Duality Theory For Distributive Lattices

In mathematics, 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 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 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.

See also

*Representation theorem
* Birkhoff's representation theorem
* Stone's representation theorem for Boolean algebras
* 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.
* {{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 , series=New Mathematical Monographs , volume=35 , location=Cambridge , isbn=9781107146723

Topology
Category theory
Lattice theory
Duality theories