A class of core inverses associated with Green's relations in semigroups

TL;DR

This paper introduces a new class of core inverses in semigroups related to Green's relations, extending existing inverse concepts and providing criteria, relations, and applications in matrix theory.

Contribution

It defines the $(b,c)$-core-EP inverse, explores its properties, relations to other inverses, and applies these results to matrix equations and existing literature.

Findings

Established criteria for $(b,c)$-core-EP invertibility.

Clarified relations among various core inverses.

Provided solutions for matrix equations involving these inverses.

Abstract

Let $S$ be a $*$-monoid and let $a,b,c$ be elements of $S$. We say that $a$ is $(b,c)$-core-EP invertible if there exist some $x$ in $S$ and some nonnegative integer $k$ such that $cax(ca)^{k}c=(ca)^{k}c$, $x{\mathcal R}(ca)^{k}b$ and $x{\mathcal L}((ca)^{k}c)^{*}$. This terminology can be seen as an extension of the $w$-core-EP inverse and the $(b,c)$-core inverse. It is explored when $(b,c)$-core-EP invertibility implies $w$-core-EP invertibility. Another accomplishment of our work is to establish the criteria for the $(b,c)$-core-EP inverse of $a$ and to clarify the relations between the $(b,c)$-inverse, the core inverse, the core-EP inverse, the $w$-core inverse, the $(b,c)$-core inverse and the $(b,c)$-core-EP inverse. As an application, we improve a result in the literature focused on $(b,c)$-core inverses. We then establish the criterion for the $(B,C)$-core-EP inverse of $A$ in complex matrices, and give the solution to the system of matrix equations.