Biordered Sets of Regular Rings

TL;DR

This paper explores the relationship between biordered sets of idempotents in regular rings and their lattice structures, aiming to characterize these sets within a specific class of regular rings.

Contribution

It provides a new characterization linking biordered sets of regular rings with complemented modular lattices, extending previous theoretical frameworks.

Findings

Biordered sets of certain regular rings can be characterized via lattice structures.

A connection between biordered sets and complemented modular lattices is established.

The work extends the understanding of the structure of regular rings through lattice theory.

Abstract

The set of idempotents of a regular semigroup is given an abstract characterization as a regular biordered set in [2], and in [4] it is shown how a biordered set can be associated with a complemented modular lattice. Von Neumann has shown earlier that any complemented modular lattice of order greater than 3 can be realized as the lattice of principal right ideals of a regular ring (see [3]). Here we try to connect these ideas to get a characterization of the biordered sets of a class of regular rings.