An Embedding Theorem for Matrices of Commutative Cancellative Semigroups

Let S be a commutative cancellative semigroup and T 0 {T_0} be a cofinal subsemigroup of S. Let h 0 {h_0} be a homomorphism of T 0 {T_0} into the semigroup of nonnegative real numbers under addition. We prove that Kobayashi’s condition [2] is necessary and sufficient for h 0 {h_0} to be extended to S. Further, we find a necessary and sufficient condition in order that the extension be unique. Related to this, the “boundedness condition” is introduced. For further study, several examples are given.

Let S be a commutative cancellative semigroup of finite rank (the free rank of the quotient group of S is finite). We give a necessary and sufficient condition for S to have a nontrivial homomorphism into the additive semigroup of nonnegative (positive) reals. We also give a counter-example in the case of infinite rank.

Different commutative semigroups may have a common saturation. We consider distinguishing semigroups with a common saturation based on their ``sparsity''. We propose to qualitatively describe sparsity of a semigroup by considering which faces of the corresponding rational polyhedral cone have saturation points. For a commutative semigroup we give a necessary and sufficient condition for determining which faces have saturation points. We also show that we can construct a commutative semigroup with arbitrary consistent patterns of faces with saturation points.

Modularity of the congruence lattice of a commutative cancellative semigroup

Necessary and sufficient conditions are given for an infinite matrix to be almost Schur and almost strongly regular in left amenable semigroups. 1. Introduction.In [7] the author discusses various methods of matrix summability in amenable semigroups.In that paper, sufficient conditions were given for an infinite matrix to be almost Schur and almost strongly regular.Examples were given to show that the conditions are not necessary.Recently, P. Schaefer gave necessary and sufficient conditions for an infinite matrix to be almost strongly regular for the additive semigroup of positive integers [8].This was also obtained by Howard T. Bell via a different approach [1].In this paper, we give necessary and sufficient conditions for an infinite matrix to be almost Schur and almost strongly regular in certain amenable semigroups. Preliminaries.We shall freely use notations and definitions in [7].Recall that a semigroup S is left amenable (LA) if the Banach space of all bounded real-valued functions on 5 with the sup norm, m(S), has a normalized positive left translation invariant linear functional.Such a linear functional is called a left invariant mean (LIM).The semigroup is said to be extremely left amenable (ELA) if the LIM is also multiplicative.It is shown by Day [2, p. 524, Theorem 1] that if S is LA, there is a net {<f>x} of finite means converging to left invariance in norm.If, in addition, S is countable then this net can be replaced by a sequence [6, p. 42, Lemma 5.1].Examples of LA semigroups are commutative semigroups, solvable groups and locally finite groups.Extremely left amenable semigroups are precisely those in which every two elements have a common right zero.For more details, we refer the reader to [2], [3], [4], [5].

Centralizers of elementary Abelian subgroups in finite p-groups

A semigroup S is called left (right) weakly commutative if, for every a, b ∈ S, there exist x ∈ S and a positive integer n such that (ab) n = bx ((ab) n = xa). A semigroup which is both left and right weakly commutative is called a weakly commutative semigroup. In this chapter we deal with left weakly commutative, right weakly commutative and weakly commutative semigroups. It is shown that a semigroup is a semilattice of left archimedean (right archimedean, t-archimedean) semigroups if and only if it is right weakly commutative (left weakly commutative, weakly commutative). It is proved that a weakly commutative 0-simple semigroup is a group with a zero adjoined. Moreover, a semigroup is weakly commutative archimedean and contains an idempotent element if and only if it is an ideal extension of a group by a nil semigroup. We get, as a consequence, that a semigroup is weakly commutative and regular if and only if it is a Clifford semigroup. We show that a right (left) weakly commutative semigroup is embeddable into a group if and only if it is cancellative. At the end of the chapter we deal with the least weakly separative congruence on weakly commutative semigroups. It is proved that if 5 is a left weakly commutative semigroup then σ defined by a σ b if and only if ab n = bn+1 and ba n = an+1 for a positive integer n is a weakly separative congruence on 5. Similarly, if S is a right weakly commutative semigroup then r defined by a τ b if and only if b n a = b n+1 and a n b = an+1 for some positive integer n is a weakly separative congruence on S. Moreover, π = σ ∩ τ is the least weakly separative congruence on a weakly commutative semigroup.

Image partition regularity of matrices over commutative semigroups

Do infinite nilpotent groups always have equipotent Abelian subgroups?

An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FA-presentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, Bruck–Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups.

Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

An interesting concept of semigroups (and also rings) of (left) quotients, based on the notion of group inverse in a semigroup, was developed by J. B. Fountain, V. Gould and M. Petrich, in a series of papers (see [5]-[12]). Among the most interesting are semigroups having a semigroup of (left) quotients that is a union of groups. Such semigroups have been widely studied. Recall from [3] that a semigroup has a group of left quotients if and only if it is right reversible and cancellative. A more general result was obtained by V. Gould [10]. She proved that a semigroup has a semilattice of groups as its semigroup of left quotients if and only if it is a semilattice of right reversible, cancellative semigroups. This result has been since generalized by A. El-Qallali [4]. He proved that a semigroup has a left regular band of groups as its semigroup of left quotients if and only if it is a left regular band of right reversible, cancellative semigroups. Moreover, he proved that such semigroups can be also characterised as punched spined products of a left regular band and a semilattice of right reversible, cancellative semigroups. If we consider the proofs of their theorems, we will observe that the principal problem treated there can be formulated in the following way: Given a semigroup S that is a band B of right reversible, cancellative semigroups Si, i ε B, to each Si, we can associate its group of left quotients Gi. When is it possible to define a multiplication of such that Q becomes a semigroup having S as its left order, and especially, that Q becomes a band B of groups Gi, i E B?Applying the methods developed in [1] (see also [2]), in the present paper we show how this problem can be solved for Qto become a strong band of groups (that is in fact a band of groups whose idempotents form a subsemigroup, by [16, Theorem 2]. Moreover, we show how Gould's and El-Quallali's constructions of semigroups of left quotients of a semilattice and a left regular band of right reversible, cancellative semigroups, can be simplified.

In this paper we discuss the question of when a variety of commutative or nilpotent semigroups consists only of finitely approximate semigroups. Moreover, we give necessary and sufficient conditions in order to have VarS = qVarS (here Var S denotes the variety generated by the semigroup S, and qVarS the quasivariety generated by S). Bibliography: 3 items.

In this paper we analyze the so-called word problem for (finite) combinatorial 0-simple semigroups and matrix semigroups from the viewpoint of computational complexity.

The paper gives a rather simple description of all maximal abelian subgroups H of the symmetric group SM acting on an arbitrary set M. In the case of finite M this result is used to determine the maximal cardinality of such an H and the maximal number of permutations without fixed points contained in an abelian subgroup of SM.