On the logical definability of some rational trace languages

We compare two notions of coding on traces, called coding and strong coding, in relation with the decision problem of the existence of a coding between two given trace monoids. We positively solve this problem if the first monoid is either a direct product of free monoids, or a free product of free commutative monoids. We show how the situation differs in the case of strong codings.

Easy multiplications I. The realm of Kleene's theorem

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.KeywordsWord ProblemInverse SemigroupCayley GraphDependence GraphDependence RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On regular trace languages

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f, where e and f are both idempotents in the free inverse monoid generated by Γ, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, it is shown that the Cayley-graph of the free inverse monoid has an undecidable monadic second-order theory.

The word problem for inverse monoids generated by a set $Gamma$ subject to relations of the form $e=f$, where $e$ and $f$ are both idempotents in the free inverse monoid generated by $Gamma$, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node $x$ to a node $y$ that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, some results on free partially commutative inverse monoids are presented.

Set Constraints in Some Equational Theories

Variétés et fonctions rationnelles

Inverse monoids: Decidability and complexity of algebraic questions

E-unitary inverse monoids and the Cayley graph of a group presentation

A Restricted Second Order Logic for Finite Structures

Noetherian Semigroup Rings with Several Objects

A totally ordered monoid—or tomonoid , for short—is a commutative semigroup with identity S equipped with a total order \les that is translation invariant, i.e. , that satisfies: \forall x, y, z∈ S, x\les y\;\Rightarrow \; x+z \les y+z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice \Z n that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell.

The Hopf algebra of skew shapes, torsion sheaves on [formula omitted], and ideals in Hall algebras of monoid representations

The regular languages in the free monoid generated by a finite alphabet A are exactly the languages that are the models of some sentence of the second-order monadic logic of one successor and a unary predicate for each letter. For trace monoids the n