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Vector lattices freely generated by distributive lattices

Published online by Cambridge University Press:  24 October 2008

W. M. Beynon
W. M. Beynon
Affiliation:
University of Warwick
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Extract

In this paper, the duality theory for vector lattices developed by the author in (2) is applied to the study of vector lattices freely generated by distributive lattices.

If L is a finite distributive lattice defined by a set of generators X, subject to a finite set of relations R, then the vector lattice FVL(L) freely generated by the elements of L subject to the relations governing intersection and union in L is a finitely generated projective vector lattice, being a quotient of the vector lattice freely generated by elements of X by the principal ideal generated by the relations R. Thus, there is a polyhedron Q, such that FVL(L) is isomorphic with the vector lattice of real valued polyhedral functions on Q, under pointwise operations, and the isomorphism class of FVL(L) determines and is determined by the polyhedral type of Q. The principal result of this paper shows that the polyhedral class to which Q belongs can be simply characterised in terms of the nerve of the poset P of join-irreducible elements of L. Explicitly, up to polyhedral equivalence, Q is a suspension of the nerve of P.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 2 , March 1977 , pp. 193 - 200
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

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