Abstract
A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space is given by very simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them.
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References
Balbes, R. and Ph. Dwinger, 1975, Distributive lattices, Univ. Miss. Press
Bigard, A., el al. ‘Groupes et Anneaux Reticules’ Lecture Notes in Mathematics 608, Springer Verlag.
Birkhoff, G., 1967, ‘Lattice Theory’, Third Edition, American Mathematical Society Colloquium Publications, 25.
Cornish, W. and P. Fowler, 1977, ‘Coproducts of De Morgan algebras’, Bull. Austral. Math. Soc. 16, 1–13.
Cornish, W. and P. Fowler, 1978, ‘Coproducts of Kleene algebras’, Bull. Austral. Math. Soc. Ser A 27 209–320.
Cornish, W., 1980, ‘Lattice ordered groups and BCK-algebras’, Math. Japonica 254, 471–476.
Chang, C. C., 1958, ‘Algebraic Analysis of many valued logics’, Trans. Am. Math. Soc. 88, 467–490.
Davey, B. and H. Werner, 1983, ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. Janos Bolyai 33, North-Holland, Amsterdam, 101–375.
Fuchs, L., 1963, Partially Ordered Algebraic Systems, Pergamon, Oxford.
Martinez, N. G., 1990, ‘The Priestley duality for Wajsberg algebras’, Studia Logica 49, 31–46.
Martinez, N. G., ‘A topological duality for a class of lattice-ordered algebraic structures including l-groups’, to appear in Algebra Universalis.
Monteiro, A., 1980, ‘Sur les algebres de Heyting symetriques’, Portugaliae Mathematica 39.
Mundici, D., 1986, ‘Interpretation of AF - C *-algebras in Lukasiewicz sentential Calculus’, Journal of Functional Analysis 65, 15–63.
Priestley, H. A., 1972, ‘Representation of distributive lattices by means of ordered Stone spaces’, Bull. London. Math. Soc. 2, 186–190.
Priestley, H. A., 1975, ‘Ordered topological spaces and the representation of distributive lattices’, Bull. London. Math.Soc. 3, 507–530.
Stone, M. H., 1937, ‘Topological representation of distributive lattices and Brouwerian logics’, Casopis Pest. Mat. 67, 1–35.
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The author is supported at the Mathematical Institute of Oxford by a grant of the Argentinian Consejo de Investigations Cientificas y Tecnicas (CONICET). The author wishes to acknowledge the CONICET and the kind hospitality of the Mathematical Institute.
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Martinez, N.G. A simplified duality for implicative lattices and l-groups. Stud Logica 56, 185–204 (1996). https://doi.org/10.1007/BF00370146
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DOI: https://doi.org/10.1007/BF00370146