  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.

Rings and Algebras