Review of Partial Orders in Rings Defined by Generalized Inverses

Dragan S. Rakić ; Dragan S. Djordjević

Zbornik radova, Tome 20 (2022) no. 28, p. 281

Résumé

A matrix $A^-$ is a generalized inverse of matrix $A$ if $AA^-A=A$ holds. Let $G$ be a function which assigns to each matrix $A$ the specific subset of the set of all generalized inverses of $A$. Then we say that $A$ is below matrix $B$ if $AA^-=BA^-$ and $A^-A=A^-B$ for some $A^-\in G(A)$. The minus, star, sharp, core and dual core partial orders are some of the well known matrix partial orders defined by appropriate choices of function $G$. This article reviews the recent known results concerning generalizations of matrix partial orders to the setting of arbitrary rings with or without involution. The article mainly consists of the published results of the authors.

Détail

Classification : NN-02, 06A06, 15A09, 16U90
Keywords: matrix partial orders, generalized inverses, ring, Von Neumann regular ring, minus partial order, star partial order, sharp partial order, core partial order, G-based order relation, Moore--Pen\-rose inverse, group inverse, core inverse