Abstract
Anyone who has ever worked with a variety \(\varvec{\mathscr {A}}\) of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one—but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category \(\varvec{\mathscr {X}}\) of such a variety, we give a characterisation, in terms of \(\varvec{\mathscr {X}}\), of finitely generated discriminator subvarieties of \(\varvec{\mathscr {A}}\). As an application of our characterisation, we give a new proof of Sankappanavar’s characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly’s characterisation of quasi-primal Ockham algebras.
Similar content being viewed by others
References
Adámek, J., H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: the Joy of Cats. Available at http://katmat.math.uni-bremen.de/acc.
Baker, K. A., and A. F. Pixley, Polynomial interpolation and the Chinese remainder theorem for algebraic systems, Math. Z. 143:165–174, 1975.
Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra. The Millennium Edition, 2012 Update. Available from https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.
Clark, D. M., and B. A. Davey, Natural Dualities for the Working Algebraist, Cambridge University Press, 1998.
Cornish, W. H., Monoids acting on distributive lattices, manuscript (invited talk at the annual meeting of the Austral. Math. Soc., May 1977).
Cornish, W. H., Antimorphic Action: Categories of Algebraic Structures with Involutions or Anti-endomorphisms, Research and Exposition in Mathematics, vol. 12, Heldermann, 1986.
Cornish, W. H., and P. R. Fowler, Coproducts of De Morgan algebras, Bull. Aust. Math. Soc. 16:1–13, 1977.
Cornish, W. H., and P. R. Fowler, Coproducts of Kleene algebras, J. Austral. Math. Soc. (Ser. A) 27:209–220, 1979.
Davey, B. A., Subdirectly irreducible distributive double p-algebras, Algebra Universalis 8:73–88, 1978.
Davey, B. A., and J. C. Galati, A coalgebraic view of Heyting duality, Studia Logica 75:259–270, 2003.
Davey, B. A., and M. Haviar, Applications of Priestley duality in transferring optimal dualities, Studia Logica 78:213–236, 2004.
Davey, B. A., L. T. Nguyen, and J. G. Pitkethly, Counting relations on Ockham algebras, Algebra Universalis 74:35–63, 2015.
Davey, B. A., and H. A. Priestley, Generalised piggyback dualities and applications to Ockham algebras, Houston J. Math. 13:151–198, 1987.
Davey, B. A., and H. A. Priestley, Optimal natural dualities, Trans. Amer. Math. Soc. 338:655–677, 1993.
Davey, B. A., and H. A. Priestley, Optimal natural dualities for varieties of Heyting algebras, Studia Logica 56:67–96, 1996.
Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, 2nd edn. Cambridge University Press, 2002.
Davey, B. A., V. J. Schumann, and H. Werner, From the subalgebras of the square to the discriminator, Algebra Universalis 28:500–519, 1991.
Esakia, L. L., Topological Kripke models, Soviet Math. Dokl. 15:147–151, 1974.
Foster, A. L., and A. Pixley, Semi-categorical algebras. I. Semi-primal algebras, Math. Z. 85:147–169, 1964.
Katriňák, T., The structure of distributive double p-algebras. Regularity and congruences, Algebra Universalis 3:238–246, 1973.
Martínez, N. G., The Priestley duality for Wajsberg algebras, Studia Logica 49:31–46, 1990.
Martínez, N. G., and H. A. Priestley, On Priestley spaces of lattice-ordered algebraic structures, Order 15:297–323, 1998.
McKenzie, R., and M. Valeriote, The Structure of Decidable Locally Finite Varieties, Progress in Mathematics, vol. 79. Birkhäuser, 1989.
Pixley, A. F., Functionally complete algebras generating distributive and permutable classes, Math. Z. 114:361–372, 1970.
Pixley, A. F., The ternary discriminator function in universal algebra, Math. Ann. 191:167–180, 1971.
Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2:186–190, 1970.
Priestley, H. A., Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24:507–530, 1972.
Priestley, H. A., Stone lattices: a topological approach, Fund. Math. 84:127–143, 1974.
Priestley, H. A., The construction of spaces dual to pseudocomplemented distributive lattices, Quart. J. Math. Oxford Ser. (2) 26:215–228, 1975.
Priestley, H. A., Ordered sets and duality for distributive lattices, in M. Pouzet and D. Richard (eds.), Orders: Description and Roles (L’Arbresle, 1982). North-Holland Math. Stud., vol. 99, pp. 39–60. North-Holland, 1984.
Priestley, H. A., Varieties of distributive lattices with unary operations. I., J. Austral. Math. Soc. Ser. A 63:165–207, 1997.
Priestley, H. A., and R. Santos, Varieties of distributive lattices with unary operations. II., Portugal. Math. 55:135–166, 1998.
Sankappanavar, H. P., Heyting algebras with dual pseudocomplementation, Pacific J. Math. 117:405–415, 1985.
Taylor, C. J., Discriminator varieties of double Heyting algebras, Rep. Math. Logic 51:3–14, 2016.
Urquhart, A., Distributive lattices with a dual homomorphic operation, Studia Logica 38:201–209, 1979.
Werner, H., Eine Charakterisierung funktional vollständiger Algebren, Arch. Math. (Basel) 21:381–385, 1970.
Werner, H., Discriminator-algebras. Algebraic Representation and Model Theoretic Properties, Studien zur Algebra und ihre Anwendungen, Band 6, Akademie, 1978.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Davey, B.A., Gair, A. Restricted Priestley Dualities and Discriminator Varieties. Stud Logica 105, 843–872 (2017). https://doi.org/10.1007/s11225-017-9713-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-017-9713-4
Keywords
- Priestley duality
- Quasi-primal algebra
- Semi-primal algebra
- Discriminator variety
- Cornish algebra
- Ockham algebra