A New Class of Partial Orders

Let R be a unital *-ring. For any a, w, b is an element of R , we apply the w-core inverse to define a new class of partial orders in R, called the w-core partial order. Suppose that a,b is an element of R are w-core invertible. We say that a is below b under the w-core partial order if a(w)(circle)a = a(w)(circle)b and awa(w)(circle) = bwa(w)(circle) , where a(w)(circle) denotes the w-core inverse of a. Characterizations of the w-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 a-core partial order, and the star partial order coincides with the a* -core partial order.