Semilattice decompositions of semigroups

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof. The results of this paper were obtained by the author between January and July of 1971, while an undergraduate at the University of California, Santa Barbara.     

Semilattice composition of ordered semigroups

We study the semilattice composition of ordered semigroups (a concept opposite to that of the semilattice decomposition), using the ideal extensions. The text was submitted by the authors in English.     

Bounded indecomposable semigroups of non-negative matrices

A semigroup \({\mathfrak{S}}\) of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists \({S\in\mathfrak{S}}\) such that (S) _ij ≠ 0. We show that if there is a pair k, l such that \({\{(S)_{kl} : S\in\mathfrak{S}\}}\) is bounded then, after a simultaneous diagonal similarity, all the entries are in [0, 1]. We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases.     

Strongly indecomposable finite groups

Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.     

Semilattice decomposition ofn (2)-permutable semigroups

Communicated by J.M. Howie     

Nonnegative matrix semigroups with finite diagonals

Let S be a multiplicative semigroup of matrices with nonnegative entries. Assume that the diagonal entries of the members of S form a finite set. This paper is concerned with the following question: Under what circumstances can we deduce that S itself is finite?     

Inverse semigroups and varieties of finite semigroups

In this paper we shall present a new construction of the free inverse monoid on a set X. Contrary to the previous constructions of [9, 11], our construction is symmetric and originates from classical ideas of language theory. The ingredients of this construction are the free group on X and the relation that associates to a word w of the free monoid on X, the set of all pairs (u, v) such that uv = w. It follows at once from our construction that the free inverse monoid on X can be naturally embedded into the Schützenberger product of two free groups of basis X. We shall also give some connections with the theory of expansions as developed by Rhodes and Birget [2, 3].     

Algebras with a finite number of classes of nonequivalent indecomposable modules

It is proved that the class of finite-dimensional algebras over fields with the property stated in the title is closed with respect to extensions by means of semi-simple artinian algebras.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 122–128, 1983.     

Computing finite commutative semigroups

Precedence results are used to improve existing algorithms for the enumeration of finite commutative semigroups. As an application 11,545,843 distinct commutative semigroups of order 9 were found.     

Sequences with idempotent products from finite regular semigroups

We find the smallest integer (n) such that for every regular semigroup S of order n, every sequence of length (n) of elements of S contains a consecutive subsequence whose product is an -element, where = ‘idempotent’, ‘core’ and ‘subgroup and core’. For arbitrary semigroups of order n, we also find (n) where = ‘regular’, ‘group’, ‘core’, ‘regular and core’ and ‘subgroup and core’.     

Graphs and finite transformation semigroups

Given a finite set X and a semigroup S of transformations of X, we study the orbitoids of S on X and on X² and, assuming S transitive, those of the statbilizer in S of an element α ∈ X. The action of S as a semigroup of endomorphisms of some relevant graphs (having X as vertex set) is also considered.     

Morita equivalence of finite semigroups

Semigroup Forum - Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup...