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 ε from R to the idempotents of R such that xR = ε(x)R and ε(xy) = ε(x)ε(xy)ε(x) for all x, y ∈ 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 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.

A non-coordinatizable sectionally complemented modular lattice with a large Jónsson four-frame

“…Definition 7.9. A Banaschewski trace on a lattice L with zero is a family (a j i | i ≤ j in Λ) of elements in L, where Λ is an upward directed partially ordered set with zero, such that (i) a k i = a j i ⊕ a k j for all i ≤ j ≤ k in Λ; (ii) {a i 0 | i ∈ Λ} is cofinal in L. We proved in [28,Theorem 6.6] that A sectionally complemented modular lattice with a large 4-frame is coordinatizable iff it has a Banaschewski trace. Hence we obtain the following result.…”

“…We proved in [28,Theorem 4.1] that Every countable complemented modular lattice has a Banaschewski function. In the present paper, we construct in Proposition 4.4 a unital regular ring S F such that L(S F ) has no Banaschewski function.…”

Generators for complemented modular lattices and the von Neumann–Jónsson coordinatization theorems

“…On the other hand, coordinatization is always possible if L is the union of a countable increasing sequence of uniquely coordinatizable principal idealssee [18,Theorem 10.3] for the ultimate proof. Recently, a thorough analysis of coordinatizability by regular rings without unit has been given by Wehrung [31,32].…”

Coordinatization of finite join-distributive lattices

“…Although there are very deep coordinatization results in Lattice Theory, see J. von Neumann [22], C. Herrmann [19], and F. Wehrung [29] for example, our investigations were motivated by simple ideas that go back to Descartes. Namely, let B be a subset of a k-dimensional Euclidian space V , and let v 1 , .…”