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arXiv · 2412.14472

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

Huihui Zhu·Bing Dong
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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.

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Huihui Zhu, Bing Dong. 2024-12-19. A class of core inverses associated with Green's relations in semigroups. https://arxiv.org/abs/2412.14472

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