Explicit formulae for the greatest least-squares and minimum norm g-inverses and the unique group inverse of matrices over commutative residuated dioids

Abstract

Motivated by Schein’s explicit formula for the maximum Thierrin-Vagner inverse of a Boolean matrix we first propose explicit formulae for the greatest least-squares and minimum norm g-inverses of a matrix A over any commutative residuated dioid. We also provide a formula for the unique group inverse of A, whenever existent. In particular, we propose formulae for the aforementioned generalized inverses of A over Boolean algebras, max-plus semirings and a class of complete and completely distributive lattices. Our main results remain valid in the context of residuated semigroups.