In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 topoi.
Definition
Let ''X'' be a topological ...
s, and
pairwise Stone space
In mathematics and particularly in topology, a pairwise Stone space is a bitopological space \scriptstyle (X,\tau_1,\tau_2) that 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, 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 ...
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 Hausdorff totally disconnected space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them i ...
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
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
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on is generated by . The spectral space is called the ''prime spectrum'' of .
The
map
A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
is a lattice
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
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'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
subsets of . In fact, each spectral space is
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
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, a pairwise Stone space is a bitopological space \scriptstyle (X,\tau_1,\tau_2) that 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 counterintuitive, as the common meanings of and are antonyms, but their mathematical def ...
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
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 equivalences
[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
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).''
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice ...
*
Stone's representation theorem for Boolean algebras
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first ha ...
*
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 ...
*
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.
*
{{refend
Topology
Category theory
Lattice theory
Duality theories