Operators and spaces associated to matrices with grades and their decompositions

Abstract

Abstract. The paper is a continuation of our previous paper on operators and spaces associated to matrices whose degrees are elements from a residuated lattice. The motivation for this study is to develop a calculus for such matrices which can be used in situations such as matrix decompositions. In this paper, we focus on row and column spaces, left and right ideals of matrices, and Green's relations. We prove basic results concerning these notions, show links to known structures, and put a new perspective on results known from Boolean matrices and their generalizations.