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Priestley spaces
Author:
P. Venugopalan
Journal:
Proc. Amer. Math. Soc. 109 (1990), 605-610
MSC:
Primary 06B35; Secondary 06B30, 54F05
MathSciNet review:
1013985
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: We give a purely order-theoretic characterization of complete lattices that are compact totally order-disconnected (Priestley) spaces with respect to the Lawson topology. We also characterize complete lattices that are Priestley spaces with respect to the interval topology.
- [1]
Gerhard
Gierz, Karl
Heinrich Hofmann, Klaus
Keimel, Jimmie
D. Lawson, Michael
W. Mislove, and Dana
S. Scott, A compendium of continuous lattices,
Springer-Verlag, Berlin, 1980. MR 614752
(82h:06005)
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G.
Gierz and J.
D. Lawson, Generalized continuous and hypercontinuous
lattices, Rocky Mountain J. Math. 11 (1981),
no. 2, 271–296. MR 619676
(82h:54069), http://dx.doi.org/10.1216/RMJ-1981-11-2-271
- [3]
G.
Gierz, J.
D. Lawson, and A.
Stralka, Quasicontinuous posets, Houston J. Math.
9 (1983), no. 2, 191–208. MR 703268
(85b:06009)
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M.
Hochster, Prime ideal structure in commutative
rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. MR 0251026
(40 #4257), http://dx.doi.org/10.1090/S0002-9947-1969-0251026-X
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H.
A. Priestley, Representation of distributive lattices by means of
ordered stone spaces, Bull. London Math. Soc. 2
(1970), 186–190. MR 0265242
(42 #153)
- [6]
H.
A. Priestley, Ordered topological spaces and the representation of
distributive lattices, Proc. London Math. Soc. (3) 24
(1972), 507–530. MR 0300949
(46 #109)
- [7]
Maurice
Pouzet and Denis
Richard (eds.), Orders: description and roles, North-Holland
Mathematics Studies, vol. 99, North-Holland Publishing Co., Amsterdam,
1984. Annals of Discrete Mathematics, 23. MR 779841
(85k:06001)
- [8]
Hilary
A. Priestley, Algebraic lattices as dual spaces of distributive
lattices, Continuous lattices and their applications (Bremen, 1982)
Lecture Notes in Pure and Appl. Math., vol. 101, Dekker, New York,
1985, pp. 237–249. MR 826005
(87i:06042)
- [9]
Albert
Stralka, A partially ordered space which is not a Priestley
space, Semigroup Forum 20 (1980), no. 4,
293–297. MR
583112 (82f:54051), http://dx.doi.org/10.1007/BF02572690
- [10]
P.
Venugopalan, Quasicontinuous posets, Semigroup Forum
41 (1990), no. 2, 193–200. MR 1057590
(91i:06018), http://dx.doi.org/10.1007/BF02573390
- [11]
P.
Venugopalan, A generalization of completely distributive
lattices, Algebra Universalis 27 (1990), no. 4,
578–586. MR 1387903
(97k:06019), http://dx.doi.org/10.1007/BF01189001
- [1]
- G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, A compendium of continuous lattices, Springer-Verlag, Berlin, Heidelberg, and New York, 1980. MR 614752 (82h:06005)
- [2]
- G. Gierz and J. D. Lawson, Generalized continuous and hypercontinuous lattices, Rocky Mountain J. Math. (1981), 271-296. MR 619676 (82h:54069)
- [3]
- G. Gierz, J. D. Lawson, and A. Stralka, Quasicontinuous posets, Houston J. Math. 9 (1983), 191-208. MR 703268 (85b:06009)
- [4]
- M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. MR 0251026 (40:4257)
- [5]
- H. A. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186-190. MR 0265242 (42:153)
- [6]
- -, Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 24 (1972), 507-530. MR 0300949 (46:109)
- [7]
- -, Orders: description and roles (M. Pouzet and M. Richard, eds.), Annals of Discrete Math., North-Holland, Amsterdam, 1984, pp. 39-60. MR 779841 (85k:06001)
- [8]
- H. A. Priestley, Algebraic lattices as duals of distributive lattices, Lecture Notes in Pure and Applied Math., vol. 101, 1985, pp. 237-249. MR 826005 (87i:06042)
- [9]
- A. R. Stralka, A partially ordered space which is not a Priestley space, Semigroup Forum 20 (1980), 293-297. MR 583112 (82f:54051)
- [10]
- P. Venugopalan, Quasicontinuous posets, Semigroup Forum (to appear). MR 1057590 (91i:06018)
- [11]
- -, A generalization of completely distributive lattices, Algebra Universalis (to appear). MR 1387903 (97k:06019)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1990-1013985-3
PII:
S 0002-9939(1990)1013985-3
Keywords:
Complete lattice,
Priestley space,
Lawson topology,
interval topology
Article copyright:
© Copyright 1990 American Mathematical Society
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