Maximal Semigroups Research Articles

A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ›(R)

An ݑŸ-potent matrix in ݑ0ݑ›(R) is an ݑ› × ݑ› matrix satisfying ݐذݑŸ = ݐ¸. A multiplicative semigroup S in ݑ0ݑ›(R) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each ݐ´ ∈ S. A semi-group S of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) is known to be decomposable if rank (ݑ†) > ݑŸ − 1 for all ݑ† in S. Further, a semigroup S in ݑ0ݑ›(R) of nonnegative matrices will be called a full semigroup if S has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative r-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of r-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of r-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.

Maximal dominated operator semigroups

Maximal operator semigroups, bounded in a certain sense, on real or complex vector spaces are studied. For any maximal semigroup M dominated by a certain pair of homogeneous functions there is an operator quasinorm for which M is exactly the semigroup of contractions in this quasinorm. Applications to Riesz spaces are given. In particular, maximal semigroups of matrices dominated by a given positive matrix are characterized. We thus answer the question implicitly posed in [2].

Maximal semigroups in semi-simple Lie groups

The maximal semigroups with nonempty interior in a semi-simple Lie group with finite' center are characterized as compression semigroups of subsets in the flag manifolds of the group. For this purpose a convexity theory, called here B-convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a B-convex set.

Minimal and Maximal Semigroups Related to Causal Symmetric Spaces

be an irreducible globally hyperbolic semisimple symmetric space, and let S⊆G be a subsemigroup containing H not isolated in S. We show that if So≠ 0 then there are H-invariant minimal and maximal cones Cmin⊆Cmax in the tangent space at the origin such that H exp Cmin⊆S⊆HZK(a)expCmax. A double coset decomposition of the group G in terms of Cartan subspaces and the group H is proved. We also discuss the case where G/H is of Cayley type.

Maximal semigroups in finitely generated nilpotent groups

We obtain a classification of maximal subsemigroups of finitely generated nilpotent groups. In the principal results of this paper we show that there is a one-to-one correspondence between these subsemigroups and the non trivial homomorphisms of the group in R.

Maximal semigroups dominated by 0–1 matrices

Maximal semigroups dominated by a 0–1 matrix of a certain type are determined. The 0–1 matrices that dominate maximal bounded and maximal commuting semigroups are given. Also semigroup modules over maximal semigroups dominated by a 0–1 matrix are discussed.

Maximality of compression semigroups

LetG be a connected semisimple Lie group andr an involution onG. Further letL be an open subgroup of the groupGr ofr-fixed points andP⊂-G a parabolic subgroup. The semigroupS(L,P)∶={g∈G∶gLP⊂-LP} is called the compression semigroup of theL-orbit of the base point in the flag manifoldG/P. We show that compression semigroups for open orbits and regular symmetric pairs are maximal semigroups.

Quantitative stability analysis for maximal monotone operators and semi-groups of contractions

Quantitative stability analysis for maximal monotone operators and semi-groups of contractions

Maximal semigroups and controllability in products of Lie groups

0. Introduction. In recent years authors from various fields of mathematics were lead to the study of subsemigroups of Lie groups (cf. [13], [2], [16, 17] [14, 15], [18] etc). In [11] a first a t tempt has been made to give a systematic approach to the study of subsemigroups of Lie groups.The basic idea is to associate with a subsemigroup S of a Lie group G a tangent object L(S) = {x ~ L(G): x = lira nx, , exp x, ~ S, n ~ N} (where L(G) is the

The largest groupoid of varieties of semigroups with zero

It is well known [i] that with respect to the operation of multiplication, the set of all varieties of semigroups forms a partial groupoid. The structure of this is very complicated and still not known. In studying this groupoid the most natural problems are the following: i) to describe all pairs of varieties whose product is a variety; 2) to find in this partial groupoid some important groupoids (in particular, semigroups) and to clarify their structure; 3) to distinguish maximal groupoids and semigroups. Some of these investigations have already been carried out. Thus, in [2] idempotents of this groupoid were described and a countable semigroup with nullary multiplication was distinguished, which consists of varieties of indempotent semigroups; in [3, 4] some conditions were given under which the product of two varieties of semigroups is a variety; in [5] a groupoid with the power of the continuum was mentioned, which consists of so-called 0-reduced varieties, that is, varieties of semigroups with zero O, having an identity basis of the form w = 0, where w is a semigroup word; in [6] a number of important properties of this groupoid G were proved: it is cancellative and is the union of two nonintersecting subgroupoids H and L, where H is the largest subsemigroup, and L is an ideal of G. The structure of H has been completely clarified [7, 8]: it is the free product of a free cormmutative semigroup of countable rank and a free semigroup of the rank of the continuum with externally adjoined unity. The main result of the paper is the following theorem. THEOREM. The groupoid G is the maximal groupoid in the partial groupoid of all varieties of semigroups. In the partial groupoid of varieties of semigroups with signature zero, G is the largest groupoid. We mention that this theorem gives a complete answer to the third question for the partial groupoid of semigroups with signature zero. Before proceeding to the proof of the theorem, we give the necessary definitions and notation. In this paper we use the generally accepted terminology (see [9, i0]). Our notation is also generally standard or follows the notation in [6]. Let X be a countable alphabet, and F a free semigroup over X. For a word a of F we denote by C(a) the set of all letters of X that occur in writing 0~, and by 151 the length of the word ~L. A letter that occurs more than once in writing OJ will be called multiple, and the letters that appear in the first and last places in writing O~ will be called the beginning and end of ~L. For the graphic coincidences of the words C5 and b of F we shall write ~L=---~. The notation ~dL means that b is a subword of /~; ~ <0~ means that ~ is

Regular, Commutative, Maximal Semigroups of Quotients

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.