Topological Semigroups of Non-Negative Matrices.

Abstract

This paper is devoted primarily to the study of topological semigroups of non-negative matrices.Usually, these semigroups are also assumed to be compact.Theorems on matrices and semigroups, which are germane to the paper, are first presented.Attention is focused on the spectrum of a non-negative matrix.It is first shown that a compact topological group of non-negative matrices is finite, by using the spectral properties of these matrices.From this theorem it follows that a clan (continuum semigroup with unit) of non-negative matrices is contractible.Some results on the existence of I-semigroups in a clan are also given.Next, the general structure of non-negative idempotents is investigated.As an application of this investigation, the set of non-negative idempotents of a fixed rank and order is shown to be arcwise connected.A similar theorem is obtained for the subset of stochastic idempotents of fixed rank and order.Commutative semigroups are next studied.The Jordan form of a matrix is used to show that any commutative semigroup of complex matrices is similar to a triangular complex matrix semigroup.This theorem, together with various algebraic and topological hypotheses, is used to obtain several sets of sufficient conditions that a semigroup be similar to a semigroup of diagonal matrices.The paper terminates with a chapter concerning topological representations of finite dimensional compact simple semigroups.It is shown that any such entity S in which the idempotents form a subsemigroup has an iseomorphic imbedding in the non negative matrices if and only if the maximal groups of S are finite.An analogous result is proved in which maximal groups are Lie groups and complex matrices are used in place of non negative real matrices.It is also shown that, if S is simple, if E is connected, and if each maximal group of S is totally disconnected, then E is a subseraigroup of S .v