Natural Partial Order on Rings with Involution

TL;DR

This paper introduces a new partial order on rings with involution, generalizing existing orders on projections, and explores its algebraic and lattice-theoretic properties, including conditions for orthomodularity and equivalence of comparability concepts.

Contribution

It defines a natural partial order on *-rings, proves its properties, and extends concepts of comparability, providing new insights into the structure of Rickart *-rings.

Findings

The natural partial order forms a sectionally semi-complemented poset.

Intervals [0,x] are orthomodular lattices in abelian Rickart *-rings.

Generalized and partial comparability are equivalent in finite abelian Rickart *-rings.

Abstract

In this paper, we introduce a partial order on rings with involution, which is a generalization of the partial order on the set of projections in a Rickart *-ring. We prove that a *-ring with the natural partial order form a sectionally semi-complemented poset. It is proved that every interval [0,x] forms an orthomodular lattice in case of abelian Rickart *-rings. The concepts of generalized comparability (GC) and partial comparability (PC) are extended to involve all the elements of a *-ring. Further, it is proved that these concepts are equivalent in finite abelian Rickart *-rings.