Partially Commutative Inverse Monoids

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.

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 presented by one idempotent relator

Inverse monoids: Decidability and complexity of algebraic questions

Trace monoids are obtained from free monoids by defining a subset I of pairs of letters that are allowed to commute. Most of the work of this theory is an attempt to relate the properties of these monoids to the properties of I. Following the work initiated by Buchi we show that when I is an equivalence relation (the trace monoid is then a free product of free commutative monoids) it is possible to define a second order logic whose models are the traces viewed as dependence graphs and which characterizes exactly the sets of traces that are rational. This logic essentially utilizes a predicate based on the ordering defined by the dependence graph and a predicate related to a restricted use of the comparison of cardinality.

On a Class of Lattice Ordered Inverse Semigroups

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

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.

On regular trace languages

Set Constraints in Some Equational Theories

On one-relator inverse monoids and one-relator groups

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.

Word problems over traces which are solvable in linear time

We study some aspects of Schein’s theory of cosets for closed inverse subsemigroups of inverse semigroups. We establish an index formula for chains of subsemigroups, and an analogue of M. Hall’s Theorem on the number of cosets of a fixed finite index. We then investigate the relationships between the following properties of a closed inverse submonoid of an inverse monoid: having finite index; being a recognizable subset; being a rational subset; being finitely generated (as a closed inverse submonoid). A remarkable result of Margolis and Meakin shows that these properties are equivalent for a closed inverse submonoid of a free inverse monoid.

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-, as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.