In mathematics, duality theory for distributive lattices provides three different (but closely related) 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 In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they fo ...

between Stone spaces and

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

s. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let . Then is a spectral space, where the

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

on is generated by . The spectral space is called the ''prime spectrum'' of . The map is a lattice

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

from onto the lattice of all

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subsets of . In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice. Similarly, if and denotes the topology generated by , then is also a spectral space. Moreover, is a pairwise Stone space. The pairwise Stone space is called the ''bitopological dual'' of . Each pairwise Stone space is bi-homeomorphic to the bitopological dual of some bounded distributive lattice. Finally, let be set-theoretic inclusion on the set of prime filters of and let . Then is a Priestley space. Moreover, is a lattice isomorphism from onto the lattice of all

clopen In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...

up-set In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...

s of . The Priestley space is called the ''Priestley dual'' of . Each Priestley space is isomorphic to the Priestley dual of some bounded distributive lattice. Let Dist denote the category of bounded distributive lattices and bounded lattice

homomorphisms In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

. Then the above three representations of bounded distributive lattices can be extended to dual equivalenceBezhanishvili et al. (2010) between Dist and the categories Spec, PStone, and Pries of spectral spaces with spectral maps, of pairwise Stone spaces with bi-continuous maps, and of Priestley spaces with Priestley morphisms, respectively: Thus, there are three equivalent ways of representing bounded distributive lattices. Each one has its own motivation and advantages, but ultimately they all serve the same purpose of providing better understanding of bounded distributive lattices.

Notes

References

* Priestley, H. A. (1970). Representation of distributive lattices by means of ordered Stone spaces. ''Bull. London Math. Soc.'', (2) 186–190. * Priestley, H. A. (1972). Ordered topological spaces and the representation of distributive lattices. ''Proc. London Math. Soc.'', 24(3) 507–530. * Stone, M. (1938)
Topological representation of distributive lattices and Brouwerian logics.
''Casopis Pest. Mat. Fys., 67 1–25. * Cornish, W. H. (1975). On H. Priestley's dual of the category of bounded distributive lattices. ''Mat. Vesnik'', 12(27) (4) 329–332. * M. Hochster (1969). Prime ideal structure in commutative rings. ''Trans. Amer. Math. Soc.'', 142 43–60 * Johnstone, P. T. (1982). ''Stone spaces''. Cambridge University Press, Cambridge. . * Jung, A. and Moshier, M. A. (2006). On the bitopological nature of Stone duality. ''Technical Report CSR-06-13'', School of Computer Science, University of Birmingham. * Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A. (2010). Bitopological duality for distributive lattices and Heyting algebras. ''Mathematical Structures in Computer Science'', 20. * {{cite book , last1=Dickmann , first1=Max , last2=Schwartz , first2= Niels , last3=Tressl , first3= Marcus , title=Spectral Spaces , year=2019 , doi=10.1017/9781316543870 , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

, series=New Mathematical Monographs , volume=35 , location=Cambridge , isbn=9781107146723 Topology Category theory Lattice theory Duality theories

See also

Notes

References