Wolfram Bentz

The conjugacy relation plays an important role in group theory. If $a$ and $b$ are elements of a group~$G$, $a$ is conjugate to $b$ if $g^{-1}ag=b$ for some $g\in G$. Group conjugacy extends to inverse semigroups in a natural way: for $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$ if $g^{-1}ag=b$ and $gbg^{-1}=a$ for some $g\in S$...

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 algebra...

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 w...

In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal...

Let Ω be a finite set and T(Ω) be the full transformation monoid on Ω. The rank of a transformation t∈T(Ω) is the natural number |Ωt|. Given A⊆T(Ω), denote by 〈A〉 the semigroup generated by A. Let k be a fixed natural number such that 2≤k≤|Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank...

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...

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 o...

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 parti...

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 $\mathca...

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 co...

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∼ of finite type. We give an explicit bound on the arities of the partial and total operations appearing in A∼. In addition, we show that the enriched partial h...

A mapping α:S→S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha :S\rightarrow S$$\end{document} is called a Cayley function if there exist an associative operatio...

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 char...

Let $$\Omega $$ be a set of cardinality $$n$$, $$G$$ be a permutation group on $$\Omega $$ and $$f:\Omega \to \Omega $$ be a map that is not a permutation. We say that $$G$$synchronizes$$f$$ if the transformation semigroup $$\langle G,f\rangle $$ contains a constant map, and that $$G$$ is a synchronizing group if $$G$$ synchronizes every non-permut...

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...

Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle G,f\rangle$ contains a constant map, and that $G$ is a \emph{synchronizing group} if $G$ synchronizes \emph{every} non-permutation....

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 str...

An algebra AA is said to be an independence algebra if it is a matroid algebra and every map α:X→Aα:X→A, defined on a basis X of AA, can be extended to an endomorphism of AA. 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 t...

Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a full transformation $f:X\to X$ preserves (or stabilizes) the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full transformations of $X$ that preserve the partition $...

The “Modularity Conjecture” is the assertion that the join of two nonmodular varieties in the lattice of interpretability types is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning n-permutability for some n, and the satisfaction of nontrivial congrue...

Suppose that a deterministic finite automata $A=(Q,\Sigma)$ is such that all but one letters from the alphabet $\Sigma$ act as permutations of the state set $Q$ and the exceptional letter acts as a transformation with non-uniform kernel. Which properties of the permutation group $G$ generated by the letters acting as permutations ensure that $A$ be...

We make a start on one of George McNulty's Dozen Easy Problems: "Which finite automatic algebras are dualizable?" We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potenti...

We address the question of the dualizability of nilpotent Mal'cev algebras, showing that nilpotent finite Mal'cev algebras with a non-abelian supernilpotent congruence are inherently non-dualizable. In particular, finite nilpotent non-abelian Mal'cev algebras of finite type are non-dualizable if they are direct products of algebras of prime power o...

The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmet...

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep...

It has been suggested that infants resonate emotionally to others' positive and negative affect displays, and that these responses become stronger towards emotions with negative valence around the age of 12-months. In this study we measured 6- and 12-month-old infants' changes in pupil diameter when presented with the image and sound of peers exper...

Let s(n) be the side length of the smallest square into which n non-overlapping unit squares can be packed. We show that s(m 2 − 3) = m for m = 4, 7, implying that the most efficient packings of 13 and 46 squares are the trivial ones. The study of packing unit squares into a square goes back to Erdős and Graham [2], who showed that large numbers of...

It is a well-known result in the study of topological groups that any T 0-topological group is also regular and satisfies the stronger separation axiom T3. The same holds for topological quasi-groups. In the area of universal algebra, the only result on condition T3 is a negative one due to Coleman, showing that congruence permutability is not stro...

In 1997, Coleman suggested that congruence modularity and n-permutability are necessary and sufficient properties for a variety to have all of its T0-topological algebras be Hausdorff. The sufficiency part of this statement was later shown by Kearnes and Sequeira. Here we show that necessity holds for a certain subclass of varieties, hence providin...

In 1997, Coleman showed that a variety V is n-permutable for some n iff every T 0-topological Algebra in V is T 1. Here we show that the implication " sober" is another such characterization for n-permutability. Other implications of a similar nature are given. For example, an n-permutable variety having a majority term satisfies "".

S : a right S-act A is strongly flat if and only if the functor A\otimes- (from the category of left S-acts into the category of sets) preserves both pullbacks and equalizers. Stenström gave two interpolation-type conditions whose conjunction describes strong flatness. In 1986, P. Normak studied these conditions separately, lablelling them (P) and...