Coordinatization of lattices by regular rings without unit and Banaschewski functions

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:

• Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism.

• Every (not necessarily unital) countable von Neumann regular ring R has a map ${\varepsilon}$ from R to the idempotents of R such that ${x{R} = \varepsilon(x){R}}$ and ${\varepsilon(xy) = \varepsilon(x)\varepsilon(xy)\varepsilon(x)}$ for all ${x, y \in R}$.

• 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 ${\mathbb{L}(R)}$ 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. Jónsson 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.