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In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in :
- (reflexive).
- If and then (transitive).
- If and then (antisymmetric).
- or .
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In mathematics, duality theory for distributive lattices provides three different representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras.
In mathematics, in the area of order theory, an upwards centered setS is a subset of a partially ordered set, P, such that any finite subset of S has an upper bound in P. Similarly, any finite subset of a downwards centered set has a lower bound. An upwards centered set can also be called a consistent set. Any directed set is necessarily centered, and any centered set is a linked set.
Wilson Alexander Sutherland was a British mathematician at the University of Oxford.
Introduction to Lattices and Order is a mathematical textbook on order theory by Brian A. Davey and Hilary Priestley. It was published by the Cambridge University Press in their Cambridge Mathematical Textbooks series in 1990, with a second edition in 2002. The second edition is significantly different in its topics and organization, and was revised to incorporate recent developments in the area, especially in its applications to computer science. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it.
References
- 1 2 Hilary Priestley at the Mathematics Genealogy Project.
- ↑ Gardam, Tim (11 July 2006). "Titles of Distinction awarded to eight Fellows" . Retrieved 11 June 2014.
- ↑ Stralka, Albert (December 1980). "A partially ordered space which is not a priestley space". Semigroup Forum . 20 (1). Springer: 293–297. doi:10.1007/BF02572690. S2CID 123310469.
- ↑ Cignoli, R.; Lafalce, S.; Petrovich, A. (September 1991). "Remarks on Priestley duality for distributive lattices". Order . 8 (3). Springer: 299–315. doi:10.1007/BF00383451. S2CID 122146613.
- ↑ Reviews of Introduction to Lattices and Order: T. S. Blyth, MR 1058437, MR 1902334; Jonathan Cohen, ACM SIGACT News, doi : 10.1145/1233481.1233488; Amy Davidow, Amer. Math. Monthly, JSTOR 2323967; Josef Niederle, Zbl 0701.06001; Václav Slavík, Zbl 1002.06001
