On morphisms of commutative monoids

The aim of this work is to study monoid morphisms between commutative monoids. Algorithms to check if a monoid morphism between two finitely generated monoids is injective and/or surjective are given. The structure of the set of monoid morphisms between a monoid and a cancellative monoid is also studied and an algorithm to obtain a system of generators of this set is provided.     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143-172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.     

The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c(S) and t(S) when M is a finitely generated commutative cancellative monoid.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.     

Quotient monoids and greatest commutative monoid images of several types

The concept of a quotient monoid modulo a subtractive subsemigroup is exploited systematically to determine necessary and sufficient conditions for a commutative semigroup to possess greatest monoid images that are separative, cancellative, power cancellative, or totally cancellative. The group of units of each such greatest image is determined and the images themselves are described via quotient monoids.     

Lorenzen Monoids: A Multiplicative Approach to Kronecker Function Rings

Using the formalism of module systems on a commutative cancellative monoid, we generalize the classical concept of Lorenzen monoids to obtain a multiplicative model for the semistar Kronecker function ring introduced by Fontana and Loper. We prove a universal mapping property and investigate the generalized Lorenzen monoid from a valuation-theoretic and an ideal-theoretic point of view.     

Rings with countably many direct summands

We consider commutative monoid amalgams having as their cores a conical cancellative monoid. In this context we present sufficient conditions for the amalgam to be strongly embeddable and for some properties of the factors to be transmitted to the amalgamated free product.     

On the linearity of ω-primality in numerical monoids

In an atomic, cancellative, commutative monoid, the ω-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the ω-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2.     

SEPARABILITY CONDITIONS IN ACTS OVER MONOIDS

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

     

M-McCoy环和M-Armendariz环的多项式扩张

研究非交换环上的相对于幺半群的McCoy环和Armendariz环的多项式扩张.对于包含无限循环子幺半群的交换可消幺半群M,证明了若R是M-McCoy（或M-Armendariz）环,则R上的洛朗多项式环R[x,x-1]是M-McCoy（或M-Armendariz）环.     

相对于幺半群的McCoy环的扩张

对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环;得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环,则~$R[M]$~是~McCoy~环;对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.     

Presentations for subsemigroups of finitely generated commutative semigroups

We introduce the concept of presentation for subsemigroups of finitely generated commutative semigroups, which extends the concept of presentation for finitely generated commutative semigroups. We show that for every subsemigroup of a finitely generated commutative semigroup there are special presentations which solve the word problem in the given subsemigroup. Some properties like being cancellative, reduced and/or torsion free are studied under this new point of view. This paper was supported by the project DGES PB96-1424.     

Strongly taut finitely generated monoids

A strongly taut monoid is a monoid in which all the powers of any element of the monoid have the same elasticity, that is, the ratio between the maximum and the minimum length of the factorizations of an element remains unchanged under powers. We give a procedure to determine if a finitely generated commutative monoid is strongly taut.     

Minimal presentations for monoids with the ascending chain condition on principal ideals

We show that the natural way to extend several key results concerning minimal presentations for finitely generated commutative cancellative reduced monoids, is to replace the finitely generated condition by the ascending chain condition on principal ideals.     

A Note on the n-Generator Property for Commutative Monoids

Let M be a cancellative, commutative monoid with integral closure . Borrowing from ring theory, we say that M has the n-generator property iff every finitely generated ideal of M can be generated by n elements, and we say M has rank n iff every ideal of M can be generated by n elements. We investigate the integral closure of such monoids. We show, in particular, that if M has the n-generator property, then is a valuation monoid, and if M has rank n, then is a principal ideal monoid.     

Atomic Commutative Monoids and Their Elasticity

We present some characterizations of properties related to factorization problems on finitely generated commutative monoids. We also give a series of algorithms for studying these problems from a presentation of a given commutative monoid.     

A Note on Almost GCD Monoids

A commutative cancellative monoid H (with 0 adjoined) is called an almost GCD (AGCD) monoid if for x,y in H, there exists a natural number n = n(x,y) so that xⁿ and yⁿ have an LCM, that is, xⁿH \cap yⁿH is principal. We relate AGCD monoids to the recently introduced inside factorial monoids (there is a subset Q of H so that the submonoid F of H generated by Q and the units of H is factorial and some power of each element of H is in F). For example, we show that an inside factorial monoid H is an AGCD monoid if and only if the elements of Q are primary in H, or equivalently, H is weakly Krull, distinct elements of Q are v-coprime in H, or the radical of each element of Q is a maximal t-ideal of H. Conditions are given for an AGCD monoid to be inside factorial and the results are put in the context of integral domains.     

The Extraction Degree of Cale Monoids

The extraction degree measures commonality of factorization between any two elements in a commutative, cancellative monoid. Additional properties of the extraction degree are developed for monoids possessing a Cale basis. For block monoids, the complete set of extraction degrees is calculated between any two elements, between any two irreducible elements, and between any irreducible element and any general element.