All semigroups and monoids except free monoids and free groups are assumed finite in this paper. A monoid M divides a monoid N if M is a quotient of a submonoid of N . The set of idempotents of a monoid M is denoted E(M). The word “variety” will be used for pseudo-variety — that is, a collection of finite monoids closed under division and finite direct product. A block group is a monoid such th...

The aim of this paper is sketch a theory of divisibility and factorisation in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely topologised topological monoids. In particular, we define the topological factorisation monoid, a generalisation of the factorisation monoid for algebraic monoids, and...

We define compact automata and show that every language has a unique minimal compact automaton. We also define recognition of languages by compact left semitopological monoids and construct the analogue of the syntactic monoid in this context. For rational languages this reduces to the usual theory of finite automata and finite monoids.

A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are satisfied [9]. Here we investigate a wider class of groups of fractions, called thin, which are those associated with monoids where minimal common multiples exist, b...

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, V the corresponding variety of languages and V̂ the smallest variety containing V and the bideterministic products of two languages of ...

We study the language-theoretic aspects of word problem, in sense Duncan & Gilman, free products semigroups and monoids. First, we provide algebraic tools for studying classes languages known as super-AFLs, which generalise e.g. context-free or indexed languages. When C is a super-AFL closed under reversal, prove that semigroup (monoid) product two (resp. monoids) with problem also has C. This ...

Overlapping tile automata and the associated notion of recognizability by means of (adequate) premorphisms in finite ordered monoids have recently been defined for coping with the collapse of classical recognizability in inverse monoids. In this paper, we investigate more in depth the associated algebraic tools that allows for a better understanding of the underlying mathematical theory. In par...

In this paper we study some combinatorial aspects of the singular Artin monoids. Firstly, we show that a singular Artin monoid SA can be presented as a semidirect product of a graph monoid with its associated Artin group A. Such a decomposition implies that a singular Artin monoid embeds in a group. Secondly, we give a solution to the word problem for the FC type singular Artin monoids. Afterwa...

We show there is a unique maximal positive variety of languages which does not contain the language (ab)∗. This variety is the unique maximal positive variety satisfying the two following conditions: it is strictly included in the class of rational languages and is closed under the shuffle operation. It is also the largest proper positive variety closed under length preserving morphisms. The or...

We investigate weighted automata with discounting and their behaviours over semirings and finitely generated graded monoids. We characterize the discounted behaviours of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product. This extends a classical result of Kleene-Schützenberger. Here we show that the very special case of Schützenberger’s res...