Schedule
Day 1
9h00 : Reception
9h30 - 11h00 : Mini-Course - Volker Diekert (Universität Stuttgart)
Title: The Local Divisor Approach for Finite Semigroups [slides]
Abstract. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method.
These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. Another application is a simplified proof of the famous Krohn-Rhodes decomposition theorem.
During the mini course we will focus on some of these applications.
The material for the talk is online available.
11h00 - 11h40 : Coffee break
11h40 - 12h00 : André Carvalho (University of Porto)
Title: Quantifying Brinkmann's problem: relative φ-order and φ-spectrum [blackboard talk]
Abstract. Brinkmann's problem is a well-known decidability question in group theory concerning the orbits of elements by endomorphisms. Despite having independent interest, it has relevant applications, as it is a crucial component of the machinery developed by Bogopolski-Martino-Maslakova-Ventura to prove the decidability of the conjugacy problem in cyclic extensions of groups.
In this talk, we will discuss the notions of relative φ-order and φ-spectrum, which can be thought as being a quantification of Brinkmann's problem. We will discuss these notions in the realm of finitely generated virtually free groups.
12h00 - 12h20 : Ana Catarina Monteiro (University of Lisbon)
Title: Formations and Fitting classes of semigroups, congruences and languages
Abstract. A class of finite groups closed under quotients and finitary subdirect products is said to be a formation of finite groups. Fitting classes of groups are the duals of formations and can be defined as classes of finite groups closed for subnormal subgroups and for groups generated by two normal subgroups which are in the class. As it happens for varieties of finite semigroups, a natural question that arises is whether there are corresponding concepts for congruences and languages. This question has been addressed for formations of groups, of monoids and of many-sorted algebras. Bijections between formations of those algebras, of congruences on the associated free object and of languages have been obtained. Recently, jointly with G. Gomes, we considered the analogous questions for the inverse case.
In this talk, I will present bijections between formations of inverse semigroups, of congruences on inverse semigroups and of congruences on free inverse semigroups. Then, by restricting ourselves to the idempotent separating world, we obtain bijection between certain classes named f-formations of inverse semigroups, i-formations of idempotent separating congruences and i-formations of languages on inverse semigroups.
Finally, based on these results, it is natural to look for dual concepts and results dual to those proved. We did so by extending the concept of Fitting class of groups to that of Fitting class of Clifford semigroups. Bijections between i-Fitting classes of Clifford semigroups that contain semilattices, i-Fitting classes of congruences and of languages on Clifford semigroups were obtained. Those results will be the focus of the second half of this talk.
12h20 - 12h40 : Duarte Ribeiro (NOVA University of Lisbon)
Title: Lattices of varieties of plactic-like monoids [slides]
Abstract. Plactic-like monoids, whose elements can be uniquely identified with combinatorial objects, have been the focus of intense study in recent years, in particular with regard to their equational theories. In particular, the hypoplactic, sylvester, Baxter, stalactic and taiga monoids have been studied by several authors, including the speaker, in joint work with Cain and Malheiro (NOVA Math), and using different techniques. In each of these families, it was shown that monoids of rank greater than or equal to 2 generate the same variety, their equational theories were fully characterised, and finite axiomatic bases and ranks were obtained for them. The speaker, in joint work with Thomas Aird (University of Manchester), also studied the combinatorial properties of factor monoids of the free monoid by meets and joins of left and right stalactic congruences, and showed that the varieties generated by these monoids are, respectively, the varietal join and meet of the varieties generated by the left and right stalactic monoids.
In this talk, we show results on lattices of varieties of plactic-like monoids, obtained in joint work with Thomas Aird. We fully characterise the equational theories and axiomatic bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the axiomatic ranks of their elements, obtain plactic-like congruences whose corresponding factor monoids generate varieties in the lattice, and determine which varieties are joins of the variety of commutative monoids and a finitely generated variety.
12h40 - 14h30 : Lunch break
14h30 - 18h00 : Proving Session! (Coffee & Cookies included)
20h00 : Dinner
Day 2
9h30 - 11h00 : Mini-Course - Volker Diekert (Universität Stuttgart)
Title: The Local Divisor Approach for Finite Semigroups [slides]
Abstract. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method.
These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. Another application is a simplified proof of the famous Krohn-Rhodes decomposition theorem.
During the mini course we will focus on some of these applications.
The material for the talk is online available.
11h00 - 11h40 : Coffee break
11h40 - 12h00 : Tânia Paulista (NOVA University of Lisbon)
Title: Commutative transformation semigroups with one idempotent
Abstract. We characterize, using a mixture of algebraic and combinatorial techniques, the maximum-order commutative subsemigroups in the full transformation semigroup T(X) that contain exactly one idempotent. We show that, with a few exceptions, the maximum-order commutative subsemigroups with one idempotent turn out to be precisely the maximum-order null subsemigroups of T(X) previously characterized by Cameron et al. [1].
[1] Peter J. Cameron, James East, Des FitzGerald, James D. Mitchell, Luke Pebody and Thomas Quinn-Gregson. Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups. Combinatorial Theory, 3(3), 2023.
12h00 - 12h20 : Neeraj Kumar (University of Porto)
Title: Counting numerical semigroups by their maximum primitive [slides]
Abstract. Numerical semigroups are the cofinite submonoids of the natural numbers. Every numerical semigroup has a unique set of minimal generators, also referred to as its primitives. The maximum of this set of primitives is called the maximum primitive of that numerical semigroup. For a given number n, we pose the problem of estimating the number A(n) of numerical semigroups with n as their maximum primitive. We discuss bounds for A(n), and give asymptotic results. We further show that this is intimately related to the well studied inverse Frobenius problem.
12h20 - 12h40 : Ricardo Guilherme (NOVA University of Lisbon)
Title: From plactic monoids to hypoplactic monoids [slides]
Abstract. Plactic monoids can be obtained from Kashiwara crystals and their tensor product by identifying words in the same position of isomorphic connected components of a crystal graph. Hypoplactic monoids can be obtained from a similar construction based on quasi-crystals and their quasi-tensor product. As quasi-crystals generalize the notion of crystals, we present a unified approach to these constructions by expressing them in the context of quasi-crystals. We show how to obtain a quasi-crystal monoid for the quasi-tensor product from a quasi-crystal monoid for the tensor product. We also establish a sufficient condition for a hypoplactic monoid to be a quotient of the plactic monoid associated to the same seminormal quasi-crystal.
12h40 - 14h30 : Lunch break
14h30 - 18h00 : Proving Session! (Coffee & Cookies included)