Relations in linear algebra

Matrices over a commutative semiring that are idempotent with respect to the Hadamard product can be identified with binary relations. These relations form an embedded structure within the semi-additive category of (finite) matrices over the semiring. In this paper we investigate this substructure and its relationship with the collection of all matrices. In particular, we are interested under which properties the idempotent matrices form a (distributive) allegory. Furthermore, we study several relational properties and their natural extension to all matrices.