obtain a precise asymptotic estimate. The paper concludes with some open problems.

The CREAM GAP package computes automorphisms, congruences, endomorphisms and subalgebras of algebras with an arbitrary number of binary and unary operations; it also decides if between two such algebras there exists a monomorphism, an epimorphism, an isomorphism or if one is a divisor of the other. Thus it finds those objects for almost all algebras used in practice (groups, quasigroups in their various signatures, semigroups possibly with many unary operations, fields, semi-rings, quandles, logic algebras, etc). As a one-size-fits-all package, it only relies on universal algebra theorems, without taking advantage of specific theorems about, eg, groups or semigroups to reduce the search space. Canon and Holt produced very fast code to compute automorphisms of groups that outperform CREAM on orders larger than 128. Similarly, Mitchell et al. take advantage of deep theorems to compute automorphisms and congruences of completely 0-simple semigroups in a very efficient manner. However these domains (groups of order above 128 and completely 0-simple semigroups) are among the very few examples of GAP code faster than our general purpose package CREAM. For the overwhelming majority of other classes of algebras, either ours is the first code computing the above mentioned objects, or the existing algorithms are outperformed by CREAM, in some cases by several orders of magnitude. To get this performance, CREAM uses a mixture of universal algebra algorithms together with GAP coupled with artificial intelligence theorem proving tools (AITP) and very delicate C implementations. As an example of the latter, we re-implement Freese's very clever algorithm for computing congruences in universal algebras, in a way that outperforms all other known implementations.

A universal algebra A with underlying set A is said to be a matroid algebra if (A,〈⋅〉), where 〈⋅〉 denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping α:X→A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n, with at least two elements. Denote by End(A) the monoid of endomorphisms of A. In the 1970s, Głazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabó in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabó to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory.

Let $\Omega$ be a finite set and $T(\Omega)$ be the full transformation monoid on $\Omega$. The rank of a transformation $t\in T(\Omega)$ is the natural number $|\Omega t|$. Given $A\subseteq T(\Omega)$, denote by $\langle A\rangle$ the semigroup generated by $A$. Let $k$ be a fixed natural number such that $2\le k\le |\Omega|$. In the first part of this paper we (almost) classify the permutation groups $G$ on $\Omega$ such that for all rank $k$ transformation $t\in T(\Omega)$, every element in $S_t:=\langle G,t\rangle$ can be written as a product $eg$, where $e^2=e\in S_t$ and $g\in G$. In the second part we prove, among other results, that if $S\le T(\Omega)$ and $G$ is the normalizer of $S$ in the symmetric group on $\Omega$, then the semigroup $SG$ is regular if and only if $S$ is regular. (Recall that a semigroup $S$ is regular if for all $s\in S$ there exists $s'\in S$ such that $s=ss's$.) The paper ends with a list of problems.

Let $G$ be a permutation group of degree $n$, and $k$ a positive integer with $k\le n$. We say that $G$ has the $k$-existential property, or $k$-et for short, if there exists a $k$-subset $A$ of the domain $\Omega$ such that, for any $k$-partition $\mathcal{P}$ of $\Omega$, there exists $g\in G$ mapping $A$ to a transversal (a section) for $\mathcal{P}$. This property is a substantial weakening of the $k$-universal transversal property, or $k$-ut, investigated by the first and third author, which required this condition to hold for all $k$-subsets $A$ of the domain. Our first task in this paper is to investigate the $k$-et property and to decide which groups satisfy it. For example, we show that, for $8\le k\le n/2$, the only groups with $k$-et are the symmetric and alternating groups; this is best possible since the Mathieu group $M_{24}$ has $7$-et. We determine all groups with $k$-et for $4\le k\le n/2$, up to some unresolved cases for $k=4,5$, and describe the property for $k=2,3$ in permutation group language. In the previous work, the results were applied to semigroups, in particular, to the question of when the semigroup $\langle G,t\rangle$ is regular, where $t$ is a map of rank $k$ (with $k

In this paper we introduce the definition of $(k,l)$-universal transversal property, which is a refinement of the definition of $k$-universal transversal property, which in turn is a refinement of the classic definition of $k$-homogeneity for permutation groups. In particular, a group possesses the $(2,n)$-universal transversal property if and only if it is primitive; it possesses the $(2,2)$-universal transversal property if and only if it is $2$-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the $(k,l)$-universal transversal property, for $k\ge 3$. Then we apply this result for studying regular semigroups of partial transformations.

Stable basis algebras were introduced by Fountain and Gould and developed in a series of articles. They form a class of universal algebras, extending that of independence algebras, and reflecting the way in which free modules over well-behaved domains generalise vector spaces. If a stable basis algebra $$\mathbb{B}$$ satisfies the distributivity condition (a condition satisfied by all the previously known examples), it is a reduct of an independence algebra $$\mathbb{A}$$ . Our first aim is to give an example of an independence algebra not satisfying the distributivity condition. Gould showed that if a stable basis algebra $$\mathbb{B}$$ with the distributivity condition has finite rank, then so does the independence algebra $$\mathbb{A}$$ of which it is a reduct, and that in this case the endomorphism monoid $${\rm End}(\mathbb{B})$$ of $$\mathbb{B}$$ is a left order in the endomorphism monoid $${\rm End}(\mathbb{A})$$ of $$\mathbb{A}$$. We complete the picture by determining when $${\rm End}(\mathbb{B})$$ is a right, and hence a two-sided, order in $${\rm End}(\mathbb{A})$$. In fact (for rank at least 2), this happens precisely when every element of $${\rm End}(\mathbb{A})$$ can be written as $${\alpha}^{\sharp} \beta$$ where $$\alpha,\beta\in{\rm End}(\mathbb{B})$$, $${\alpha}^{\sharp}$$ is the inverse of $$\alpha$$ in a subgroup of $${\rm End}(\mathbb{A})$$ and $$\alpha$$ and $$\beta$$ have the same kernel. This is equivalent to $${\rm End}(\mathbb{B})$$ being a special kind of left order in $${\rm End}(\mathbb{A})$$ known as straight.

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in particular we show that $2$-homogeneous groups are $2$-generated. We also describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts, classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partitions are invariant under the normalizer of $G$ in $S_n$, and describe the normalizers of $2$-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

We show that every finite affine algebra A admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure (A)under-tilde of finite type. We give an explicit bound on the arities of the partial and total operations appearing in (A)under-tilde. In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.

We show that every finite affine algebra A admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure (A)under-tilde of finite type. We give an explicit bound on the arities of the partial and total operations appearing in (A)under-tilde. In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.

Malcev described the congruences of the monoid $T_n$ of all full transformations on a finite set $X_n=\{1, \dots,n\}$. Since then, congruences have been characterized in various other monoids of (partial) transformations on $X_n$, such as the symmetric inverse monoid $In_n$ of all injective partial transformations, or the monoid $PT_n$ of all partial transformations. The first aim of this paper is to describe the congruences of the direct products $Q_m\times P_n$, where $Q$ and $P$ belong to $\{T, PT,In\}$. Malcev also provided a similar description of the congruences on the multiplicative monoid $F_n$ of all $n\times n$ matrices with entries in a field $F$, our second aim is provide a description of the principal congruences of $F_m \times F_n$. The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and a fairly large number of open problems.

We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either explicitly determine the automorphism group or we show that the graph is a “normal” circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups.

A mapping \(\alpha :S\rightarrow S\) is called a Cayley function if there exist an associative operation \(\mu :S\times S\rightarrow S\) and an element \(a\in S\) such that \(\alpha (x)=\mu (a,x)\) for every \(x\in S\). The aim of the paper is to give a characterization of Cayley functions in terms of their directed graphs. This characterization is used to determine which elements of the centralizer of a permutation on a finite set are Cayley functions. The paper ends with a number of problems.

Let $s(n)$ be the side length of the smallest square into which $n$ non-overlapping unit squares can be packed. In 2010, the author showed that $s(13)=4$ and $s(46)=7$. Together with the result $s(6)=3$ by Keaney and Shiu, these results strongly suggest that $s(m^2-3)=m$ for $m\ge 3$, in particular for the values $m=5,6$, which correspond to cases that lie in between the previous results. In this article we show that indeed $s(m^2-3)=m$ for $m=5,6$, implying that the most efficient packings of 22 and 33 squares are the trivial ones. To achieve our results, we modify the well-known method of sets of unavoidable points by replacing them with continuously varying families of such sets.

Let Omega be a set of cardinality n, G be a permutation group on Omega and f : Omega -> Omega be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup < G, f > contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation.A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation.The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately root n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group.The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n - 1 and n - 2, and here this is extended to n - 3 and n -4.In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically root|A|.Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

We show that every finite Abelian algebra A from congruence-permutable varieties admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure of finite type. We give an explicit bound on the arities of the partial and total operations appearing in the dualizing structure. In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.

Abstract Let $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$ ) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$ . In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$ ), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture. The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$ ), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem. The paper ends with a number of problems for experts in group and semigroup theories.