A new class of partial orders

TL;DR

This paper introduces a new class of partial orders called the $w$-core partial order in unital $*$-rings, based on the $w$-core inverse, and explores its properties and relationships with existing partial orders.

Contribution

It defines the $w$-core partial order using the $w$-core inverse and characterizes its properties and connections to other known partial orders.

Findings

The $w$-core partial order is well-defined and characterized.

It shows the $w$-core partial order coincides with the core partial order.

Relationships with several existing partial orders are established.

Abstract

Let $R$ be a unital $*$-ring. For any $a,w,b\in R$, we apply the defined $w$-core inverse to define a new class of partial orders in $R$, called the $w$-core partial order. Suppose $a,b\in R$ are $w$-core invertible. We say that $a$ is below $b$ under the $w$-core partial order, denoted by $a\overset{\tiny{\textcircled{\#}}}\leq_w b$, if $a_w^{\tiny{\textcircled{\#}}} a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}} =bwa_w^{\tiny{\textcircled{\#}}}$, where $a_w^{\tiny{\textcircled{\#}}}$ denotes the $w$-core inverse of $a$. Characterizations of the $w$-core partial order are given. Also, the relationships with several types of partial orders are considered. In particular, we show that the core partial order coincides with the $a$-core partial order, and the star partial order coincides with the $a^*$-core partial order.