A Non-coordinatizable Sectionally Complemented Modular Lattice with a Large Jónsson Four-frame

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; 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 א1. 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 א1 and index of nilpotence 3 such that L(R) has no Banaschewski function.