Abstract
The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice L σ of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-Čech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; X Y denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of L P is determined, where P is a poset and L a bounded distributive lattice.
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References
BakerKirby A. and HalesAlfred W. (1974) From a lattice to its ideal lattice, Algebra Universalis 4, 250–258.
BalbesRaymond and DwingerPhilip (1974) Distributive Lattices, University of Missouri Press, Columbia, MO.
Bowen, Will (1981) Lattice theory and topology, D. Phil. Thesis, University of Oxford.
Cignoli, Roberto (1991) Distributive Lattice Congruences and Priestley Spaces, in Proceedings of the First “Dr. Antonio A. R. Monteiro” Congress on Mathematics, Universidad Nacional del Sur, Bahía Blanca, pp. 81–84.
CignoliR., LafalceS., and PetrovichA. (1991) Remarks on Priestley duality for distributive lattices, Order 8, 299–315.
CrawleyPeter and DilworthRobert P. (1973) Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, New Jersey.
DaveyB. A. and PriestleyH. A. (1990) Introduction to Lattices and Order, Cambridge University Press, Cambridge.
DaveyBrian A. and RivalIvan (1982) Exponents of lattice-ordered algebras, Algebra Universalis 14, 87–98.
Farley, Jonathan David (1994) Priestley powers of lattices and their congruences: a problem of E. T. Schmidt, Acta Scientiarum Mathematicarum, to appear.
GehrkeMai and JónssonBjarni (1994) Bounded distributive lattices with opertors, Mathematica Japonica 40, 207–215.
GierzG., HofmannK. H., KeimelK., LawsonJ. D., MisloveM., and ScottD. S. (1980) A Compendium of Continuous Lattices, Springer-Verlag, Berlin.
GrafSiegfried (1977) A selection theorem for Boolean correspondences, J. Reine Angew. Math. 295, 169–186.
GrätzerGeorge (1978) General Lattice Theory, Academic Press, New York.
HalmosPaul R. (1954) Algebraic logic I: monadic Boolean algebras, Compositio Mathematica 12, 217–249.
HansoulG. (1983) The Stone-Čech compactification of a pospace, Coll. Math. Soc. János Bolyai 43, 161–176.
HunsakerWorthen N. (1980) Extensions of continuous increasing functions, Topology Proc. 5, 105–110.
KelleyJ. C. (1963) Bitopological spaces, Proc. London Math. Soc. 13, 71–89.
LawsonJimmie D. (1988) The versatile continuous order, in: M.Main, A.Melton, M.Mislove, and D.Schmidt (eds), Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science 298, Springer-Verlag, Berlin, pp. 134–160.
McDowellRobert H. (1958) Extensions of functions from dense subspaces, Duke Math. J. 25, 297–304.
NachbinLeopoldo (1965) Topology and Order, D. Van Nostrand, Princeton.
PriestleyH. A. (1972) Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 24, 507–530.
PriestleyH. A. (1984) Catalytic distributive lattices and compact zero-dimensional topological lattices, Algebra Universalis 19, 322–329.
PriestleyH. A. (1984) Ordered sets and duality for distributive lattices, in MauricePouzet and DenisRichard (eds), Orders: Description and Roles, Annals of Discrete Mathematics 23, North-Holland, Amsterdam, pp. 39–60.
Salbany, Sergio (1974) Bitopological Spaces, Compactifications and Completions, The University of Cape Town.
SchmidtJürgen (1972) Universal and internal properties of some extensions of partially ordered sets, J. Reine Angew. Math. 253, 28–42.
StoneM. H. (1936) The theory of representations for Boolean algebras, Tran. Amer. Math. Soc. 40, 37–111.
WrightFred B. (1957) Some remarks on Boolean duality, Portugaliae Mathematica 16, 109–117.
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Communicated by Ivan Rival
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Farley, J.D. Priestley duality for order-preserving maps into distributive lattices. Order 13, 65–98 (1996). https://doi.org/10.1007/BF00383968
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DOI: https://doi.org/10.1007/BF00383968