Regular nonnegative matrices

Abstract

Regular elements in the semigroup Jfn of all nxn nonnegative matrices are characterized.The regular ^-classes in Vn are investigated and an alternate proof is given for a theorem of Flor characterizing the maximal subgroups. I. Introduction.Several authors in recent years have investigated the algebraic structure of semigroups of nonnegative matrices.In particular, maximal subgroups of such semigroups have been thoroughly studied.Brown [2] has shown that such subgroups are finite if compact and Flor [4] has a simple proof of the Brown result.Let jV" denote the semigroup of all n x n nonnegative matrices, let Sf' " denote the nxn stochastic matrices and let 2" denote the nxn doubly stochastic matrices.Flor has also shown that the maximal subgroups of jVn are isomorphic to complete nomomial groups over the reals.Schwarz [7] has characterized the maximal subgroups of Sf n as groups isomorphic to full symmetric groups while he [8] and Farahat [3] have independently shown that the maximal subgroups of D,n are isomorphic to direct products of full symmetric groups.In [5] the Green's relations and regularity in the semigroup Q." of doubly stochastic matrices were investigated.The purpose of this note is to characterize regularity in ^V" and to provide an alternate proof of the Flor result concerning the maximal subgroups.The techniques used here involve only elementary matrix properties and elementary semigroup theory. II. Regularity.The following concepts from the algebraic theory of semigroups will be used.The definitions and notation follow those in [1].Let T denote a semigroup and let a, b e T. Then the relation 3# [JS?] is defined on Thy the rule a%b [aJfb] if and only if a and b generate the same principal right [left] ideal in T. The relation Jt is defined to be &r\3l.