Coordinatization of lattices by regular rings without unit and Banaschewski functions

TL;DR

This paper explores the structure of lattices and rings, establishing conditions under which lattices can be coordinatized by regular rings and introducing Banaschewski functions and traces as key tools.

Contribution

It introduces the concept of Banaschewski traces, linking lattice properties to ring coordinatization, and extends Jönsson's theorem to broader classes of lattices.

Findings

Countable complemented modular lattices have unique Banaschewski functions.

Regular rings admit maps to idempotents satisfying specific algebraic properties.

Lattices with a large 4-frame are coordinatizable iff they have a Banaschewski trace.

Abstract

A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: (1) Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. (2) Every (not necessarily unital) von Neumann regular ring R has a map e from R to the idempotents of R such that xR=e(x)R and e(xy)=e(x)e(xy)e(x) for all x,y in R. (3) Every sectionally complemented modular lattice with a ``Banaschewski trace'' (a weakening of the notion of a Banaschewski function) embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice. This applies, in particular, to any sectionally complemented modular lattice with a countable cofinal subset. A sectionally complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of all principal right ideals of a von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame, if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. J\'onsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable. We prove that a sectionally complemented modular lattice with a large 4-frame is coordinatizable iff it has a Banaschewski trace.