Strong regularity of matrices — a survey of results

Abstract

Let G = ( G , ⊗, ≤) be a linearly ordered, commutative group and u ⊕ v = max( u, v ) for all u, v ϵ G . Extend ⊕, ⊗ in the usual way on matrices over G . An m × n matrix A is said to have strongly linearly independent (SLI) columns, if for some b the system of equations A ⊗ x = b has a unique solution. If, moreover, m = n then A is said to be strongly regular (SR). This paper is a survey of results concerning SLI and SR with emphasis on computational complexity. We present also a similar theory developed for a structure based on a linearly ordered set where ⊕ is maximum and ⊗ is minimum.