2009/03/27 by Friedrich Wehrung, Wehrung, Friedrich
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Rings, Modules, and Algebras #math.RA #msc:03C20 #msc:06C05 #msc:06C20 #msc:16E50
paper · pdf · doi:10.48550/arxiv.0903.4756
To appear in Algebra Universalis
arxiv created 2009/06/05 · arxiv updated 2009/12/01
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 (a0,a1,a2,a3) such that the neutral ideal generated by a0 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.