Errata to "Formations of Monoids, Congruences, and Formal Languages"

[1] A. Ballester-Bolinches, E. Cosme-Llópez, R. Esteban Romero, and J.J.M.M. Rutten. Formations of monoids, congruences, and formal languages. Scientific Annals of Computer Science, 25(2):171–209, 2015. doi:10.7561/SACS.2015.2.171

Group extensions and graphs

NOTICE: this is the author’s version of a work that was accepted for publication in Expositiones Mathematicae. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Expositiones Mathematicae, [Volume 34, Issue 3, 2016, Pages 327-334] DOI#10.1016/j.exmath.2015.07.005¨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 phi: 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.This work has been supported by the grant MTM-2014-54707-C3-1-P of the Ministerio de Economia y Competitividad (Spain). The first author is also supported by Project No. 11271085 from the National Natural Science Foundation of China. The second author is supported by the predoctoral grant AP2010-2764 (Programa FPU, Ministerio de Educacion, Spain).Ballester Bolinches, A.; Cosme-Llópez, E.; Esteban Romero, R. (2016). Group extensions and graphs. Expositiones Mathematicae. 34(3):327-334. https://doi.org/10.1016/j.exmath.2015.07.005S32733434

A description based on languages of the final non-deterministic automaton

The study of the behaviour of non-deterministic automata has traditionally focused on the languages which can be associated to the different states. Under this interpretation, the different branches that can be taken at every step are ignored. However, we can also take into account the different decisions which can be made at every state, that is, the branches that can be taken, and these decisions might change the possible future behaviour. In this case, the behaviour of the automata can be described with the help of the concept of bisimilarity. This is the kind of description that is usually obtained when the automata are regarded as labelled transition systems or coalgebras. Contrarily to what happens with deterministic automata, it is not possible to describe the behaviour up to bisimilarity of states of a non-deterministic automaton by considering just the languages associated to them. In this paper we present a description of a final object for the category of non-deterministic automata, regarded as labelled transition systems, with the help of some structures defined in terms of languages. As a consequence, we obtain a characterisation of bisimilarity of states of automata in terms of languages and a method to minimise non-deterministic automata with respect to bisimilarity of states. This confirms that languages can be considered as the natural objects to describe the behaviour of automata

Allegories: decidability and graph homomorphisms

International audienceAllegories were introduced by Freyd and Scedrov; they form a fragment of Tarski's calculus of relations. We show that their equational theory is decidable by characterising it in terms of a specific class of graph homomorphisms. We actually do so for an extension of allegories which we prove to be conservative, namely allegories with top. This generalisation makes it possible to exploit the correspondence between terms and K4-free graphs, for which isomorphism was known to be finitely axiomatisable