# Xarrada «Les màquines del llenguatge» d’Enric Cosme Llópez a la Setmana Cultural de la Facultat de Ciències Matemàtiques, 18/03/2021, 12.00

 Mar ’21 18 11:00

Enric Cosme Llópez presentarà la xarrada «Les màquines del llenguatge» el dia 18/03/2021 a les 12.00 dins del programa d’activitats de la Setmana Cultural de Matemàtiques 2021.

https://www.uv.es/coslloen/Arxiu/Fitxers/Beamers/UV21.pdf

# Paper «Congruence-based proofs of the recognizability theorems for free many-sorted algebras» published in J. Logic Comput.

The following paper has been published:
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J. Climent Vidal and E. Cosme Llópez.
J. Logic Comput., 30(2):561–633, 2020.

doi:10.1093/logcom/exz032

Abstract

We generalize several recognizability theorems for free single-sorted algebras to free many-sorted algebras and provide, in a uniform way and without using either regular tree grammars or tree automata, purely algebraic proofs of them based on congruences.

Keywords: free many-sorted algebra, recognizability, congruence

# Paper «A characterization of the n-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem» in Quaest. Math.

The following paper has been published:
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J. Climent Vidal, E. Cosme Llópez.
A characterization of the $n$-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem.
Quaest. Math., 42:1427-1444, 2019.

Abstract:

A theorem of single-sorted algebra states that, for a closure space (A, J ) and a natural number n, the closure operator J on the set A is n-ary if and only if there exists a single-sorted signature Σ and a Σ-algebra A such that every operation of A is of an arity ≤ n and J = SgA, where SgA is the subalgebra generating operator on A determined by A. On the other hand, a theorem of Tarski asserts that if J is an n-ary closure operator on a set A with n ≥ 2, then, for every i, j ∈ IrB(A, J ), where IrB(A, J ) is the set of all natural numbers which have the property of being the cardinality of an irredundant basis ( minimal generating set) of A with respect to J , if i < j and {i + 1, . . . , j − 1} ∩ IrB(A, J ) = Ø, then j − i ≤ n − 1. In this article we state and prove the many-sorted counterparts of the above theorems. But, we remark, regarding the first one under an additional condition: the uniformity of the many-sorted closure operator.

2020 Mathematics Subject Classification: 06A15, 54A05.

Keywords: S-sorted set, delta of Kronecker, support of an S-sorted set, n-ary many-sorted closure operator, uniform many-sorted closure operator, irredundant basis with respect to a many-sorted closure operator.

# Paper «Eilenberg theorems for many-sorted formations» published in Houston J. Math.

The following paper has been published:
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El següent article ha sigut publicat:

Juan Climent Vidal, Enric Cosme Llópez.
Eilenberg theorems for many-sorted formations.
Houston J. Math., 45(2):321-369, 2019.

Abstract:

A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts S and a fixed S-sorted signature Σ, the concepts of formation of congruences with respect to Σ and of formation of Σ-algebras, we prove that the algebraic lattices of all Σ-congruence formations and of all Σ-algebra formations are isomorphic, which is an Eilenberg’s type theorem. Moreover, under a suitable condition on the free Σ-algebras and after defining the concepts of formation of congruences of finite index with respect to Σ, of formation of finite Σ-algebras, and of formation of regular languages with respect to Σ, we prove that the algebraic lattices of all Σ-finite index congruence formations, of all Σ-finite algebra formations, and of all Σ-regular language formations are isomorphic, which is also an Eilenberg’s type theorem.

2020 Mathematics Subject Classification: 08A68, 08A70, 68Q70

Keywords: Many-sorted algebra, support, many-sorted congruence, sat-
uration, cogenerated congruence, many-sorted (finite) algebra formation, many-sorted
(finite index) congruence formation, many-sorted regular language formation.

# Paper «Some contributions to the theory of transformation monoids» published in J. Algebra

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A. Ballester-Bolinches, E. Cosme-Llópez, P. Jiménez-Seral.

Some contributions to the theory of transformation monoids

J. Algebra., 522:31-60, 2019

doi:10.1016/j.jalgebra.2018.12.005

Abstract

The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg (2010)  have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture.

2010 Mathematics Subject Classification: 16W22, 20M30

Keywords: monoid theory, monoid action, transitive, faithful, primitive

# Paper «When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?» published in Log. J. IGPL

The following paper has been published:
El siguiente artículo ha sido publicado:
El següent article ha sigut publicat:

J. Climent Vidal, E. Cosme Llópez.
When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?
Log. J. IGPL, 26(4):381-407, 2018.

Abstract:

For a set of sorts S and an S-sorted signature Σ we prove that a profinite Σ-algebra, i.e. a projective limit of a projective system of finite Σ-algebras, is a retract of an ultraproduct of finite Σ-algebras if the family consisting of the finite Σ-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction.

2020 Mathematics Subject Classification: 03C20, 08A68, 18A30.

Keywords: support of a many-sorted set, family of many-sorted algebras with constant support, profinite, retract, projective limit, inductive limit, ultraproduct.

# Paper «K_4-free graphs as a free algebra» published in LIPIcs. Leibniz Int. Proc. Inform., 83

The following paper has been published:
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E. Cosme Llópez, Damien Pous.
K_4-free graphs as a free algebra.
42nd International Symposium on Mathematical Foundations of Computer Science, Art. No. 76, 14 pp., LIPIcs. Leibniz Int. Proc. Inform., 83, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2017.

Abstract:

Graphs of treewidth at most two are the ones excluding the clique with four vertices as a minor. Equivalently, they are the graphs whose biconnected components are series-parallel.

We turn those graphs into a free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting them: in addition to series and parallel compositions, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equational presentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph.’

2020 Mathematics Subject Classification: 68R10, 68Q45.

Keywords: universal Algebra, graph theory, axiomatisation, graph minors.

# Paper “On subgroup functors of finite soluble groups” published in Sci. China Math.

The following paper has been published:

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Adolfo Ballester-Bolinches, Enric Cosme-Llópez, Sergey Fedorovich Kamornikov

On subgroup functors of finite soluble groups.

Sci. China Math., 60(3):439–448, 2017

https://doi.org/10.1007/s11425-015-0330-9

Abstract

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups. We prove that they form a complemented and non-modular lattice containing two relevant sublattices. This is the answer to a question (Question 1.2.12) proposed by Skiba (1997) in the finite soluble universe.

2010 Mathematics subject classification: 20D10; 20D30

Keywords: finite group; soluble group; lattices of subgroups; subgroup functors; formations

# Paper “Group extensions and graphs” published in Expo. Math.

The following paper has been published

El siguiente paper ha sido publicado

El següent paper ha sigut publicat

A. Ballester-Bolinches, E. Cosme-Llópez, and R. Esteban-Romero.

Group extensions and graphs.

Expo. Math., 34(3):327–334, 2016

https://doi.org/10.1016/j.exmath.2015.07.005

Abstract

A classical result of Gaschütz affirms that given a finite A-generated group G and a prime p , there exists a group G^# and an epimorphism φ:G→G^# whose kernel is an elementary abelian p-group which is universal among all groups satisfying this property. This Gaschütz universal extension has also been described in the mathematical literature with the help of the Cayley graph. We give an elementary and self-contained proof of the fact that this description corresponds to the Gaschütz universal extension. Our proof depends on another elementary proof of the Nielsen–Schreier theorem, which states that a subgroup of a free group is free.

2010 Mathematical Subject Classification: Primary 20F65; Secondary 05C25, 20D20, 20E22, 20F05, 20F10

Keywords: Group, Group extension, Graph

# Actualització: Defensa tesi doctoral Enric Cosme 22/10/2015, 12.00, sala graus Química

 Oct ’15 22 12:00 El proper dijous 22 d’octubre de 2015, a les 12.00, a la sala de graus de la Facultat de Química de la Universitat de València (carrer de Vicent Andrés Estellés, s/n, Burjassot), es procedirà a la defensa de la tesi doctoral d’Enric Cosme i Llópez, dirigida per Adolfo Ballester Bolinches i Jean-Éric Pin i amb títol

«Some contributions to the algebraic theory of automata».

Esteu convidats a assistir a aquest acte.

Actualització 16/10/2015: La defensa es durà a terme a la sala de graus de la Facultat de Química i no a la de Farmàcia, com inicialment estava previst.