Free submonoids and minimal ω-generators of Rω

Let A be an alphabet and let R be a language in A+. An (¿-generator of -R" is a language G such that G" = R". The language Stab(-R") = {u G A* : ttiZ" Ç R"} is a submonoid of A*. We give results concerning the wgenerators for the case when Stab(Ru ) is a free submonoid which are not available in the general case. In particular, we prove that every ((»-generator of 22" contains at least one minimal w-generator of R". Furthermore these minimal w-generators are codes. We also characterize the w-languagea having only finite languages as minimal u-generators. Finally, we characterize the w- languages »-generated by finite prefix codes

Markov chains, R\mathscr R-trivial monoids and representation theory

We develop a general theory of Markov chains realizable as random walks on $\mathscr R$-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via M\"obius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.Comment: Dedicated to Stuart Margolis on the occasion of his sixtieth birthday; 71 pages; final version to appear in IJA

Reduced languages as omega-generators

International audienceWe consider the following decision problem: “Is a rational omega-language generated by a code ?” Since 1994, the codes admit a characterization in terms of infinite words. We derive from this result the definition of a new class of languages, the reduced languages. A code is a reduced language but the converse does not hold. The idea is to “reduce" easy-to-obtain minimal omega-generators in order to obtain codes as omega-generators

Arithmetic-arboreal residue structures induced by Prufer extensions : An axiomatic approach

We present an axiomatic framework for the residue structures induced by Prufer extensions with a stress upon the intimate connection between their arithmetic and arboreal theoretic properties. The main result of the paper provides an adjunction relationship between two naturally defined functors relating Prufer extensions and superrigid directed commutative regular quasi-semirings.Comment: 56 page

A Theory of Transformation Monoids: Combinatorics and Representation Theory

The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular, we obtain a generalization of a lemma of P. Neumann, from the theory of synchronizing groups, concerning the partition associated to a transformation of minimal rank

Graph expansions of semigroups

We construct a graph expansion from a semigroup with a given generating set, thereby generalizing the graph expansion for groups introduced by Margolis and Meakin. We then describe structural properties of this expansion. The semigroup graph expansion is itself a semigroup and there is a map onto the original semigroup. This construction preserves many features of the original semigroup including the presence of idempotent/periodic elements, maximal group images (if the initial semigroup is E-dense), finiteness, and finite subgroup structure. We provide necessary and sufficient graphical criteria to determine if elements are idempotent, regular, periodic, or related by Green’s relations. We also examine the relationship between the semigroup graph expansion and other expansions, namely the Birget and Rhodes right prefix expansion and the monoid graph expansion. If S is a -generated semigroup, its graph expansion is generally not -generated. For this reason, we introduce a second construction, the path expansion of a semigroup. We show that it is a -generated subsemigroup of the semigroup graph expansion. The semigroup path expansion possesses most of the properties of the semigroup graph expansion. Additionally, we show that the path expansion construction plays an analogous role with respect to the right prefix expansion of semigroups that the group graph expansion plays with respect to the right prefix expansion of groups

New Approach to Arakelov Geometry

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with "exotic" objects such as F_1 ("field with one element"), Z_\infty ("real integers"), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models.Comment: 568 pages, with hyperlink

New approach to Arakelov geometry

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with "exotic" objects such as F_1 ("field with one element"), Z_\infty ("real integers"), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models