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

Abstract

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; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a non-coordinatizable sectionally complemented modular lattice L with a large 4-frame; it has cardinality ℵ 1 . Furthermore, L is an ideal in a complemented modular lattice L ′ with a spanning 5-frame (in particular, L ′ is coordinatizable).Our proof uses Banaschewski functions. A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented modular lattice has a Banaschewski function. We prove that there exists a unit-regular ring R of cardinality ℵ 1 and index of nilpotence 3 such that L(R) has no Banaschewski function.

Infinite Combinatorial Issues Raised by Lifting Problems in Universal Algebra

“…-There exists a non-coordinatizable sectionally complemented modular lattice (without unit), of cardinality ℵ 1 , with a large 4-frame (cf. Wehrung [49]). -There exists a lattice of cardinality ℵ 1 , in the variety generated by the five-element modular non-distributive lattice M 3 , without any congruence n-permutable, congruence-preserving extension for any positive integer n (cf.…”

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 lattices by regular rings without unit and Banaschewski functions

“…We shall prove in [19] that there exists a non-coordinatizable sectionally complemented modular lattice L with a large 4-frame. Hence L does not have a Banaschewski trace as well.…”

“…We also prove (Corollary 4.8) that such a Boolean range is uniquely determined up to isomorphism. In a subsequent paper [19], we shall prove that the countability assumption is needed.…”

“…Then we extend the notion of a Banaschewski function to non-unital lattices, thus giving the notion of a Banaschewski measure (Definition 5.5) and the more general concept of a Banaschewski trace (Definition 5.1)-first allowing the domain to be a cofinal subset and then replacing the function by an indexed family of elements. It follows from [19,Lemma 5.2] that every Banaschewski measure on a cofinal subset is a Banaschewski trace. Banaschewski measures are proved to exist on any countable sectionally complemented modular lattice (Corollary 5.6), and every sectionally complemented modular lattice with a Banaschewski trace embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice (Theorem 5.3).…”