Permanent Semigroups

A permanent semigroup is a semigroup of n × n matrices on which the permanent function is multiplicative. If the underlying ring is an infinite integral domain with characteristic p > n or characteristic 0 we prove that any permanent semigroup consists of matrices with at most one nonzero diagonal. The same result holds if the ring is a finite field with characteristic p > n and at least n²+n elements. We also consider the Kronecker product of permanent semigroups and show that the Kronecker product of permanent semigroups is a permanent semigroup if and only if the pennanental analogue of the formula for the determinant of a Kronecker product of two matrices holds. This latter result holds even when the matrix entries are from a commutative ring with unity.     

Prime rings with semigroup generalized identity

We show that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field.     

Semigroups of partial isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.     

Connected Components of Open Semigroups in Semi-Simple Lie Groups

This paper studies connected components of open subsemigroups of non-compact semi-simple Lie groups by using control sets in the flag manifolds and their coverings. The concept of recurrent component is introduced and a method is given by which their number can be computed. It is shown that the union of all recurrent components is a semigroup. The idea of mid-reversibility comes up to show that an open semigroup has just one semigroup component if the identity belongs to its closure. A necessary and sufficient condition for mid-reversibility is proved showing that e.g. in a complex group any open semigroup is mid-reversible.     

Completely Simple Semigroups,Lie Algebras,and the Road Coloring Problem

Consider a semigroup generated by matrices associated with an edge-coloring of a strongly connected, aperiodic digraph. We call the semigroup Lie-solvable if the Lie algebra generated by its elements is solvable. We show that if the semigroup is Lie-solvable then its kernel is a right group. Next, we study the Lie algebra generated by the kernel. Lie algebras generated by two idempotents are analyzed in detail. We find that these have homomorphic images that are generalized quaternion algebras. We show that if the kernel is not a direct product, then the Lie algebra generated by the kernel is not solvable by describing the structure of these algebras. Finally, we discuss an infinite class of examples that are shown to always produce strongly connected aperiodic digraphs having kernels that are not right groups.     

Local and Global Properties of Nonautonomous Dynamical Systems and Their Application to Competition Models

We develop the inheritance principle for local properties by the global Poincare mapping of nonautonomous dynamical systems. Namely, if a semigroup property is uniformly locally universal then it is enjoyed by the global Poincare mapping. In studying the global dynamics of competitors in a periodic medium, the crucial role is played by the multiplicative semigroup of the so-called sign-invariant matrices. We give geometric criteria for stability of equilibria (periodic solutions) in competition models.     

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.     

Matrix semigroup homomorphisms into higher dimensions

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. We prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices to the semigroup of (n + 1)-by-(n + 1) matrices when n ? 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices to the semigroup of 5-by-5 matrices is reducible.     

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.     

On Prime Rings Satisfying a Certain Generalized Polynomial Identity

Abstract

Recently, Beidar, Fong and Bokut proved that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field. We shall extend their result to the case when generalized polynomial contains three summands.     

紧致交换矩阵半群

通过将矩阵同时对角化或同时上三角化的方法,给出有关紧致Abel矩阵半群以及紧致Hermite矩阵半群中矩阵的特征值的一些很好的刻画,证明了由可逆的Hermite矩阵构成的紧致矩阵半群中每个矩阵的特征值都是±1,Hermite矩阵单半群相似于对角矩阵半群,紧致交换矩阵半群的谱半径不超过1,等等.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

The doubly stochastic matrix equations x:aty= A and X Aty=t

For a doubly stochastic matrix A, each of the equations x:a^ty= A and X A^ty=^t is shown to have doubly stochastic solutions X and Y if and only if A lies in a subgroup of the semigroup of all doubly stochastic matrices of a given order. All elements of this semigroup which are left regular, right regular, or intra-regular are identified.     

Positive matrix semigroups with binary diagonals

We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in S{\mathcal{S}} satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.     

Generation results forB-bounded semigroups

In [3]A. Bellini-Morante defined and analysed a new one-parameter family of bounded operators which he called a B-bounded semigroup. The definition was motivated by an example from the transport theory where the evolution generated by an operator A was in a certain sense controlled by another operator B. In this paper we show that a given pair (A, B) generates a B-bounded semigroup if and only if in a certain extrapolation space related to the operator B, the closure of A generates a semigroup and we also address some related topics.     

Inverse subsemigroups and classes of finite aperiodic semigroups

We prove a number of results related to finite semigroups and their inverse subsemigroups, including the following. (1) A finite semigroup is aperiodic if and only if it is a homomorphic image of a finite semigroup whose inverse subsemigroups are semilattices. (2) A finite inverse semigroup can be represented by order-preserving mappings on a chain if and only if it is a semilattice. Finally, we introduce the concept of pseudo-small quasivariety of finite semigroups, generalizing the concept of small variety.     

Cancellative Orders

left order in Q and Q is a semigroup of left quotients of S if every q∈Q can be written as q=a^*b for some a, b∈S where a^* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the {\cal D}-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S. The authors are grateful to the ARC for its generous financial support.     

Factorization properties of (<Emphasis Type="Italic">n</Emphasis> × <Emphasis Type="Italic">n</Emphasis>) Boolean matrices

It is proved that every (n × n) Boolean matrix can be expressed as a product of primes and elementary matrices in the semigroup of Boolean matrices.     

The analyticity of abstract parabolic and correct systems

LetA be the matrix associated with an abstract parabolic system in the sense of Shilov or correct system in the sense of Petrovskii. We show that if the spectrum of its symbol is contained in a sector including some negative real axis, thenA generates an analytic regularized semigroup. The corresponding result related to numerical range conditions is also showed. Moreover, these results are applied to matrices of partial differential operators on many function spaces.     

Irreducible Numerical Semigroups Having Toms Decomposition

In this article we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.