Totally Ordered Commutative Monoids

A totally ordered monoid—or tomonoid , for short—is a commutative semigroup with identity S equipped with a total order \les that is translation invariant, i.e. , that satisfies: \forall x, y, z∈ S, x\les y\;\Rightarrow \; x+z \les y+z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice \Z ⁿ that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell. April 15, 1999     

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.     

Ideals of Finitely Generated Commutative Monoids

Abstract. We extend the concept of presentation of finitely generated commutative monoids to ideals of finitely generated commutative monoids and give algorithms to obtain information about an ideal from a given presentation.     

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.     

关于交换序半群的一个注记

给出了正则交换序半群的与素理想结构有关的若干性质，还给出了为有限个主理想的并的交换序半群类中的诺特性，阿基米德性，正则性以及有限生成性之间的一些关系。     

Note on the Construction of Free Monoids

We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains. In particular this will be the case if tensoring preserves sifted colimits.     

A Note on the Almansi Property

The first goal of this note is to study the Almansi property on an m-dimensional model in the sense of Greene and Wu and, more generally, in a Riemannian geometric setting. In particular, we shall prove that the only model on which the Almansi property is verified is the Euclidean space \({\mathbb R}^m\). In the second part of the paper we shall study Almansi’s property and biharmonicity for functions which depend on the distance from a given submanifold. Finally, in the last section we provide an extension to the semi-Euclidean case \({\mathbb R}^{p,q}\) which includes the proof of the classical Almansi property in \({\mathbb R}^m\) as a special instance.     

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 the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

关于算子空间逼近性质的一个注记

考虑算子空间和C*-代数的算子空间逼近性质,强箅子空间逼近性质与分片映射性质之间的某些关系.     

Commutative Non-Noetherian Rings with the Diamond Property

A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.

     

A Note on the Universal Factorization Property of Groups

We give criteria when a full subcategory D of the category of groups has C-universal factorization property (C-UFP) or C-strong universal factorization property(C-SUFP) for a certain category of groups C.As a byproduct,we give affirmative answers to three unsettled questions in[S.W.Kim,J.B.Lee,Universal factorization property of certain polycyclic groups,J.Pure Appl.Algebra 204 (2006) 555-567].     

A Note on the V-shaped Property in One-Machine Scheduling

This note deals with a one-machine scheduling problem to minimize the sum of the total earliness and total tardiness penalties with respect to a common due date, which is a decision variable. We show that the V-shaped property holds for a general case. This result is also true if the common due date is a prespecified value.     

具有F_k性质的环的交换性

利用根性,幂零性,结合零因子,正则元,中心及亚直不可约环以及密度定理等相关知识,研究了某些满足可变恒等式条件的环,特别是对具有强F_k性质的环进行了讨论,研究并推导出了八个具有F_k性质的环的若干交换性条件.     

关于有界闭凸集上的弱滴性质的注记

证明了“若K是Banach空间X中的有界闭凸集,0∈intK,则K有弱滴性质当且仅当K⁰有(WS)性质.”从而证明了文[1]中提出的一个问题.