Isolation numbers of integer matrices and their preservers

Bull. Korean Math. Soc. Published online February 14, 2020

Leroy B Beasley, Kyung-Tae Kang, and Seok-Zun Song
Utah State University, Jeju National University, Mathematics, Natural Science College, Jeju National University

Abstract : Let A be an m x n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k if and only if T is a (P,Q)-operator, that is, for fixed permutation matrices P and Q, m x n matrix A, T(A) = PAQ or, m=n and T(A) = PA^t Q where A^t is the transpose of A.

Keywords : Isolation number, upper ideal, linear operator, (P,Q)-operator.