ω-terms over finite aperiodic semigroups

This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that are given by w-terms, that is that can be obtained from the free generators using only multiplication and the w-power. A necessary and sufficient condition for this property to hold turns out to be given by the conjunction of two rather simple finiteness conditions: the nonexistence of infinite anti-chains of factors and the rationality of the language of McCammond normal forms of w-terms that define factors.FCT through the Centro de Matemática da Universidade do Minho and the Centro de Matemática da Universidade do PortoEuropean Community Fund FEDE

The linear nature of pseudowords

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.The work of the first, third, and fourth authors was partly supported by the Pessoa French-Portuguese project “Separation in automata theory: algebraic, logical, and combinatorial aspects”. The work of the first three authors was also partially supported respectively by CMUP (UID/MAT/ 00144/2019), CMUC (UID/MAT/00324/2019), and CMAT (UID/MAT/ 00013/2013), which are funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The work of the fourth author was partly supported by ANR 2010 BLAN 0202 01 FREC and by the DeLTA project ANR-16-CE40-000

Iterated periodicity over finite aperiodic semigroups

This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that can be described from the free generators using only the operations of multiplication and omega-power. A necessary and sufficient condition for this property to hold turns out to be given by the conjunction of two rather simple finiteness conditions: the nonexistence of infinite anti-chains of factors and the rationality of the language of McCammond normal forms of omega-terms that define factors of the given pseudoword. The relationship between pseudowords with this property and arbitrary pseudowords is also investigated.Projeto PTDC/MAT/65481/2006 financiado em parte pelo European Community Fund FEDERPrograma AutoMathA da European Science Fundation (ESF)Projeto Egide-Grices 11113YMFundação para a Ciência e a Tecnologia (FCT

Free profinite locally idempotent and locally commutative semigroups

This paper is concerned with the structure of semigroups of implicit operations on the pseudovariety LSl of finite locally idempotent and locally commutative semigroups. We depart from a general result of Almeida and Weil to give two descriptions of these semigroups: the first in terms of infinite words, and the second in terms of infinite and bi-infinite words. We then derive some applications.North Atlantic Treaty Organization (NATO) - Gabinete de Relações Internacionais da Ciência e do Ensino Superior (INVOTAN) - 4/A/94/PO.Fundação para a Ciência e a Tecnologia (FCT) - Centro de Matemática da Universidade do Minho - Algebra, Geometry and Combinatory (AGC) - Praxis/2/2.1/MAT/63/94