mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via

Priestley space In mathematics, a Priestley space is an partial order, ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study ...

spectral space In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Definition Let ''X'' be a topological ...

s, and

pairwise Stone space In mathematics and particularly in topology, pairwise Stone space is a bitopological space \scriptstyle (X,\tau_1,\tau_2) which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional. Pairwise Stone spaces are a bitopological v ...

s. This duality, which is originally also due to

Marshall H. Stone Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras. Biography Stone was the son of Harlan Fiske Stone, who wa ...

, 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 space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in t ...

s 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 in e ...

s. Let be a bounded distributive lattice, and let denote the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of prime filters of . For each , let . Then is a spectral space, where the

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

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

map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...

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 is ...

from onto the lattice of all

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...

subsets of . In fact, each spectral space is

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

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

. 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

. 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 set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but ...

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 A university press is an academic publishing hou ...

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

See also

Notes

References