Coordinatization of lattices by regular rings without unit and Banaschewski functions

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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) countable 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 (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.

Mots clés: Lattice; complemented; sectionally complemented; modular; coordinatizable; frame; neutral; ideal; Banaschewski function; Banaschewski measure; ring; von Neumann regular; idempotent

Domaines: Anneaux et algèbres [math.RA]; Mathématiques générales [math.GM]

Friedrich Wehrung. Coordinatization of lattices by regular rings without unit and Banaschewski functions. Algebra Universalis, 2010, 64 (1), pp.49--67. ⟨10.1007/s00012-010-0088-x⟩. ⟨hal-00371268v2⟩