Английская Википедия: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,[1] generalizes the well-known Stone duality between Stone spaces and Boolean algebras.
Let Шаблон:Math be a bounded distributive lattice, and let Шаблон:Math denote the set of prime filters of Шаблон:Math. For each Шаблон:Math, let Шаблон:Math. Then Шаблон:Math is a spectral space,[2] where the topology Шаблон:Math on Шаблон:Math is generated by Шаблон:Math. The spectral space Шаблон:Math is called the prime spectrum of Шаблон:Math.
The map Шаблон:Math is a lattice isomorphism from Шаблон:Math onto the lattice of all compact open subsets of Шаблон:Math. In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice.[3]
Similarly, if Шаблон:Math and Шаблон:Math denotes the topology generated by Шаблон:Math, then Шаблон:Math is also a spectral space. Moreover, Шаблон:Math is a pairwise Stone space. The pairwise Stone space Шаблон:Math is called the bitopological dual of Шаблон:Math. Each pairwise Stone space is bi-homeomorphic to the bitopological dual of some bounded distributive lattice.[4]
Finally, let Шаблон:Math be set-theoretic inclusion on the set of prime filters of Шаблон:Math and let Шаблон:Math. Then Шаблон:Math is a Priestley space. Moreover, Шаблон:Math is a lattice isomorphism from Шаблон:Math onto the lattice of all clopen up-sets of Шаблон:Math. The Priestley space Шаблон:Math is called the Priestley dual of Шаблон:Math. Each Priestley space is isomorphic to the Priestley dual of some bounded distributive lattice.[5]
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[6] 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. Шаблон:ISBN.
- 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.
- Шаблон:Cite book
