We develop an algebraic theory for languages of data words. We prove that, under certain conditions, a language of data words is definable in first-order logic if and only if its syntactic monoid is aperiodic. 1 Introduction This paper is an attempt to combine two fields. The first field is the algebraic theory of regular languages. In this theory, a regular language is represented by its synta...

Let M be a submonoid of the free monoid A∗, and let X ⊆ M be a variable length code (for short a code). X is weakly M-complete if any word in M is a factor of some word in X∗ [J. Néraud, C. Selmi, Free monoid theory: maximality and completeness in arbitrary submonoids, Internat. J. Algorithms Comput. 13(5) (2003) 507–516]. Given a code X ⊆ M , we are interested in the construction of a weakly M...

Let Σ = X ∪ X = {x1, x2, ..., xm, x −1 1 , x 2 , ..., x m } and let G be a group with set of generators Σ. Let L(G) = {ω ∈ Σ∗ | ω ≡ e (mod G)} ⊆ Σ∗ be the group language representing G, where Σ∗ is a free monoid over Σ and e is the identity in G. The problem of determining whether a context-free language is subset of a group language is discussed. Polynomial algorithms are presented for testing...

A language is torsion (resp. bounded torsion, aperiodic, bounded aperiodic) if its syntactic monoid is torsion (resp. bounded torsion, aperiodic, bounded aperiodic). We generalize the regular language theorems of Kleene, Sch utzenberger and Straubing to describe the classes of torsion, bounded torsion, aperiodic and bounded aperiodic languages. These descriptions involve taking limits of sequen...

A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is...

Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k > 2q 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q. r = q (mod 2)). It is also proved that every simple graph with maximal degree A > 2q 2 and average degree d > ((2q 2)/(2q l))(A + 1), where q is a prime power, co...

We develop a counterpart to Garside’s analysis of the braid monoid B n relevant for the monoid MLD that describes the geometry of the left self-distributivity identity. The monoid MLD extends B ∞, of which it shares many properties, with the exception that it is not a direct limit of finitely generated monoids. By introducing a convenient local version of the fundamental elements ∆, we prove th...