Extreme Sets of Rank Inequalities Over Boolean Matrices and Their Preservers

Abstract

Abstract. 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 .

Extreme Preservers of Term Rank Inequalities Over Nonbinary Boolean Semiring

“…For the case S k = B k see [12,Lemma 2.4]. In the more general case, the same proof applies with the observation that a line is mapped into a weighted line (not all entries are 1, but rather nonzero).…”

“…An operator T : M m,n (S) → M m,n (S) is called a (P, Q, B)-operator if there exist permutation matrices P and Q, and a matrix B ∈ M m,n (S) with no zero entries, such that In [4] linear preservers of extremal cases of classical matrix inequalities over fields were characterized. On the other hand, linear preservers for various rank functions over semirings have been the object of much study during the last 30 years, see for example [2]- [12]. In particular term rank was investigated in the last years, see for example [5,6,7].…”

“…There are many papers on linear operators that preserve some properties of matrices ( [2]- [12]). We call such a topic of research "Linear Preserver Problems".…”

“…However, in the case of a nonbinary Boolean semiring, all elements except 0 and 1 in nonbinary Boolean semirings are zero-divisors. So there are few results on linear preserver problems for matrices over nonbinary Boolean semirings ([11]- [12]). Kirkland and Pullman characterized the linear operators that preserve the rank of matrices over nonbinary Boolean semirings in [8].…”