McCammond's normal forms for free aperiodic semigroups revisited

This paper revisits the solution of the word problem for omega-terms interpreted over finite aperiodic semigroups, obtained by J. McCammond. The original proof of correctness of McCammond's algorithm, based on normal forms for such terms, uses McCammond's solution of the word problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness proof of McCammond's algorithm, based on properties of certain regular languages associated with the normal forms. This method leads to new applications.Pessoa French-Portuguese project Egide-Grices 11113YMEuropean Regional Development Fund, through the programme COMPETEEuropean Community Fund FEDERANR 2010 BLAN 0202 01 FRE

Canonical forms for free k-semigroups

The implicit signature kappa consists of the multiplication and the (omega-1)-power. We describe a procedure to transform each kappa-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond's normal form algorithm for omega-termsninterpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Sigma of pseudoidentities. As an application, we deduce that the variety of kappa-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Sigma and that the free kappa-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each omega-term has a unique omega-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, representatives for omega-terms interpreted over that pseudovariety.European Regional Development Fund, through the programme COMPETEFundação para a Ciência e a Tecnologia (FCT), under the project PEst-C/MAT/UI0013/2011

The omega-inequality problem for concatenation hierarchies of star-free languages

The problem considered in this paper is whether an inequality of omega-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing-Th\'erien hierarchy

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

Some structural properties of the free profinite aperiodic semigroup

Profinite semigroups provide powerful tools to understand properties of classes of regular languages. Until very recently however, little was known on the structure of "large" relatively free profinite semi- groups. In this paper, we present new results obtained for the class of all finite aperiodic (that is, group-free) semigroups. Given a finite alphabet X, we focus on the following problems: (1) the word problem for ω-terms on X evaluated on the free pro-aperiodic semigroup, and (2) the computation of closures of regular languages in the ω-subsemigroup of the free pro-aperiodic semigroup generated by X.FCT through the Centro de Matemática da Universidade do Minho and the Centro de Matemática da Universidade do PortoEuropean Community Fund FEDERESF programme “Automata: from Mathematics to Applications (AutoMathA)”Pessoa Portuguese-French project Egide-Grices 11113Y

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201