This paper is concerned with the many deep and far reaching consequences of Ash’s positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite mono...

The increasing competition in global markets forces companies to optimise their manufacturing and logistic processes. Tied supply chains, just in time production are means to reduce costs while shortening the needed “time to market”. Customers are attracted not only by the product prize, but properties like quality and safety. In case of higher prized goods the ability to offer individualised p...

We study monoids generated by Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C listed by Sekiguchi (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaußian and, hence, are neither Garside nor Artin (Theorem 2). However, we introduce, similarly to Artin monoids,...

Given a variety of finite monoids V, a subset of a monoid is a V-subset if its syntactic monoid belongs to V. A function between two monoids is V-preserving if it preservesV-subsets under preimages and it is hereditary V-preserving if it is W-preserving for every subvariety W of V. The aim of this paper is to study hereditary V-preserving functions when V is one of the following varieties of fi...

We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the partition ...

We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating finite case by various structural observations counterexamples, also highlighting role played positive orderability. Moreover, we construct a non-left-orderable, positively right-orderable submonoid of free product cyclic group order 7 with on 4 generators. Any extension that mon...

We consider a compressed form of the word problem for finitely presented monoids, where the input consists of two compressed representations of words over the generators of a monoid M, and we ask whether these two words represent the same monoid element of M. Words are compressed using straight-line programs, i.e., context-free grammars that generate exactly one word. For several classes of fin...

This paper presents a survey of recent results in the theory of rational sets in arbitrary monoids. Main topics considered here are : the so-called Kleene monoids (i.e. monoids where Kleene's theorem holds), rational functions and relations, rational sets in partially commutative monoids, and rational sets in free groups.

monoid can be defined. Monoidal categories are ubiquitous: cartesian categories (in particular, the category of sets), free word-category over any category, endofunctors category over any category, category of R-modules over a commutative ring R, etc. The abstract monoids definable in the monoidal category incarnate in the usual monoids, triples (or monads), R-algebras, etc. 2 Given a monoidal ...