Abstract
The paper discusses “regularisation” of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of constants within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities.
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References
Banaschewski, B., 1971, ‘Projective covers in categories of topological spaces and topological algebras’, General Topology and its Relation to Modern Analysis and Algebra, (Proc. Kanpur Topology Conference 1968), Academia, Prague, 63–91.
Berman, J., 1983, ‘Free spectra of 3-element algebras’, Universal Algebra and Lattice Theory, R.S. Freese and O.C. Garcia, eds., Springer, Berlin, 10–53.
Birkhoff, G., 1967, Lattice Theory (3rd.ed.), American Mathematical Society, Providence, R.I.
Bochvar, D. A., 1938, ‘On a three-valued logical calculus and its application to the analysis of contradictions’, (Russian), Mat. Sb. 46, 287–308, English translation in Hist. Philos. Logic 2, 1981, 87–112.
Davey, B. A., 1992, ‘Duality theory on ten dollars a day’, La Trobe University Mathematics Research Paper 131 (92–3).
Davey, B. A. and H. Werner, 1980, ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. J. Bolyai 33, 101–275.
Gierz, G. and A. Romanowska, 1991, ‘Duality for distributive bisemilattices’, J. Austral. Math. Soc. Series A 51, 247–275.
Guzmán, F., 1992, Three-valued logics in the semantics of programming languages, preprint.
Hewitt, E. and H. S. Zuckerman, 1956, ‘The l 1-algebra of a commutative semigroup’, Trans. Amer. Math. Soc. 83, 70–97.
Hofmann, K. H., M. Mislove and A. Stralka, 1974, The Pontryagin Duality of Compact O-Dimensional Semilattices and its Applications, Springer, Berlin.
Howie, J. M., 1976, An Introduction to Semigroup Theory, Academic Press, London.
Idziak, P. M., 1989, ‘Sheaves in universal algebra and model theory’, Reports on Math. Logic 23, 39–65.
Johnstone, P. T., 1982, Stone Spaces, Cambridge University Press, Cambridge.
Kleene, S. C., 1952, Introduction to Metamathematics, Van Nostrand, Princeton.
Libkin, L., 1993, Towards a theory of edible powerdomains, preprint.
Mac Lane, S., 1971, Categories for the Working Mathematician, Springer, Berlin.
Mac Lane, S. and I. Moerdijk, 1992, Sheaves in Geometry and Logic, Springer, Berlin.
Mel'nik, I. I., 1971, ‘Normal closures of perfect varieties of universal algebras’, Ordered Sets and Lattices, (Russian), Izdat. Sarat. Univ., 56–65.
Płonka, J., 1967, ‘On distributive quasi-lattices’, Fund. Math. 60, 191–200.
Płonka, J., 1967, ‘On a method of construction of abstract algebras’, Fund. Math. 61, 183–189.
Płonka, J., 1969, ‘On equational classes of algebras defined by regular equations’, Fund. Math. 64, 241–247.
Płonka, J., 1984, ‘On the sum of a direct system of universal algebras with nullary polynomials’, Alg. Univ. 19, 197–207.
Płonka, J., 1985, ‘On the sum of a i-semilattice ordered system of algebras’, Studia Scient. Math. Hungar. 20, 301–307.
Płonka, J., 1991, ‘Characteristic algebras in some varieties defined by symmetrically regular identities’, Contributions to General Algebra 7, D. Dorninger, G. Eigenthaler, H. K. Kaiser and W. B. Müller, eds., Hölder-Pichler-Tempsky, Vienna, 265–276.
Płonka, J. and A. Romanowska, 1992, ‘Semilattice sums’, Universal Algebra and Quasigroup Theory, A Romanowska and J.D.H. Smith, eds., Heldermann, Berlin, 123–158.
Priestley, H. A., 1970, ‘Representation of distributive lattices by means of ordered Stone spaces’, Bull. Lond. Math. Soc. 2, 186–190.
Priestley, H. A., 1972, ‘Ordered topological spaces and the representation of distributive lattices’, Proc. Lond. Math. Soc. 24, 507–530.
Puhlmann, H., 1993, ‘The snack powerdomain for database semantics’, Mathematical Foundations of Computer Science, A. M. Borcyszkowski, S. Sokołowski (eds), 650–659.
Romanowska, A. B., 1986, ‘On regular and regularized varieties’, Alg. Univ. 23 215–241.
Romanowska, A. B. and J. D. H. Smith, 1985, ‘From affine to projective geometry via convexity’, Universal Algebra and Lattice Theory, S.D. Comer, ed., Springer, Berlin, 255–269.
Romanowska, A. B. and J. D. H. Smith, 1985, Modal Theory, Heldermann, Berlin.
Romanowska, A. B. and J. D. H. Smith, 1989, ‘Subalgebra systems of idempotent entropie algebras’, J. Algebra 120, 247–262.
Romanowska, A. B. and J. D. H. Smith, 1991, ‘On the structure of semilattice sums’, Czech. J. Math. 41, 24–43.
Romanowska, A. B. and J. D. H. Smith 1993, ‘Duality for semilattice representations’, preprint.
Semadeni, Z. and A. Wiweger, 1979, Einführung in die Theorie der Kategorien und Funktoren, Teubner, Leipzig.
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Romanowska, A.B., Smith, J.D.H. Semilattice-based dualities. Stud Logica 56, 225–261 (1996). https://doi.org/10.1007/BF00370148
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DOI: https://doi.org/10.1007/BF00370148