A New Class of Partial Orders

Abstract

Let [Formula: see text] be a unital [Formula: see text]-ring. For any [Formula: see text], we apply the [Formula: see text]-core inverse to define a new class of partial orders in [Formula: see text], called the [Formula: see text]-core partial order. Suppose that [Formula: see text] are [Formula: see text]-core invertible. We say that [Formula: see text] is below [Formula: see text] under the [Formula: see text]-core partial order if [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the [Formula: see text]-core inverse of [Formula: see text]. Characterizations of the [Formula: see text]-core partial order are given, and its relationships with several types of partial orders are also considered. In particular, we show that the core partial order coincides with the [Formula: see text]-core partial order, and the star partial order coincides with the [Formula: see text]-core partial order.