Duality theory for distributive lattices
Encyclopedia
In mathematics
, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices
via Priestley space
s, spectral space
s, and pairwise Stone space
s. This generalizes the well-known Stone duality
between Stone spaces and Boolean algebras.
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 on is generated by . The spectral space is called the prime spectrum of .
The map
is a lattice isomorphism
from onto the lattice of all compact 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
. 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 equivalence
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
via Priestley space
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 distributive lattices...
s, spectral space
Spectral space
In mathematics, a spectral space is a topological space which is homeomorphic to the spectrum of a commutative ring.-Definition:Let X be a topological space and let K\circ be the set of allquasi-compact and open subsets of X...
s, and pairwise Stone space
Pairwise Stone space
In mathematics and particularly in topology, pairwise Stone space is a bitopological space \scriptstyle which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional....
s. This generalizes the well-known Stone duality
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 form a natural generalization of Stone's representation...
between Stone spaces and Boolean algebras.
Let be a bounded distributive lattice, and let denote the set of prime filters
Ideal (order theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set . Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion...
of . For each , let . Then is a spectral space, where the topology on is generated by . The spectral space is called the prime spectrum of .
The map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
is a lattice isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
from onto the lattice of all compact open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
subsets of . In fact, each spectral space is homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
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
Pairwise Stone space
In mathematics and particularly in topology, pairwise Stone space is a bitopological space \scriptstyle which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional....
. 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
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 distributive lattices...
. Moreover, is a lattice isomorphism from onto the lattice of all clopen
Clopen set
In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
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
Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...
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
- Birkhoff's representation theoremBirkhoff's representation theoremIn mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets...
- Stone dualityStone dualityIn 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 form a natural generalization of Stone's representation...
- Stone's representation theorem for Boolean algebrasStone's representation theorem for Boolean algebrasIn mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Stone...
- Esakia dualityEsakia dualityIn 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....