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.     

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.     

Periodic subsemigroups of endomorphism monoids

If S is a periodic subsemigroup of the endomorphism monoid of a polycyclic group, then Endimioni (Mediterr J Math 8:307–313, 2011) proved that S is locally finite. Here we present an alternative proof that also extends the result to groups with suitable rank restrictions. Further we give an alternative proof of McNaughton and Zalcstein’s (J Algebra 34:292–299, 1975) theorem that periodic multiplicative subsemigroups of a matrix ring over a field are also locally finite. Finally we extend the latter to periodic subsemigroups of the endomorphism ring of a finitely generated module over a commutative ring.     

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD(B _n ) of partial contracting transformations of a Boolean, the semigroup TD(B _n ) of full contracting transformations of a Boolean, and the inverse semigroup ISD(B _n ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD(B _n ) and TD(B _n ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD(B _n ), we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases.     

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.     

The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras

In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P ^<∞(Λ) if and only if P ^<∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]).     

Generators and relations for subsemigroups via boundaries in Cayley graphs

Given a finitely generated semigroup S and subsemigroup T of S, we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.     

Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices

In this paper we bring together results about the density of subsemigroups of abelian Lie groups, the minimal number of topological generators of abelian Lie groups and a result about actions of algebraic groups. We find the minimal number of generators of a finitely generated abelian semigroup or group of matrices with a dense or a somewhere dense orbit by computing the minimal number of generators of a dense subsemigroup (or subgroup) of the connected component of the identity of its Zariski closure.     

Automatic structures for subsemigroups of Baumslag–Solitar semigroups

This paper studies automatic structures for subsemigroups of Baumslag–Solitar semigroups (that is, semigroups presented by 〈x,y∣(yx ^m ,x ⁿ y)〉 where $m,n \in\mathbb {N}$ ). A geometric argument (a rarity in the field of automatic semigroups) is used to show that if m>n, all of the finitely generated subsemigroups of this semigroup are (right-) automatic. If m<n, all of its finitely generated subsemigroups are left-automatic. If m=n, there exist finitely generated subsemigroups that are not automatic. An appendix discusses the implications of these results for the theory of Malcev presentations. (A Malcev presentation is a special type of presentation for semigroups embeddable into groups.)     

Howson's Property for Semidirect Products of Semilattices by Groups

An inverse semigroup S is a Howson inverse semigroup if the intersection of finitely generated inverse subsemigroups of S is finitely generated. Given a locally finite action θ of a group G on a semilattice E, it is proved that E*_θG is a Howson inverse semigroup if and only if G is a Howson group. It is also shown that this equivalence fails for arbitrary actions.     

On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S _α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S _α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups. January 21, 2000     

On optimal factorization of free semigroups into free subsemigroups

Optimal factorizations of a subsemigroup into very pure and weakly prefix subsemigroups are considered. It is shown that the intersection of the family of very pure (weakly prefix) subsemigroups containing a given subset A of X⁺ is itself a very pure (weakly prefix) subsemigroup and procedures for constructing the basis of such minimal subsemigroups are provided.     

On the Howson Property of HNN-Extensions with Nilpotent Base Group and Amalgamated Free Products of Nilpotent Groups

We give necessary and sufficient conditions under which an amalgamated free product of finitely generated nilpotent groups is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated). Also we prove that if G = ? t, K | t ^?1 At = B ?, where K is a finitely generated and infinite nilpotent group and A, B non-trivial infinite proper subgroups of K, then G is not a Howson group. The problem of deciding when an ascending HNN-extension of a finitely generated nilpotent group is a Howson group is still open.     

The subvariety lattice of the join of two semigroup varieties

The join A ∨ B of two semigroup varieties A and B is investigated. The latrine of subvarieties of A ∨ B is completely described, It is shown that this lattice is finite and non-modular and that all varieties in it are finitely based and finitely generated.     

A finite interval in the subsemigroup lattice of the full transformation monoid

In this paper we describe a portion of the subsemigroup lattice of the full transformation semigroup Ω ^Ω , which consists of all mappings on the countable infinite set Ω. Gavrilov showed that there are five maximal subsemigroups of Ω ^Ω containing the symmetric group \(\operatorname {Sym}(\varOmega )\) . The portion of the subsemigroup lattice of Ω ^Ω which we describe is that between the intersection of these five maximal subsemigroups and Ω ^Ω . We prove that there are only 38 subsemigroups in this interval, in contrast to the \(2^{2^{\aleph_{0}}}\) subsemigroups between \(\operatorname {Sym}(\varOmega )\) and Ω ^Ω .     

Module valuations and representation theory of orders I: Algebraic aspects

ABSTRACT

In this article, we study finitely generated reflexive modules over coherent GCD-domains and finitely generated projective modules over polynomial rings. In particular, we give a sufficient condition for a finitely generated reflexive module over a coherent GCD-domain to be a free module. By use of this result, we prove that every finitely generated projective R + [X]-module can be extended from R if R is a commutative ring with gl.dim(R) ≤ 2.     

On Hereditary Coreflective Subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(AB) is again FG.     

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.