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Baer semigroup coordinatizations of modular lattices*

Published online by Cambridge University Press:  14 November 2011

T. S. Blyth
Affiliation:
Mathematical Institute, University of St Andrews

Synopsis

In the coordinatization of lattices by Baer semigroups, two notable gaps that remain to be filled concern the coordinatization of modular and distributive lattices. In this paper we present coordinatizations of modular lattices. In a subsequent paper we shall deal with the distributive case. Here we show that a bounded lattice L is modular if and only if L can be coordinatized by a Baer semigroup S such that if eS, fSR(S) then there exist idempotents ē, S such that ēS = eS, S = fS and e¯, commute; equivalently, if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) with e idempotent then there is an idempotent such that S = fS and e = ee.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Blyth, T. S. and Janowitz, M. F.Residuation theory (Oxford: Pergamon, 1972).Google Scholar