�-Stone Lattices Academic Article uri icon

abstract

  • 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 �-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 = {bL|a ∧ b = 0, for all aA} is a principal ideal generated by an element of the centre of L.In this paper we characterize �-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 �-Stone lattice, the embedding being a left adjoint to the forgetful functor.

publication date

  • December 1972