Principal ideals of finitely generated commutative monoids

We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.     

Automatic subsemigroups of free products

We consider the automaticity of subsemigroups of free products of semigroups, proving that subsemigroups of free products, with all generators having length greater than one in the free product, are automatic. As a corollary, we show that if S is a free product of semigroups that are either finite or free, then any finitely generated subsemigroup of S is automatic. In particular, any finitely generated subsemigroup of a free product of finite or monogenic semigroups is automatic.     

Subdirect decompositions of finitely generated commutative semigroups

All finitely generated commutative semigroups which do not have proper finite subdirect decompositions are determined. This yields subdirect decompositions of finitely generated commutative semigroups and some idea of their structure.     

Ideals of finitely generated commutative monoids

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.     

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.     

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups

We prove that the class of the lattices embeddable into subsemigroup lattices of n-nilpotent semigroups is a finitely based variety for all n < ω. Repnitski? showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitski? result not appealing to the Bredikhin-Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.     

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。     

On semigroups admitting ring structure

A multiplicative semigroup S with 0 is said to be a R-semigroup if S admits a ring structure. Isbell proved that if a finitely generated commutative semigroup is a R-semigroup, then it should be finite. The non-commutative version of this theorem is unsettled. This paper considers semigroups, not necessarily commutative, which are principally generated as a right ideal by single elements and semigroups which are generated by two independent generators and describes their structure. We also prove that if a cancellative 0-simple semigroup containing an identity is a R-semigroup, then it should be a group with zero. Communicated by A. H. Clifford     

Commutative semigroups whose homomorphic images in groups are groups

In this paper we study commutative semigroups whose every homomorphic image in a group is a group. We find that for a commutative semigroup S, this property is equivalent to S being a union of subsemigroups each of which either has a kernel or else is isomorphic to one of a sequence T₀, T₁, T₂, ... of explicitly given, countably infinite semigroups without idempotents. Moreover, if S is also finitely generated then this property is equivalent to S having a kernel.     

FINITELY GENERATED COMMUTATIVE ARCHIMEDEAN SEMIGROUPS

In this work commutative Archimedean finitely generated semigroups are characterized in terms of ideal extensions.     

Finite union of commutative power joined semigroups

A commutative semigroup is called power joined if for every element a, b there are positive integers m, n such that a^m=bⁿ. A commutative power joined semigroup is archimedean (p. 131, [3]) and cannot be decomposed into the disjoint union of more than one subsemigroup. Every commutative semigroup is uniquely decomposed into the disjoint union of power joined subsemigroups which are called the power joined components. This paper determines the structure of commutative archimedean semigroups which have a finite number of power joined components. The number of power joined components of commutative archimedean semigroups is one or three or infinity. The research for this paper was supported in part by NSF Grant GP-11964.     

On the Finite and Non-finite Generation of Finitary Power Semigroups

The finitary power semigroup of a semigroup S, denoted P_f(S), is the set of finite subsets of S with respect to the usual set multiplication. Semigroups with finitely generated finitary power semigroups are characterised in terms of three other properties. From this statement there are drawn several corollaries. It follows that P_f(S) is not finitely generated if S is infinite and in any of the following classes: commutative; Bruck-Reilly extensions; inverse semigroups that contain an infinite group; completely zero-simple; completely regular.     

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.     

Finitely Generated Commutative Regular Semigroups

Abstract

In this paper we give a construction that allows us to describe up to isomorphisms all finitely generated regular commutative semigroups.     

A deterministic algorithm to decide if a finitely presented abelian monoid is cancellative

In this paper we give an algorithm to compute a finite presentation for any finitely generated commutative cancellative monoid, and in particular we apply it to derive an algorithm to decide whether a finitely presented commutative monoid is cancellative or not.     

On finitely related semigroups

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid. This answers a question by Davey, Jackson, Pitkethly, and Szabó from Davey et al. (Semigroup Forum, 83(1):89–122, 2011).     

Categorical semigroups

This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc. 33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups. Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc. 170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain. It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.