March 3rd
9h00 : Reception
9h30 - 10h10 : Mini-Course - Gracinda Gomes
Title: From the structure of an inverse semigroup to the study of associated classes
Abstract. In this course we shall look at inverse semigroups.
First, we look at an inverse semigroup as an object whose structure is deeply associated to semilattices and groups. We recall the celebrated theorems of McAlister [8] and O’Carroll [9].
10h10 - 10h35 : José Carlos Costa (U. Minho)
Title: On ω-identities over finite aperiodic block groups
Abstract. An ω-term is a formal expression obtained from the letters of an alphabet using the operations of multiplication and ω-power.
We will begin by briefly reviewing the word problem for ω-terms over the pseudovariety A∩ECom, of all finite aperiodic semigroups whose idempotents commute. This problem is reduced to considering only ω-terms of rank 1. The decidability of the problem, proven in collaboration with Mário Branco, is obtained by associating to each of these ω-terms α a certain automaton that, after a reduction process, produces an automaton that characterizes the ω-term α over the pseudovariety A∩ECom.
Next, we will talk about the ω-word problem over the pseudovariety A∩BG, of all finite aperiodic block groups. Recall that a block group is a semigroup whose elements have at most one inverse. A block group can also be defined as a semigroup in which each R-class and each L-class have at most one idempotent. Therefore, A∩ECom is a subpseudovariety of A∩BG. In the case A∩BG, the ω-word problem reduces to consider only ω-terms of ranks 1 and 2. The proof of the decidability of this problem is an ongoing joint work with Conceição Nogueira and M. Lurdes Teixeira, and also uses automata associated to these ω-terms in order to characterize them over the pseudovariety A∩BG.
10h35 - 11h00 : Tânia Paulista (U. Nova de Lisboa)
Title: Classes of semigroups and their commuting graphs
Abstract. The commuting graph of a finite non-commutative semigroup S is a simple graph whose vertices are the non-central elements of S, and where two distinct vertices x,y ∈ S\Z(S) are adjacent if and only if xy = yx. In 2011 Araújo, Kinyon and Konieczny [AKK11] showed that for each integer n > 2 there exists a (finite non-commutative) semigroup whose commuting graph has diameter n. In 2022 Cutolo [Cut22] proved the same result for groups. In 2016 Bauer and Greenfeld [BG16] and, independently, Giudici and Kuzma [GK16] characterized the graphs that arise as commuting graphs of semigroups, which led to the proof that for each integer n > 1 (respectively, n > 3) there is a (finite non-commutative) semigroup whose commuting graph has clique number/chromatic number (respectively, girth) equal to n. In this talk we will see if, by restricting our options to a specific class of semigroups, the possible values for the properties of a commuting graph are still the same. For instance, if we consider the class of completely simple semigroups, is it still true that for each integer n > 3 there is a completely simple semigroup whose girth is n? We will show what the possible values are for some of the properties of a commuting graph when we consider the classes of completely simple, completely 0-simple and inverse semigroups.
[AKK11] João Araújo, Michael Kinyon, and Janusz Konieczny. Minimal paths in the commuting graphs of semigroups. European Journal of Combinatorics, 32(2):178–197, 2011.
[BG16] Tomer Bauer and Be’eri Greenfeld. Commuting graphs of boundedly generated semigroups. European Journal of Combinatorics, 56:40–45, 2016.
[Cut22] Giovanni Cutolo. On a construction by Giudici and Parker on commuting graphs of groups. Journal of Combinatorial Theory, Series A, 192, 2022. Article no. 105666.
[GK16] Michael Giudici and Bojan Kuzma. Realizability problem for commuting graphs. Journal of the Australian Mathematical Society, 101(3):335–355, 2016
11h00 - 11h30 : Coffee break
11h30 - 12h10 : Mini-Course - Gracinda Gomes
Title: From the structure of an inverse semigroup to the study of associated classes
Abstract. We consider congruences on inverse semigroups [10], in particular the least group congruence and the idempotent separating ones [3]. These last lead to the concept of i-normal subsemigroup. Varieties, i-formations and i-Fitting classes of finite inverse semigroups are defined, and we compare examples of these classes with associated classes of finite groups [6, 5, 2].
12h10 - 12h35 : André Carvalho (U. Porto)
Title: Linguistic subsets of groups and epiC groups
Abstract. We will discuss general properties of subsets of groups defined by natural language theoretic conditions, unifying previous approaches by several authors in a general framework. As an application, we prove that the class of epi-C groups is closed under taking finite index subgroups if C is any full semi-AFL, answering a recent question posed by Al Kohli, Bleak & Elliot. This is joint work with Carl-Fredrik Nyberg-Brodda (KIAS).
12h35 - 14h30 : Lunch break
14h30 - 18h00 : Proving Session
(Coffee & Cookies included)
20h00 : Journey Dinner
March 4th
9h30 - 10h10 : Mini-Course - Gracinda Gomes
Title: From the structure of an inverse semigroup to the study of associated classes
Abstract. After recalling Thrien’s theory of varieties of regular languages [11] by means of varieties of congruences of finite index on free monoids, which inspired the analogue for formations by Ballester-Bolinches et al [1], we look at i-formations of congruences and of languages on inverse semigroups [5], and show that they are in bijection with i-formations of inverse semigroups that contain all those that are fundamental [7].
10h10 - 10h35 : Ambroise Grau (U. Nova de Lisboa)
Title: Cancellativity properties in the stylic monoid
Abstract. Given an finite alphabet A of size n, the stylic monoid of rank n can be described as a finite quotient of the plactic monoid of rank n and whose presentation is given by the Knuth relations together with the relations a^2≡ a for all a ∈ A. In a similar fashion to the plactic monoid, elements of the stylic monoid are in bijection with the so-called N-tableaux (special semistandard Young tableaux in which each row is strictly increasing and contained in the previous one) under an adapted version of Schensted algorithm for left and right insertion.
In this talk, we will present some work in progress, looking at understanding when two elements u and v of the stylic monoid are L*-related, where L* is the extended Green’s relation which relate two words u and v whenever ux ≡ uy if and only if vx ≡ vy for all words x and y. In other words, we want our words u and v to have the same cancellativity property on the right. Unexpectedly, what is important to describe this relation is to consider the values which are not present in the respective tableaux of the considered elements, rather than looking at the values which are there. We will give some examples as well as some key ideas of why this holds.
This is a joint work with Duarte Ribeiro.
10h35 - 11h00 : Célia Borlido (U. Coimbra)
Title: On monoids with additional structure
Abstract. Very often one is led to consider monoids equipped with (compatible) additional structure, topological in nature. Classical examples are partially ordered and topological monoids. In this talk, I will present a framework suited to a unified treatment of all such enriched monoids and state some related open questions.
11h00 - 11h30 : Coffee break
11h30 - 12h10 : Mini-Course - Gracinda Gomes
Title: From the structure of an inverse semigroup to the study of associated classes
Abstract. To conclude, we discuss the concepts of i-Fitting classes of congruences and of languages in a way that we obtain bijective correspondences with classes of inverse semigroups that are both i-formations and i-Fitting [4].
[1] A. Ballester-Bolinches, J.'Eric Pin, and X. Soler-Escrivà. Formations of finite monoids and formal languages: Eilenberg’s variety theorem revisited. Forum Mathematicum, 26(6):1737–1761, 2014.
[2] K. Doerk and T. O. Hawkes. Finite soluble groups, volume 4. Walter de Gruyter, 2011.
[3] G. M. S. Gomes. Orthodox congruences on regular semigroups. Semigroup Forum, 37:149–166, 1998.
[4] G. M. S. Gomes and A.-C. C. Monteiro. i-fitting-formations on inverse semigroups. submitted.
[5] G. M. S. Gomes and A.-C. C. Monteiro. Formations and i-Fitting classes of inverse semigroups, congruences and languages. Semigroup Forum, 109:87–115, 2024.
[6] G. M. S. Gomes and I. Nobre. On formations of inverse semigroups. Semigroup Forum, 105:217–243, 2022.
[7] J. M. Howie. Fundamentals of Semigroup Theory. Clarendon Press, 1999.
[8] D. Mcalister. Groups, semilattices and inverse semigroups. Transactions of the American Mathematical Society, 192:227–244, 01 1974.
[9] L. O’Carroll. Embedding theorems for proper inverse semigroups. Journal of Algebra, 42:26–40, 1976.
[10] M. Petrich. Inverse Semigroups. Wiley-Interscience, 1984.
[11] D. Thérien. Classification of Regular Languages by Congruences. PhD thesis, 1980. AAI0533628.
This work was developed within the activities of Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa and Centro de Matemática Computacional e Estocástica, CEMAT, within the projects UIDB/04621/2020 and UIDP/04621/2020, DOI 10.54499/UIDB/04621/2020, financed by Fundação para a Ciência e a Tecnologia, FCT.
12h10 - 12h35 : Ana Catarina Monteiro (U. Nova de Lisboa)
Title: Product of formations and Fitting classes on groups and some generalisations
Abstract. In this talk, we will focus on formations and Fitting classes of groups. A formation of groups is a class of groups closed under quotients and subdirect products of finite families, while a Fitting class of groups is a class of groups closed under normal subgroups and products of two normal subgroups belonging to the class.
In [1], different definitions of the product of classes of groups have been presented, and studied, particularly regarding the preservation of properties as being a formation or a Fitting class.
Furthermore, research has been conducted with the aim of extending these concepts to congruences and languages on groups, leading to the introduction and study of the concepts of formations and Fitting classes of congruences and languages (see, for example, [2], [3], [4]).
This naturally raises the question of what would be a suitable definition for the product of formations or Fitting classes of congruences and languages. In this talk, we will explore this question for groups and discuss possible generalizations to other algebraic structures, such as Clifford semigroups and inverse semigroups.
This is joint work with Gracinda Gomes.
[1] K. Doerk and T. Hawkes (1992) Finite Soluble Groups, Walter de Gruyter.
[2] G. Gomes and A.-C. Monteiro (2024) Formations and i-Fitting classes of inverse semigroups, congruences and languages, Semigroup Forum.
[3] Ballester-Bolinches, J.-E. Pin, and X. Soler-Escrivà (2015) Languages associated with saturated formations of groups, Forum Mathematicum, 27(3):1471–1505.
[4] A. Ballester-Bolinches, E. Cosme Llópez, R. Esteban-Romero and J. Rutten (2015) Formations of monoids, congruences, and formal languages. Scientific Annals of Computer Science, 25:171–209.
12h35 - 14h30 : Lunch break
14h30 - 16h30 : Proving Session
16h30 - 17h00 : Farewell
(Coffee & Cookies included)