Abstract: In this talk I will begin with a brief introduction to the free left adequate monoids. I will then discuss which identities are satisfied in the monogenic case. Finally, we will consider identities in the enriched (2,1,0) signature and explore which of these are satisfied.
Abstract: Dubuc showed in 1998 that all synchronizing automata having one letter acting cyclically in all vertices satisfy Cerny's conjecture. Pin had already established in 1978 this property for the special case where the cycle has prime length.
A one-cluster automaton A is a complete deterministic automaton with a state set Q and a distinguished letter x such that, for some nonnegative integer L, the letter x acts cyclically on the set C=Qx^L. The least of such L is called the ``level of A'', and the size of C is the ``length cycle''. Dubuc's result concerns the special case of one-cluster automata of level 0. Inspired by Pin's and Dubuc's techniques, Steinberg extended in 2011 Pin's result to all synchronizing one-cluster automata with prime length cycle.
Dubuc's proof relies on a very delicate induction scheme that makes it challenging to adapt to other situations. In this talk, we present a framework towards an extension of Dubuc's proof-scheme to one-cluster automata of strictly positive level. Within this framework, we are able to establish Cerny's conjecture for all synchronizing one-cluster automata such that |C|+1=|Q|. This special class of one-cluster automata includes several notable examples of slowly synchronizing automata that deserved special attention from Volkov and his co-authors, in their paper ``Primitive digraphs with large exponents and slowly synchronizing automata'' . Some elements of our framework work for other one-cluster automata, specially those where |C|+L=|Q| (that is, the case where the configuration of the edges labeled by the distinguished letter x is that of a frying pan). This is ongoing joint work with Benjamin Steinberg.
Abstract: Cayley's Theorem states that any finite monoid can be faithfully represented as a semigroup of transformations (self-maps) of a finite set. The minimum size of such a set is the \emph{(minimum transformation) degree} of the monoid.
We obtain formulae for the degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid $\mathcal P_n$ has degree $1 + \frac{B(n+2)-B(n+1)+B(n)}2$ for $n\geq2$, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.
This is joint work with Reinis Cirpons and James Mitchell, both at Univ St Andrews.
Abstract: Diagram monoids have their origins in theoretical physics and have found applications in many fields of mathematics such as representation, invariant and knot theories. There are various families of diagram monoids including the partition, transformation, Brauer, Temperley-Lieb and symmetric inverse monoids to name a few. The full-domain partition monoid arose naturally in two separate research programs: the work of East and Gray on Ehresmann structures in partitions monoids, and that of Stokes on constellations and restriction monoids. This talk will present joint work with Luka Carroll and James East where we developed presentations for the full-domain partition monoid and its singular ideal. The key ingredient being the decomposition of the full-domain partition monoid into the product of the transformation monoid and the semilattice of equivalences, and action pair results for restriction monoids by Carson, Dolinka, East, Gould and Zenab.
Abstract: Restriction monoids are the one of the most studied non-regular generalizations of inverse monoids. These are monoids equipped with a unary operation $a\mapsto a^+$ which models the operation of taking the domain idempotent of a partial function. $F$-restriction monoids are restriction monoids where every $\sigma$-class has a maximum element. They are natural generalizations of $F$-inverse monoids and form a variety with respect to the extended signature $(\cdot,\,^+, {\bf m}, 1)$, where the unary operation ${\bf m}$ maps each element $a$ to the maximum element of its $\sigma$-class. We adapt our previous construction of $F$-inverse group expansions~[1] which is based on the Margolis-Meakin group expansion, but involves some additional features such as a suitable extended set of generators and a certain identification of subgraphs intended to capture the ${\bf m}$-operation. Given an $X$-generated monoid $S$, we provide a construction of an $F$-restriction monoid $F(S,X)$ which is the initial object in the category of all $X$-generated $F$-restriction monoids with the maximum reduced quotient $S$. As a special case, we provide a model of the free $X$-generated $F$-restriction monoid and its strong and perfect analogues. This is joint work with Ganna Kudryavtseva.
Abstract: There is a number of important combinatorial matrix sets such as indecomposable matrices, primitive matrices, scrambling matrices, chainable matrices, etc., that are useful both in fundamental investigations and in applications. One of the approaches to study these sets is via certain their numerical invariants. To list a few of them, we mention matrix exponent, scrambling index and solidarity index.
In the talk we will define the extensions of these invariants and sets to matrix semigroups and investigate them. As an application, we obtain the analogs of the theorems by Perron, Frobenius, and Wielandt for semigroups of non-negative matrices.
This is a joint work with Yu. A. Alpin, A. M. Maksaev, E. R. Shafeev.
Abstract: An r-regular family F of permutations on a set V contains, for each pair of elements u, v ∈ V , exactly r permutations φ mapping u to v. Although the concept was introduced as a generalization of the concept of quasi-Cayley graphs first considered by Gauyacq in the 1990’s, its applications are interesting not just for researchers in the area of Algebraic Graph Theory, but to group theorists as well, and also have many interesting computational aspects. The key problem addressed in our presentation is that of the enumeration of r-regular families in transitive permutation groups; in particular in Sn. Due to lack of suitable constructions, the problem appears at the moment computationally hard, and the question of the existence of even a single r-regular family in a randomly chosen set of permutations in Sn exhibits connections to several well-known NP-complete problems. The aim of our presentation is to offer a quick overview of (computational) results obtained so far as well as to explore connections to other well-known group theoretical and combinatorial problems.
Abstract: Dedicated to Professor Mikhail Volkov on his 70th birthday
Mikhail Volkov is a brilliant mathematician who has obtained a number of fundamental results and is the author of well-known surveys on various aspects of semigroup theory. His encyclopedic knowledge, erudition, and talent have placed him among the world leaders in the field. The authors have known the jubilarian since their student days—more than 50 years. Many of their results were achieved thanks to his support, advice, and friendship.
This talk presents a survey of the authors’ work devoted to the equational theory of various classes of completely 0-simple semigroups. It is worth noting that Mikhail Volkov has consistently shown interest in these questions. His general criterion for the non-finite basis property of equational theories (see [1]), which significantly generalizes the method used by the second author in [2], is one of the earliest and most important results in this area.
The study of identities in completely 0-simple semigroups enables the derivation of identity bases for certain transformation semigroups and matrix semigroups (see [1, 7]). The first examples of finite completely 0-simple semigroups that possess neither a finite basis nor an independent basis of identities were presented by the second author in [2].
The class of all 0-simple periodic semigroups over groups of exponent n is denoted by RS_n. In 1991, G. Mashevitzky established that the following three identities:
x^2 = x; xyx = (xy)^{n+1}x; (xhz)^nxyz = xyz(xhz)^n
form an identity basis for the variety RS_n (see [6, 3]).
Following S. Kublanovsky, varieties satisfying these three identities are called Rees–Sushkevich varieties. The first author identified a series of 13 indicator finite semigroups whose absence in a given variety serves as a characteristic feature of Rees–Sushkevich varieties (see [4]). Descriptions of various properties of Rees–Sushkevich varieties were obtained by the first author in [5]. In particular, it was shown that there exists a polynomial-time algorithm which, given a finite basis of identities for a periodic variety, determines whether the variety is generated by completely 0-simple semigroups.
References
[1] L. N. Shevrin, M. V. Volkov, “Identities of semigroups”, Izv. Vyssh. Uchebn. Zaved. Mat., 1985, no. 11, 3–47; Soviet Math. (Iz. VUZ), 29:11 (1985), 1–64
[2] G.I. Mashevitzky, An example of a finite semigroup without an irreducible basis of identities in the class of completely 0-simple semigroups, Russian Math. Surveys, 1983, v. 38, p. 192 - 193 (translated from Uspehi mathem. nauk).
[3] T. E Hall, S. I. Kublanovskii, S. Margolis, M. V. Sapir, P. G. Trotter, Algorithmic problems for finite groups and finite 0-simple semigroups, Journal of Pure and Applied Algebra 119 (1), 75-96
[4] S. Kublanovsky, On the Rees - Sushkevich variety, arXiv : 1009, 29821v [math. GR]
[5] S. I. Kublanovsky, “On the rank of Rees–Sushkevich varieties”, Algebra i Analiz, 23:4 (2011), 59–135; St. Petersburg Math. J., 23:4 (2012), 679–730
[6] G.I. Mashevitzky, Varieties generated by completely 0-simple semigroups, in Semigroups and their homomorphisms, E. S. Lyapin (ed.), Leningrad, 1991, 53-62 (in Russian).
[7] G. Mashevitzky Bases of identities of semigroups of a bounded rank transformations of a set, Israel Journal of Mathematics, 191 (2012), 451 - 481
Abstract: We define and study automata of ${\tilde{R}}$-classes of restriction monoids which are analogous to Sch\"utzenberger automata of ${\mathcal{R}}$-classes of inverse monoids. We develop an analogue of Stephen's procedure which allows us (in some cases) to construct these automata. We also provide applications to the word problem for restriction monoids.
Abstract: I will discuss a spectral sequence arising from a natural filtration on the Kobayashi resolution associated to a complete rewriting system, leading to new proofs of several results, including the Lyndon--Hochschild--Serre spectral sequence as well as the cohomology of special monoids due to Gray & Steinberg (2021). I will then apply the sequence to compute the cohomological dimension of many one-relation monoids.
Abstract: Starting from any semigroup S with idempotents E one can define the so-called free idempotent-generated semigroup IG(E), which reflects the essential idempotent structure of S. These semigroups originated from the seminal work of Nambooripad on the structure of regular semigroups, and have been subject of intensive research over the past couple of decades. If S is a regular *-semigroup with projections P one can associate a second natural semigroup, the free projection-generated semigroup PG(P). This has recently been introduced by the above authors, and will be treated in the invited lecture by Azeef Muhammed. For both semigroups IG(E) and PG(P) the nature of their maximal subgroups is key to understanding their structure. In my talk I will report on the very recent results determining the maximal subgroups of both IG(E) and PG(P) in the case when S=P_n, the partition monoid of degree n.
Abstract: Let A be a family of m transformations on a set of n elements. The diameter of the semigroup generated by A is the smallest k such that every element of this semigroup is a product of at most k transformations from A. Denote by max-diam(n, m) the maximum diameter of a semigroup among all semigroups generated by m transformations on a set of n elements. I will discuss the precise growth of max-diam(n, m) depending on the relations between n and m, and will explain the connections with careful synchronisation of automata and the membership problem in semigroups of nonnegative integer matrices.
Abstract: Working with weighted automata naturally leads to the problem of the definition of the (Kleene) star operation on the semiring of (formal power) series. And proving that rational series are realised by weighted finite automata (one direction of Kleene Theorem) requires, in one way or another, to establish the identity: (s_0 + s_p)* = (s_0)* (s_p (s_0)*)* where s_0 and s_p are respectively the 'constant term' and the 'proper part' of a series s .
In my book 'Elements of Automata Theory', I give a proof of the above identity under the hypothesis that the weight semiring has the property that the product of two summable families is a summable family. I call such semirings 'strong' and even though all semirings that I know are strong, I stated the conjecture that there should exist some semirings which are not strong.
My colleague David Madore gave me an example of a semiring which is not strong and that I shall present.
Abstract: In the 1880s, Dedekind introduced the concept of the group determinant for finite groups and, together with Frobenius, began to study it in depth.
This study involved examining the determinant of a \(G \times G\) matrix, where the entry at position \((g, h)\) is \(x_{gh}\), with \(G\) being a finite group and the \(x_k\) variables corresponding to each \(k \in G\). One application of the semigroup determinant for finite semigroups is the extension of the MacWilliams theorem for codes over finite fields to chain rings.
Steinberg, in [1], provided a factorization of the semigroup determinant for commutative semigroups. Building on this, in [3], the determinant of semigroups in the pseudovariety \(\sf{ECom}\) is studied. This pseudovariety consists of semigroups whose idempotents commute. As shown by Ash, \(\sf{ECom}\) is generated by finite inverse semigroups-a broader class than the semigroups with central idempotents considered in [1].
In [4], the study was extended to semigroups beyond \(\sf{ECom}\), focusing on finite semigroups possessing pairs of non-commutative idempotents. The focus was on a class of semigroups which, while not in \(\sf{ECom}\), satisfy certain structural properties called \(\ll\)-smooth.
However, a fundamental question remains open: Are there other natural classes of semigroups or monoids that satisfy these conditions, and what are their determinant properties?
In this talk, we mention a class of monoids, which we call \emph{Layered Catalan Monoids} (\(LC_n\)), that satisfy the conditions for \(\ll\)-smoothness as defined in~\cite{Sha-Det2}. These monoids are characterized by specific algebraic identities, including those defining Catalan monoids, with some exceptions. We determine the canonical form of the elements in \(LC_n\). Using the theoretical framework developed in [4], we then compute the determinant of \(LC_n\). Our main result shows that the determinant is nonzero for \(1 \leq n \leq 7\), but becomes zero for \(n \geq 8\).
The key to computing the determinant of the semigroups studied in [1,3,4] lies in equipping the semigroup \(S\) with a partially ordered set structure, along with a map \(Z\) defined based on this order. The map \(Z\) sends each element of \(S\) to the sum of all elements in \(S\) that are less than it with respect to the partial order. A crucial aspect of this method is that \(Z\) induces an isomorphism of \(\mathbb{C}\)-algebras, thereby establishing a connection between the determinants of the original semigroup algebra and its image under \(Z\).
Since the determinant of the image algebra is typically much easier to compute, this isomorphism enables the calculation of the determinant of the original semigroup algebra via its image.
For the semigroups considered in [1,3,4], such a connection has been firmly established. In particular, [4] demonstrates the existence of this connection for \(\ll\)-transitive semigroups. However, for semigroups that are not \(\ll\)-transitive, this question remains open. In this talk, we also explore a broader class of semigroups in which the connection still holds, despite the absence of \(\ll\)-transitivity.
References:
[1] B. Steinberg. Factoring the Dedekind-Frobenius determinant of a semigroup. {\it J. Algebra}, 605:1--36, 2022.
[2] J. A. Wood. Factoring the semigroup determinant of a finite commutative chain ring. {\it In Coding theory, cryptography and related areas (Guanajuato, 1998)}, pages 249--259. Springer, Berlin, 2000.
[3] M.H. Shahzamanian, The determinant of semigroups of the pseudovariety \sf{ECom}. {\it J. Algebra Appl.}, 2025.
[4] M.H. Shahzamanian, A step to compute the determinant of finite semigroups not in \sf{ECom}, {\it arXiv:2407.00083}.
Abstract: A quiver presentation is a standard method for presenting an associative algebra. It consists of a generating graph Q and a set of relations imposed on the paths within Q.
Given a finite monoid M and a field K, constructing a quiver presentation for the monoid algebra KM is a natural and fundamental problem in the representation theory of finite monoids.
In this talk, we focus on the monoid algebra of OD_n, the monoid of all functions on an n-element set that are either order-preserving or order-reversing.
We will show that the induced left Schützenberger modules associated with Green's J-classes are projective and provide a complete description of the homomorphisms between them.
Using this description, we derive a quiver presentation for the algebra KOD_n, where K is a field of characteristic not equal to 2.
We show that the quiver consists of two straightline paths, and that all compositions of consecutive arrows equal to 0.
If time permits, we will also discuss related monoids and further connections between the representation theory of finite monoids and generalized Green's relations.
Abstract: In 1970 Weissglass gave sufficient conditions for the semigroup ring RS of an inverse semigroup to be von Neumann regular. In 1984 proved that over a field of characteristic 0 Weissglass’s sufficient conditions are necessary using analytic means. He also provided some necessary conditions and was able to handle the case of inverse semigroups with dcc on principal ideals. Recently, we have completely resolved the question over arbitrary rings for the more general situation of ample groupoid ring. Here I’ll sketch the proof idea that Weissglass’s sufficient conditions are necessary for inverse semigroups.
Abstract: A classical results states that free groups $F_n$, $n\geq 2$, enjoy the Howson property: intersection of finitely generated subgroups are, again, finitely generated. We prove that this is the only obstruction in the lattice of subgroups of $F_n$: there exists subgroups $H_1,\ldots ,H_m$ whose partial intersections are/are not finite generated, according to any prefixed pattern, as soon as it does not violate the Howson property. We will also discuss similar results for $F_nxZ^m$ and $BS(1,n)$ (results partially joint with J. Delgado, M. Roy, or A. Massegú).
Abstract: Following the work of J. Almeida and E. Rodaro in the context of the Černý conjecture and the Wedderburn-Artin theory, we show some relations between the Radical of a DFA and its congruences. In particular, we outline some techniques to construct the Radical of a DFA and we study its Wedderburn-Artin decomposition. Furthermore, we claim that it is sufficient to prove the Černý conjecture just for semisimple automata, and show this reduction in some particular cases.
Abstract: A countable structure A is said to be homomorphism-homogeneous if every homomorphism between its finitely generated substructures extends to an endomorphism of A. This notion was introduced by Cameron and Nesetril, serving as a homomorphism-based analogue of homogeneity, though the resulting theory differs significantly. In this talk, we will give a complete and explicit classification of homomorphism-homogeneous finite semilattices, addressing an open problem posed by Dolinka and Masulovic in 2011.