m-Stone Lattices

Davey, B. A.

Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1027-1032

Résumé

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.

Détail

DOI : 10.4153/CJM-1972-104-x

Bibliographie

[1] 1. Chen, C. C. and G., Grätzer, Stone lattices. I: construction theorems, Can. J. Math. 21 (1969), 884–894. Google Scholar

[2] 2. Frink, O., Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505–514. Google Scholar

[3] 3. Kist, J., Compact spaces of minimal prime ideals, Math. Z. III (1969), 151-158. Google Scholar

[4] 4. Mitchell, B., Theory of categories (Academic Press, New York, 1965). Google Scholar

[5] 5. Peercy, D. E., Complexes and the complete Baer extension of a commutative ring (to appear). Google Scholar

[6] 6. Speed, T. P., A note on commutative semigroups, J. Austral. Math. Soc. 8 (1968), 731–736. Google Scholar

[7] 7. Speed, T. P., Some remarks on a class of distributive lattices, J. Austral. Math. Soc. 9 (1969), 289–296. Google Scholar

[8] 8. Speed, T. P., A note on commutative Baer rings III (to appear). Google Scholar

[9] 9. Swan, G., The theory of sheaves (University of Chicago Press, Chicago, 1964). Google Scholar