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Algebra universalis

, Volume 64, Issue 1–2, pp 49–67 | Cite as

Coordinatization of lattices by regular rings without unit and Banaschewski functions

  • Friedrich Wehrung
Article

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 \({\varepsilon}\) from R to the idempotents of R such that \({x{R} = \varepsilon(x){R}}\) and \({\varepsilon(xy) = \varepsilon(x)\varepsilon(xy)\varepsilon(x)}\) for all \({x, y \in R}\).

  • 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 \({\mathbb{L}(R)}\) 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.

2000 Mathematics Subject Classification

06C20 06C05 03C20 16E50 

Key words and phrases

lattice complemented sectionally complemented modular coordinatizable frame neutral ideal Banaschewski function Banaschewski measure Banaschewski trace ring von Neumann regular idempotent 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.LMNO, CNRS UMR 6139, Département de MathématiquesUniversité de CaenCaen cedexFrance

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