A new class of partial orders

Abstract

Let $R$ be a unital $\ast$-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 $a{\stackrel{\stackrel{◯}{\#}}{\le }}_{w}b$, if $a_w^{\stackrel{◯}{\#}}a=a_w^{\stackrel{◯}{\#}}b$ and $aw{a}_w^{\stackrel{◯}{\#}}=bw{a}_w^{\stackrel{◯}{\#}}$, where $a_w^{\stackrel{◯}{\#}}$ 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 $a^{\ast }$-core partial order.