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On the Triple Characterization for Stone Algebras

  • Raymond Balbes (a1)

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In [1], C. C. Chen and G. Grâtzer developed a method for studying Stone algebras by associating with each Stone algebra L, a uniquely determined triple (C(L), D(L), ɸ (L)), consisting of a Boolean algebra C(L), a distributive lattice D(L), and a connecting map ɸ(L). This approach has been successfully exploited by various investigators to determine properties of Stone algebras (e.g. H. Lakser [9] characterized the injective hulls of Stone algebras by means of this technique). The present paper is a continuation of this program.

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References

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1. Chen, C. C. and Grâtzer, G., Stone lattices: Construction theorems, Can. J. Math. 21 (1969), 884894.
2. Stone lattices: Structure theorems, Can. J. Math. 21 (1969), 895903.
3. Grâtzer, G., Lattice theory: First concepts and distributive lattices (W. H. Freeman Co., San Francisco, 1971).
4. Grâtzer, G. and Lakser, H., The structure of pseudocomplemented distributive lattices, II: Congruence extension and amalgamation, Trans. Amer. Math. Soc. 156 (1971), 343348.
5. Grâtzer, G. and Lakser, H., The structure of pseudo complemented distributive lattices, III: Infective and absolute subretracts, Trans. Amer. Math. Soc. 157 (1972), 475487.
6. Jonsson, B., A Boolean algebra without proper automorphisms, Proc. Amer. Math. Soc. 2 (1951), 766770.
7. Katrinâk, T., A new proof of the construction theorem for Stone algebras, Proc. Amer. Math. Soc. 40 (1973), 7579.
8. Lakser, H., The structure of pseudocomplemented distributive lattices; I: Subdirect decomposition, Trans. Amer. Math. Soc. 156 (1971), 335342.
9. Lakser, H., Infective hulls of Stone algebras, Proc. Amer. Math. Soc. 24 (1970), 524529.
10. Nachbin, L., Une Propriété caractéristique des algebras Booléinnes, Portugal. Math. 6 (1947), 115118.
11. Stone, M. H., Topological representations of distributive lattices and Brouwerian logics, Casopis Pest. Mat. 67 (1937), 125.
12. Varlet, J., On the characterization of Stone lattices, Acta Sci. Math. (Szeged) 27 (1966), 8184.
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On the Triple Characterization for Stone Algebras

  • Raymond Balbes (a1)

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