Partial orders in rings based on generalized inverses – unified theory

Abstract

MSC: 06A06 15A09 16U99Keywords: Matrix partial ordering G-based order relation Minus partial order Star partial order Sharp partial order Core partial order Generalized inverse Moore-Penrose inverse Group inverse Core inverse Ring Dedekind finite ringThe unified theory for matrix partial orders based on generalized inverses has already been done by Mitra. We consider a special kind of ring R, which generalizes the ring of linear operators on finite dimensional vector space, and extend Mitra's approach to it. Thus, we find necessary and sufficient condition for a G-based relation to be a partial order on R. Also, some known results are generalized and some new results are proved.

1MP and MP1 inverses and one-sided star orders in a ring with involution

“…Proof. Since the g-map G(a) = a{1, 2, 3} is semi-complete, the proof is a direct consequence of Theorem 4.5 and Theorem 3.2 and Corollary 3.3 in [11].…”

“…The unified theory of these relations was introduced by Mitra in [9]. This unified theory has recently been generalized in an arbitrary ring context by Rakić in [11]. By the end of this section we will consider our relations through the prism of this unified theory.…”

“…First we need to introduce some notions which is given in [9] and [11]. If P(R) denotes the power set of R then a function G : R → P(R) is called a g-map if for every a ∈ R, G(a) is a certain subset of a{1}.…”

“…Then we show that a < − † b if and only if a * < b when a, b ∈ R (1,3) , (see below for notation). We will examine this order through the Mitra's unified theory of partial orders based on generalized inverses, [9] and [11]. In particular, we present in Theorem 4.10 very useful simultaneous diagonalizations of a and b when a * < b.…”

Weighted binary relations involving core-EP inverse

“…A binary relation is a pre-order if it is reflexive and transitive; if it is also antisymmetric, then it is a partial order. The theory of partial orders (preorders) based on various generalized inverses has been increasingly investigated, such as, * -partial order [2], minus partial order [7,16], sharp partial order [12], Drazin pre-order [11,13], core partial order [1,17,19] and core-EP preorder [5,14,18]. Meanwhile, weighted Drazin pre-order and one-sided weighted Drazin pre-order were studied by Hernández et al [8,9].…”