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.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

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).     

Identity Bases for Some Non-exact Varieties

Let A₂ be the variety generated by the five-element non-orthodox 0-simple semigroup. This paper presents the identity bases for several subvarieties of A₂ that are not generated by any completely 0-simple or completely simple semigroups. It will be shown that several subvarieties of A₂, including the variety generated by the five-element Brandt semigroup, are hereditarily finitely based.     

Finite degree: algebras in general and semigroups in particular

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p _n -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.     

On the Finite and Nonfinite Generation of Diagonal Acts

The diagonal right act of a semigroup S is the set S × S, with S acting by componentwise multiplication from the right. The diagonal left act and diagonal bi-act of S are defined analogously.

Necessary and sufficient conditions are found for the finite generation of the diagonal bi-acts of completely zero-simple semigroups and completely simple semigroups. It is also proved that various classes of semigroups do not have finitely generated or cyclic diagonal right, left, or bi-acts.     

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     

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.     

Identities of semigroup rings over semilattices of completely (0-) simple semigroups

LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.     

Finite state wreath powers of transformation semigroups

The finite state wreath power of a transformation semigroup is introduced. It is proved that the finite state wreath power of nontrivial semigroup is not finitely generated and in some cases even does not contain irreducible generating systems. The free product of two monogenic semigroups of index 1 and period m is constructed in the finite state wreath power of corresponding monogenic monoid.     

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     

Globals of Idempotent Semigroups

If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.     

Properties of Integers and Finiteness Conditions for Semigroups

Let h and k be integers greater than 1; we prove that the following statements are equivalent: 1) the direct product of h copies of the additive semigroup of non-negative integers is not k-repetitive; 2) if the direct product of h finitely generated semigroups is k-repetitive, then one of them is finite. Using this and some results of Dekking and Pleasants on infinite words, we prove that certain repetitivity properties are finiteness conditions for finitely generated semigroups.     

Levi’s Commutator Theorems for Cancellative Semigroups

For every semigroup of finite exponent whose chains of idempotents are uniformly bounded we construct an identity which holds on this semigroup but does not hold on the variety of all idempotent semigroups. This shows that the variety of all idempotent semigroups E is not contained in any finitely generated variety of semigroups. Since E is locally finite and each proper subvariety of E is finitely generated [1, 3, 4], the variety of all idempotent semigroups is a minimal example of an inherently non-finitely generated variety.     

Guarded and Banded Semigroups

The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g_⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g_⋆ ≅ L is obtained.     

Row reduced matrices and annihilator semigroups 1

Throughout we are discussing matrices with entries from a field K. It was first proved in [1] that a product of row reduced matrices is row reduced. This means that the set of row reduced matrices in any matrix ring form a semigroup. It is also the case that every matrix A ? M_n(K)has the property that it has the same right annihilator as its row reduced form, and distinct row reduced matrice have distinct right annihilators. Let R be a ring. Motivated by these observations, we call a multiplicative semigroup S in R a right annihilator semigroup for R if every element in R has the same right annihilator as exactly one element in S. Reasoning that row reduced matrices are very important we study semigroups that share their formal properties. Ultimately we would like to know all right annihilator semigroups in M_n(K).This seems to be a formidable task. Here we determine all right annihila-tor semigroups in M₃(K) up to a change of basis, that is conjugation.     

Ergodic semigroups of positivity preserving self-adjoint operators

We prove that an ergodic semigroup of positivity preserving self-adjoint operators is positivity improving. We also present a new proof (using Markov techniques) of the ergodicity of semigroups generated by spatially cutoff P(?)₂ Hamiltonians.     

Generators and relations of abelian semigroups and semigroup rings

The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Apéry [1]. This result applies to algebraic geometry: An affine space-curve C with the parametric equations x=t^a, y=t^b, z=t^c, a, b, c natural numbers with greatest common divisor 1, is a global idealtheoretic complete intersection, if and only if the semigroup S generated by a, b, c is symmetric.This paper forms part of the author's thesis, submitted at Lousiana State University.The writing of this paper was partially supported by NSF grant GP-6388 in which the author participated as a junior assistant at Purdue University.     

Sublattices of the Lattices of Strong P-Congruences on P-Inversive Semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C_P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C_P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C_P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C_P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C_P(S) and the greatest elements of these sublattices are given.