Enlargements, semiabundancy and unipotent monoids 1

Mathematika, 1999

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green's relations f and Jf coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids. PROPOSITION

Semigroup Forum, 2018

Numerical monoids (cofinite, additive submonoids of the non-negative integers) arise frequently in additive combinatorics, and have recently been studied in the context of factorization theory. Arithmetical numerical monoids, which are minimally generated by arithmetic sequences, are particularly well-behaved, admitting closed forms for many invariants that are difficult to compute in the general case. In this paper, we answer the question "when does omitting generators from an arithmetical numerical monoid S preserve its (well-understood) set of length sets and/or Frobenius number?" in two extremal cases: (i) we characterize which individual generators can be omitted from S without changing the set of length sets or Frobenius number; and (ii) we prove that under certain conditions, nearly every generator of S can be omitted without changing its set of length sets or Frobenius number.

Glasgow Mathematical Journal, 1993

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.

Notes on Number Theory and Discrete Mathematics, 2020

In this note we consider identities in the alphabet X = {x, y}. This note is selfcontained and the aim is to describe gradually the identities partition (with three parameters) of the free semigroup X + for the class of monoids B n = a, b | ba = b n (n > 0).

2021

Repeated idempotent elements are commonly used to characterise iterable behaviours in abstract models of computation. Therefore, given a monoid $M$, it is natural to ask how long a sequence of elements of $M$ needs to be to ensure the presence of consecutive idempotent factors. This question is formalised through the notion of the Ramsey function $R_M$ associated to M, obtained by mapping every positive integer $k$ to the minimal integer $R_M(k)$ such that every word $u$ in $M^*$ of length $R_M(k)$ contains $k$ consecutive non-empty factors that correspond to the same idempotent element of $M$. In this work, we study the behaviour of the Ramsey function $R_M$ by investigating the regular $D$-length of $M$, defined as the largest size $L(M)$ of a submonoid of $M$ isomorphic to the set of natural numbers $\{1,2, ..., L(M)\}$ equipped with the Max operation. We show that the regular $D$-length of $M$ determines the degree of $R_M$, by proving that $k^{L(M)} \leq R_M(k) \leq (k|M|^4)^{L...

Arxiv preprint arXiv: …, 2010

Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

2012

Let n ∈ N, Γ ⊆ N and define Γn = {x ∈ Zn | x ∈ Γ} the set of residues of elements of Γ modulo n. If Γn is multiplicatively closed we may define the following submonoid of the naturals: HΓn = {x ∈ N | x = γ, γ ∈ Γn}∪{1} known as a congruence monoid (CM). Unlike the naturals, many CMs enjoy the property of non-unique factorization into irreducibles. This opens the door to the study of arithmetic invariants associated with nonunique factorization theory; most important to us will be the concept of elasticity. In particular we give a complete characterization of when a given CM has finite elasticity. Throughout we explore the arithmetic properties of HΓn in terms of the arithmetic and algebraic properties of Γn.

Acta Mathematica Hungarica, 1990

International Journal of Algebra and Computation, 2011

We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid are related to the formal language properties of its idempotent problem. We show that if an inverse monoid is either E-unitary or has a finite set of idempotents, then its idempotent problem is regular if and only if the inverse monoid is finite. We also give examples of inverse monoids with context-free idempotent problems, including all Bruck–Reilly extensions of finite groups.

2021

Graph products of monoids provide a common framework for direct and free products, and graph monoids (also known as free partially commutative monoids). If the monoids in question are groups, then any graph product is, of course, a group. For monoids that are not groups, regularity is perhaps the first and most important algebraic property that one considers; however, graph products of regular monoids are not in general regular. We show that a graph product of regular monoids satisfies the related weaker condition of being abundant. More generally, we show that the classes of left abundant and left Fountain monoids are closed under graph product. The notions of abundancy and Fountainicity and their one-sided versions arise from many sources, for example, that of abundancy from projectivity of monogenic acts, and that of Fountainicity (also known as weak abundancy) from connections with ordered categories. As a very special case we obtain the earlier result of Fountain and Kambites t...