Let M be a commutative cancellative atomic monoid. We use unions of sets of lengths in M to construct the V-Delta set of M . We first derive some basic properties of V-Delta sets and then show how they offer a method to investigate the asymptotic behavior of the sizes of unions of sets of lengths. A central focus of number theory is the study of number theoretic functions and their asymptotic b...

Let M be a (commutative cancellative) monoid. A nonunit element q ∈ M is called almost primary if for all a, b ∈ M , q|ab implies that there exists k ∈ N such that q|a or q|b. We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties ...

We study the problem of testing whether a context-free language is included in a fixed set L0, where L0 is the language of words reducing to the empty word in the monoid defined by a complete string rewrite system. We prove that, if the monoid is cancellative, then our inclusion problem is polynomially reducible to the problem of testing equivalence of straight-line programs in the same monoid....

We introduce the growth partition function ZΓ,G(t) associated with any cancellative infinite monoid Γ with a finite generator system G. It is a power series in t whose coefficients lie in integral Lie-like space LZ(Γ, G) in the configuration algebra associated with the Cayley graph (Γ, G). We determine them for homogeneous monoids admitting left greatest common divisor and right common multiple...

— Let M be an infinité commutative monoid. Suppose that M has a Church-Rosser présentation. If M is cancellative or if the présentation is special then M is either the free cyclic group or the free cyclic monoid. Résumé. — Soit M un monoide commutatif infini. Supposons que M possède une présentation finie ayant la propriété de « Church-Rosser ». Si M est simplifiable ou si la présentation est s...

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 ...

Given a set-theoretic solution (X, r) of the Yang–Baxter equation, we denote by M = M(X, structure monoid and A A(X, r), respectively A0 left, right, derived r). It is shown that there exist left action on right 1-cocycles π 0 with coefficients in respect to these actions, respectively. We investigate when are injective, surjective, or bijective. In case X finite, it turns out bijective if only...

On Graphs With A Given Endomorphism Monoid Benjamin Shemmer Hedrĺın and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. We will give a construction G(M) for a graph with prescribed endomorphism monoid M. Using this construction we derive bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism mo...

It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lower common multiples...

If A is a stable basis algebra of rank n, then the set Sn−1 of endomorphisms of rank at most n−1 is a subsemigroup of the endomorphism monoid of A. This paper gives a number of necessary and sufficient conditions for Sn−1 to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left T -sets of finite rank where T is can...