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Congruences on direct products of transformation and matrix monoids (2018)
Journal Article
Araújo, J., Bentz, W., & Gomes, G. M. S. (2018). Congruences on direct products of transformation and matrix monoids. Semigroup Forum, 97(3), 384–416. https://doi.org/10.1007/s00233-018-9931-8

Malcev described the congruences of the monoid Tn of all full transformations on a finite set Xn={1,…,n}. Since then, congruences have been characterized in various other monoids of (partial) transformations on Xn, such as the symmetric inverse monoi... Read More

Orbits of primitive k-homogenous groups on (n-k)-partitions with applications to semigroups (2017)
Journal Article
Araújo, J., Bentz, W., & Cameron, P. J. (2018). Orbits of primitive k-homogenous groups on (n-k)-partitions with applications to semigroups. Transactions of the American Mathematical Society, 371, 105-136. https://doi.org/10.1090/tran/7274

The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we examine the k-homogeneous permutation gro... Read More

Automorphism groups of circulant digraphs with applications to semigroup theory (2017)
Journal Article
Araújo, J., Bentz, W., Dobson, E., Konieczny, J., & Morris, J. (2018). Automorphism groups of circulant digraphs with applications to semigroup theory. Combinatorica, 38(1), 1-28. https://doi.org/10.1007/s00493-016-3403-0

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" ci... Read More

Primitive groups, graph endomorphisms and synchronization (2016)
Journal Article
Aroújo, J., Bentz, W., Cameron, P. J., Royle, G., & Schaefer, A. (2016). Primitive groups, graph endomorphisms and synchronization. Proceedings of the London Mathematical Society, 113(6), 829-867. https://doi.org/10.1112/plms/pdw040

© 2016 London Mathematical Society. Let Ω be a set of cardinality n, G be a permutation group on Ω and f : Ω → Ω 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 th... Read More

Directed graphs of inner translations of semigroups (2016)
Journal Article
Araújo, J., Bentz, W., & Konieczny, J. (2017). Directed graphs of inner translations of semigroups. Semigroup Forum, 94(3), 650-673. https://doi.org/10.1007/s00233-016-9821-x

A mapping α: S → S is called a Cayley function if there exist an associative operation µ: S x S → S and an element a ϵ S such that α(x) = µ(a, x) for every x ϵ S. The aim of the paper is to give a characterization of Cayley functions in terms of thei... Read More

The rank of the semigroup of transformations stabilising a partition of a finite set (2015)
Journal Article
Araújo, J., Bentz, W., Mitchell, J., & Schneider, C. (2015). The rank of the semigroup of transformations stabilising a partition of a finite set. Mathematical proceedings of the Cambridge Philosophical Society, 159(02), 339-353. doi:10.1017/S0305004115000389

Let P be a partition of a finite set X. We say that a full transformation f:X→X preserves (or stabilizes) the partition P if for all P∈P there exists Q∈P such that Pf⊆Q. Let T(X,P) denote the semigroup of all full transformations of X that preserve t... Read More

The commuting graph of the symmetric inverse semigroup (2015)
Journal Article
Araújo, J., Bentz, W., & Janusz, K. (2015). The commuting graph of the symmetric inverse semigroup. Israel journal of mathematics, 207(1), 103-149. doi:10.1007/s11856-015-1173-9

The commuting graph of a finite non-commutative semigroup S, denoted G(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 I(X) be the symmetric inverse semigroup of parti... Read More

The largest subsemilattices of the endomorphism monoid of an independence algebra (2014)
Journal Article
Araújo, J., Bentz, W., & Konieczny, J. (2014). The largest subsemilattices of the endomorphism monoid of an independence algebra. Linear algebra and its applications, 458, 60-79. doi:10.1016/j.laa.2014.05.041

An algebra A is said to be an independence algebra if it is a matroid algebra and every map α:X→A, defined on a basis X of A, can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and... Read More

Taylor's modularity conjecture holds for linear idempotent varieties (2014)
Journal Article
Bentz, W., & Sequeira, L. (2014). Taylor's modularity conjecture holds for linear idempotent varieties. Algebra universalis, 71(2), 101-107. doi:10.1007/s00012-014-0273-4

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... Read More

Dualizability of automatic algebras (2013)
Journal Article
Bentz, W., Davey, B., Pitkethly, J., & Willard, R. (2014). Dualizability of automatic algebras. Journal of pure and applied algebra, 218(7), 1324-1345. doi:10.1016/j.jpaa.2013.11.020

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 dualiz... Read More

Supernilpotence prevents dualizability (2013)
Journal Article
Bentz, W., & Mayr, P. (2014). Supernilpotence prevents dualizability. Journal of the Australian Mathematical Society, 96(01), 1-24. https://doi.org/10.1017/S1446788713000517

We address the question of the dualizability of nilpotent Mal’cev algebras, showing that nilpotent finite Mal’cev algebras with a nonabelian supernilpotent congruence are inherently nondualizable. In particular, finite nilpotent nonabelian Mal’cev al... Read More