J. Korean Math. Soc. 41 (2004), No. 2, pp. 397–406 RANK AND PERIMETER PRESERVERS OF BOOLEAN RANK-1 MATRICES Seok-Zun Song, LeRoy B. Beasley, Gi-Sang Cheon and Young-Bae Jun Abstract. For a Boolean rank-1 matrix A = ab t , we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the Boolean linear operators that preserve rank and perimeter of Boolean rank-1 matrices. 1. Introduction and preliminaries The Boolean algebra consists of the set B = {0, 1} equipped with two binary operations, addition and multiplication. The operations are defined as usual except that 1 + 1 = 1. There are many papers on linear operators that preserve the rank of matrices over several semirings. Boolean matrices also have been the subject of research by many authors. Beasley and Pullman [1] ob- tained characterizations of rank-preserving operators of Boolean matri- ces. They were unable to find necessary and sufficient conditions for a Boolean operator to preserve the rank of Boolean rank-1 matrices. This remains open until now. We consider this problem by adding conditions on perimeter of the Boolean rank-1 matrices. Let M m,n (B) denote the set of all m × n matrices with entries in the Boolean algebra B. The usual definitions for adding and multiplying matrices apply to Boolean matrices as well. Throughout this paper, we shall adopt the convention that m ≤ n unless otherwise specified. If an m × n Boolean matrix A is not zero, then its Boolean rank, b(A), is the least k for which there exist m × k and k × n Boolean matrices B and C with A = BC . The Boolean rank of the zero matrix is 0. It is Received February 27, 2003. 2000 Mathematics Subject Classification: Primary 15A03, 15A04. Key words and phrases: Boolean linear operator, perimeter, (U,V)-operator. This work was supported by Korea Research Foundation Grant KRF-2002-042- C00001.