Abstract: Instead of giving a conventional talk about the latest research results, I will take a look back, a tiny survey of sorts, of my research collaboration with M. V. Volkov in the last 15+ years and some results obtained in this period. The topics covered will revolve mainly around the Finite Basis Problem for
involution semigroups, including criteria for a (finite) involution semigroup to be INFB, and then various applications, such as the FBP for matrix semigroups
equipped with transposition and Moore-Penrose inverse, diagram (partition) monoids, as well as identities in power structures of groups and inverse semigroups.
Abstract: The lattice of varieties of semigroups has been the subject of intensive examination since the 1960s. In contrast, although the first paper concerning the lattice of varieties of monoids was also published back in the 1960s, this lattice has received much less attention over the years, and many questions remain open. In this talk, I will survey my recent results in this area.
Abstract: Alfred Tarski asked whether the familiar index laws for exponentiation (along with the even more familiar commutative semiring laws for addition and multiplication) are logically complete for all exponential equations of the natural numbers, while in 1982, George McNulty and Caroline Shallon asked something similar for the basic combinatorial operations (binomial coefficients, factorial and fixed base exponentiation).
These questions are remarkable for their apparent simplicity and accessibility, but there are many surprising results, striking open problems, and an uncommon mix of model theory, analysis, number theory and universal algebra, with occasional seasoning from semigroups, semirings, Heyting algebras and order theory.
We survey some of the many results in this area, focussing on a number of recent developments by Tumadhir Alsulami and the speaker.
Abstract: A semigroup endowed with a self-inverse anti-automorphism is an involution semigroup. This talk surveys some counterintuitive examples of involution semigroups and their constructions. These examples include involution semigroups satisfying properties different from those of their semigroup reducts; two or more involution semigroups sharing the same semigroup reduct but satisfying contrasting properties; and pairs of involution semigroups sharing the same semigroup reduct and generating varieties that bound an uncountable chain of varieties.
Abstract: Plactic-like monoids -- such as the hypoplactic, stalactic, taiga, sylvester, and Baxter monoids -- arise naturally in algebraic combinatorics and exhibit rich algebraic behavior in terms of the identities they satisfy and the varieties they generate. This talk presents a unified approach to understanding the equational theories of these monoids, including complete characterizations of their identities and the construction of finite bases for the corresponding varieties.
A central aspect of this work is the use of embeddings into direct products of lower-rank monoids, which allows the reduction of identity questions to the rank-two case. These structural results are complemented by faithful representations over tropical and other semirings, which shed light on the algebraic properties of these monoids and clarify their place within the lattice of semigroup varieties.
Recent developments also include the study of new plactic-like structures arising from natural congruence operations, as well as further analysis of the varieties they generate. Together, these results contribute to a broader understanding of the interplay between combinatorial constructions and semigroup identities.
These results are part of joint work with Alan Cain, Marianne Johnson, Mark Kambites, and Duarte Ribeiro
Abstract: (joint work with John Rhodes)
The theory of flows is a crucial tool in Krohn-Rhodes complexity theory.
In this talk we give the foundations of the theory of flows and give a united approach to the Presentation Lemma and its relations to flows and the Slice Theorem. We describe semigroups having a flow over the trivial semigroup and connect this to classical results in inverse semigroup theory. We reinterpret Tilson's Theorem on the complexity of small monoids in terms of flows. We conclude with examples of semigroups built from the character table of Abelian Groups that have aperiodic flows.
Abstract: Graph inverse semigroups were introduced by Ash and Hall in 1975 in connection with their work on the poset of J -classes of a semigroup. We discuss several properties of graph inverse semigroups, including a calculation of the universal groups of the local submonoids of these monoids.
The theory of Leavitt path algebras has been developed over the past 20 years. Leavitt path algebras are linear associative algebras associated with directed graphs.
They are closely related to graph inverse semigroups and graph C*-algebras. Their development is an outgrowth of the work of W.G. Leavitt who showed in the 1960’s that non-commutative rings fail to have the invariant basis number property (in a very strong sense).
With each directed graph we introduce an inverse semigroup that we refer to as the Leavitt inverse semigroup of the graph. The Leavitt inverse semigroup of a directed graph is a natural subsemigroup of the multiplicative semigroup of the corresponding Leavitt path algebra. We show that two directed graphs that have isomorphic Leavitt inverse semigroups have isomorphic Leavitt path algebras. By contracting spanning trees of certain subgraphs of a directed graph Γ to a point, we obtain a new directed
graph Γ with the property that the Leavitt inverse semigroups of Γ and ̄ Γ are strongly Morita equivalent and the Leavitt path algebras of Γ and Γ are Morita equivalent. We make use of this construction to give necessary and sufficient conditions for two graphs to have isomorphic Leavitt inverse semigroups. As a consequence, we study some structural properties of Leavitt inverse semigroups and Leavitt path algebras, and we show in particular that Leavitt path algebras are 0-retracts of certain matrix algebras.
This is joint work with David Milan and Zhengpan Wang.
Abstract: Regular ∗-semigroups occupies something of a ‘sweet spot’ between the important classes of inverse and regular semigroups, and contains many natural examples like diagram monoids. Recently, by focussing on the projection algebra structure of the semigroup, it was shown that the category RSS of regular ∗-semigroups is isomorphic to the category of so-called chained projection groupoids. This generalisation of the celebrated ESN (Ehresmann-Schein-Nambooripad) Theorem also leads to the construction of certain free (idempotent- and projection-generated) regular ∗-semigroups PG(P). Free idempotent generated semigroups has been a hot topic in the area of algebraic theory of semigroups.
In this talk, we shall discuss the construction of PG(P) using its chained projection groupoid structure, and show how they form free objects in the category RSS. The construction involves the notion of linked diamonds in projection algebras, which are the regular∗-analogues of Nambooripad’s singular squares in biordered sets. This is a recent joint work with James East, Robert Gray and Nik Ruskuc.
Abstract: Synchronizing automata model situations on which we need to regain control over a device whose current state is unknown. The concept of adversarial synchronization captures scenarios in which the device can be accessed by an adversary who strives to prevent synchronization or, if it is unavoidable, to delay it as long as possible.
While synchronizability of an automaton is a property of its transition monoid, synchronizability in the presence of an adversary is not. Nevertheless, certain properties of transition monoids turn out to be essential for adversarial synchronization. Among these properties are those familiar from the theory of finite monoid varieties, such as the property that regular D-classes are subsemigroups. We show that if the transition monoid of a synchronizing automaton possesses this property, then the automaton can be synchronized even in the face of adversarial opposition.