PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 119. Number 4, December 1993 LINEAR OPERATORS THAT PRESERVE COLUMN RANK OF BOOLEAN MATRICES SEOK-ZUNSONG (Communicated by Eric Friedlander) Abstract. We study the extent to which certain theorems on rank preservers of Boolean matrices carry over to column rank preservers. We characterize the linear operators that preserve the column rank of Boolean matrices. I. Introduction and preliminaries There are many papers on the study of linear operators that preserve the rank of matrices over several semirings. Boolean matrices also have been the subject of research by many authors. But there are few papers on column rank preservers of the matrices over semirings. Recently Beasley and Song [ 1 ] ob- tained characterizations of column rank preservers of matrices over nonnegative integers. In this paper, we obtain characterizations of the linear operators that preserve the column rank of Boolean matrices. We let Mm<n(B) denote the set of all m x n matrices with entries in B = {0, 1}, the two element Boolean algebra. Arithmetic in B follows the usual rules except that 1 + 1 = 1. Throughout this paper, we shall adopt the conven- tion that m < n . If V is a nonempty subset of Bk = Mkx(B) containing 0 which is closed under addition, then V is called a Boolean vector space. If V and W are vector spaces with V c W, then V is called a subspace of W. We identify Mmj„(B) with Bmn in the usual way when we discuss it as a Boolean vector space and consider its subspaces. Let V be a Boolean vector space. If S is a subset of V, then (S) denotes the intersection of all subspaces of V containing 5, which is a subspace of V too, called the subspace generated by S. If S = {Vx, V2, ... , Vr}, then (S) = {Y?i=xxivi\xi e B} > tne set °f linear combinations of elements in S. Received by the editors April 8, 1992; presented at the conference of the International Linear Algebra Society at the University of West Florida, March 17-20, 1993. 1991 Mathematics Subject Classification. Primary 15A03, 15A04. Key words and phrases. Column rank, congruence operator. This paper was supported by the NON-DIRECTEDRESEARCH FUND, Korea ResearchFoun- dation, 1991,and in part by KOSEF-TGRC. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 1085 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use