Biordered sets of lattices and homogeneous basis

TL;DR

This paper explores the structure of biordered sets derived from complemented modular lattices, introduces an operation based on sandwich elements, and establishes a coordinatization theorem analogous to von Neumann's, enhancing understanding of lattice representations.

Contribution

It introduces a new operation on biordered sets from complemented modular lattices and proves a coordinatization theorem using biordered sets of idempotents.

Findings

Defined an operation using sandwich elements of biordered sets

Identified conditions for homogeneous bases in complemented modular lattices

Established a coordinatization theorem similar to von Neumann's

Abstract

In this paper we discuss the properties of the biordered set obtained from a complemented modular lattice and defines an operation using the sandwich elements of the biordered set. Further we describe a biordered subset satisfying certain conditions, so that the complemented modular lattice admits a homogeneous basis. Finally analogous to von Neumann coordinatization theorem we describe the coordinatization theorem for complemented modular lattice using the biordered set of idempotents.