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.