Moore-Penrose Equations in Involutive Residuated Semigroups and Involutive Quantales

Abstract

In this paper we provide procedures for testing the existence of various types of generalized inverses in involutive residuated semigroups
and involutive quantales defined by the Moore-Penrose equations, and for computing the extreme inverses whenever they exist.~We also determine certain instances when $a^\dag=a^*$, whenever $a^\dag $ exists. The obtained results can be applied to a wide class of quantales of fuzzy relations and fuzzy matrices, as well as to Gelfand quantales.