In
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 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 of distributiv ...
s,
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 ...
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 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 ...
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 i ...
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 ...
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 i ...
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'' ( ...
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
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 ...
. 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
In mathematics, a Priestley space is an 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 of distributiv ...
. 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-sets 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. Then the above three representations of bounded distributive lattices can be extended to
dual equivalence[Bezhanishvili 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.
See also
*
Representation theorem
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.
Examples
Algebra
* Cayley's theorem states that every group i ...
*
Birkhoff's representation theorem
*
Stone's representation theorem for Boolean algebras
*
Stone duality
*
Esakia duality In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces.
Let Esa denote t ...
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 King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pre ...
, series=New Mathematical Monographs , volume=35 , location=Cambridge , isbn=9781107146723
Topology
Category theory
Lattice theory
Duality theories