Reverse order law for the inverse along an element

TL;DR

This paper introduces new generalized inverses in $*$-monoids, explores their properties, and establishes reverse order laws and formulas for inverses of product elements, extending the theory to rings.

Contribution

It presents the concept of left and right g-MP inverses, investigates their relation to inverse along an element, and derives reverse order laws in monoids and rings.

Findings

Established reverse order law for inverse along an element

Derived existence criteria for inverses of product elements

Extended inverse concepts to rings

Abstract

In this paper, we introduce a new concept called left (right) g-MP inverse in a $*$-monoid. The relations of this type of generalized inverse with left inverse along an element are investigated. Also, the reverse order law for the inverse along an element is studied. Then, the existence criteria and formulae of the inverse of the product of triple elements along an element are investigated in a monoid. Finally, we further study left and right g-MP inverses, the inverse along an element in the context of rings.