Perimeter preserver of matrices over semifields

Journal of the Korean Mathematical Society

Proceedings of the American Mathematical Society, 1993

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.

Information Processing Letters, 1984

Journal of the Korean Mathematical Society

The Boolean rank of a nonzero m × n Boolean matrix A is the least integer k such that there are an m × k Boolean matrix B and a k × n Boolean matrix C with A = BC. We investigate the structure of linear transformations T : Mm,n → Mp,q which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, 2 ≤ k ≤ min{m, n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.

Discussiones Mathematicae - General Algebra and Applications, 2000

The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.

The Korean Mathematical Society, 2017

A Boolean rank one matrix can be factored as uv t for vectors u and v of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in u plus the number of nonzero entries in v. A Boolean matrix of Boolean rank k is the sum of k Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix A of Boolean rank k is the minimum over all Boolean rank one decompositions of A of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.

2007

Abstract. There are many papers on linear operators that preserve commuting pairs ofmatrices over fields or semirings. From these research works, we have a motivation to theresearch on the linear operators that preserve commuting pairs of matrices over nonneg-ative integers. We characterize the surjective linear operators that preserve commutingpairs of matrices over nonnegative integers.1. Introduction and preliminariesLet Z + be the nonnegative part of the ring of integers Z and let M n (Z + ) denote theset of all n × n matrices over Z + . Similarly let B = {0,1} be the binary Boolean algebraand let M n (B) denote the set of all n × n matrices over B. We denote the n × n identitymatrix by I n and the n×n zero matrix by O n . The n×n matrix all of whose entries arezero except its (i,j)th, which is 1, is denoted E i,j . We call E i,j a cell. We denote the n×nmatrix all of whose entries are 1 by J n . We omit the subscripts on I,O, and J when theyare implied by the context. If A and B...

Communications of the Korean Mathematical Society, 2013

We consider the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of T (X) = P XP T with some permutation matrix P .

Journal of Applied Mathematics and Computing, 2004

An open problem proposed by Safavi-Naini and Seberry in IEEE transactions on information theory(1991) can be reduced to a combinatorial problem on partitioning a subset of binary matrices. We solve the generalized Naini-Seberry's open problem by considering a certain class of binary matrices. Thus a subliminal channel of r > 1 bit capacity is systematically established for Naini-Seberry's authentication schemes. We also construct concrete examples.

Linear Algebra and its Applications, 1988

Characterizations are obtained of those linear operators over certain semirings that preserve (1) the r th coefficient of the rook polynomial, those that preserve (2) the term rank of matrices with term rank r, and those that preserve (3) the rth elementary symmetric permanental function. Our results apply to many algebraic systems of combinatorial interest, including the nonnegative integer matrices, Boolean matrices, and fuzzy matrices.

Linear Algebra and its Applications, 2010

Let M m,n (B) be the semimodule of all m × n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2 k min (m, n). Let B (m, n, k) denote the subsemimodule of M m,n (B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B (m, n, k), then there exist permutation matrices P and Q such that T(A) = PAQ for all A ∈ B (m, n, k) or m = n and T (A) = PA t Q for all A ∈ B (m, n, k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B (m, n, k) that preserve both the weight of each matrix and rank one matrices of weight k 2. Here the weight of a Boolean matrix is the number of its non-zero entries.

Filomat, 2019

We define right and left invariant matrices as Boolean matrices that are solutions to certain systems of matrix equations and inequalities over additively idempotent semirings. We provide improved algorithms for computing the greatest right and left invariant equivalence and quasi-order matrices. The improvements are based on the usage of the well-known partition refinement technique. Afterwards, we present the application of right invariant matrices in the determinization of weighted automata over additively idempotent, commutative and zero-divisor free semirings. In particular, we provide improvements of the well-known determinization method of weighted automata over tropical semirings given by Mohri [Computational Linguistics 23 (2) (1997) 269-311].

Mathematics

We study some properties of arctic rank of Boolean matrices. We compare the arctic rank with Boolean rank and term rank of a given Boolean matrix. Furthermore, we obtain some characterizations of linear operators that preserve arctic rank on Boolean matrix space.

2017

Czechoslovak Mathematical Journal, 2011

The set of all m × n Boolean matrices is denoted by Mm,n. We call a matrix A ∈ Mm,n regular if there is a matrix G ∈ Mn,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on Mm,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} 2, then all operators on Mm,n strongly preserve regular matrices, and if min{m, n} 3, then an operator T on Mm,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T (X) = U XV for all X ∈ Mm,n, or m = n and T (X) = U X T V for all X ∈ Mn.