On periodic points of free inverse monoid endomorphisms

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.Comment: 18 page

On the insertion of n-powers

In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1\le x^n$ in which both sides are regular elements with respect to the upper bound

Adding modular predicates to first-order fragments

We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with modular predicates. Our approach aims toward the most generic statements that we could achieve, which successfully covers the quantifier alternation hierarchy of first order logic and some of its fragments. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its two-variable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order and the successor. For two-variable fragments we also treat the case of the signature containing only the order and modular predicates.Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragmentwhile in the case of the first order logic the question remains open for levels greater than two.The main ingredients of the proofs are syntactic transformations of first order formulas as well as the algebraic framework of finite categories

Commutative positive varieties of languages

We study the commutative positive varieties of languages closed under various operations: shuffle, renaming and product over one-letter alphabets

The semaphore codes attached to a Turing machine via resets and their various limits

We introduce semaphore codes associated to a Turing machine via resets. Semaphore codes provide an approximation theory for resets. In this paper we generalize the set-up of our previous paper "Random walks on semaphore codes and delay de Bruijn semigroups" to the infinite case by taking the profinite limit of $k$-resets to obtain $(-\omega)$-resets. We mention how this opens new avenues to attack the P versus NP problem.Comment: 28 pages; Sections 3-6 appeared in a previous version of arXiv:1509.03383 as Sections 9-12 (the split of the previous paper was suggested by the journal); Sections 1-2 and 7 are ne

Green's J-order and the rank of tropical matrices

We study Green's J-order and J-equivalence for the semigroup of all n-by-n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between tropical convex sets. We establish connections between the J-order, isometries of tropical convex sets, and various notions of rank for tropical matrices. We also study the relationship between the relations J and D; Izhakian and Margolis have observed that $D \neq J$ for the semigroup of all 3-by-3 matrices over the tropical semiring with $-\infty$, but in contrast, we show that $D = J$ for all full matrix semigroups over the finitary tropical semiring.Comment: 21 pages, exposition improve