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 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 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. 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 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 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
*
Birkhoff's representation theorem
*
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
*
Esakia duality
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