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

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

A sectionally complemented modular lattice L is coordinatizable if it is isomorphic to the lattice L(R) of all principal right ideals of some 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 aleph one. Furthermore, L is an ideal in a (necessarily coordinatizable) complemented modular lattice with a spanning 5-frame. 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 aleph one and index of nilpotence 3 such that L(R) has no Banaschewski function.

Mots clés: lattice; complemented; sectionally complemented; modular; coordinatizable; frame; entire; neutral; ideal; Banaschewski function; Banaschewski measure; ring; von Neumann regular; idempotent; larder; Condensate Lifting Lemma

Domaines: Mathématiques générales [math.GM]; Anneaux et algèbres [math.RA]; Catégories et ensembles [math.CT]; Logique [math.LO]

Friedrich Wehrung. A non-coordinatizable sectionally complemented modular lattice with a large Jónsson four-frame. Advances in Applied Mathematics, 2011, 47 (1), pp.173--193. ⟨10.1016/j.aam.2010.07.001⟩. ⟨hal-00462951v2⟩