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𝔪-Stone Lattices

Published online by Cambridge University Press:  20 November 2018

B. A. Davey
B. A. Davey*
Affiliation:
Monash University, Clayton, Australia
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A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.

Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.

In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1027 - 1032
Copyright
Copyright © Canadian Mathematical Society 1972

References

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