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It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic problems using their representation in the correspondent topological space.

Natural duality theory provides a general framework that encompasses many important and well-known representation theorems in algebra. In fact, when dualities are useful, the duals are simpler structures: the dual of a free algebra is a direct exponent.

In Pontryagin duality, the structure of the dual space is the same when adding the topology. As the semilattice duality by Hofmann–Mislove–Stralka, the Pontryagin duality is for algebras for which the basic operations are homomorphisms. The same happens with vector spaces.

Gelfand Duality is a representation theorem for commutative Banach algebras describing them as algebras of continuous functions. Moreover, for commutative $C^{*}$-algebras, this representation is an isometric isomorphism. It is also a classical example of the contributions of lattice duality to topology.

A famous example is the proof of Tychonoff theorem without using the axiom of choice: if you pass from topological spaces over to the algebraic side and use locales, then one can show that a product of compact locales is compact, without choice (cf. Peter Johnstone's book "Stone spaces").

Otherwise, the main use of lattice dualities seems to be rather one-sided, i.e., (citing "Natural Dualities for the Working Algebraist") to translate algebraic problems, usually stated in an abstract symbolic language, into dual, topological problems, where geometric intuition comes to our help.

Are there other clear examples of problems of topological nature solved by lattice duality? Thank you in advance.

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    $\begingroup$ Pontryagin duality should not be referred to as a lattice duality since locally compact abelian groups do not have a lattice structure. $\endgroup$
    – Joseph Van Name
    Commented Oct 17, 2013 at 21:17

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The idea that these dualities are only used in the direction of proving algebraic results using topological spaces is not correct. Any sort of completion or compactification process (that I can think of at least) can be constructed using special knids of filters or ultrafilters on lattices. These constructions include the Stone-Cech compactification, the Hewitt realcompactification, the completion of a uniform space, the Smirnov compactification of a proximity space, and the completion of the hyperspace of a uniform space and other constructions. There are countless examples of using ultrafilters and filters on Boolean algebras and lattices to prove results about topology.

For example, one can obtain the characterization of weakly compact cardinals given in [1] using a Boolean algebraic characterization of when a uniform space generated by equivalence relations is supercomplete. This result says that a cardinal $\kappa$ is weakly compact if and only if the space $2^{\kappa}$ with an appropriate uniformity is supercomplete. Furthermore, one can use the same technique to obtain characterizations of strongly compact cardinals.

  1. Artico, Giuliano (I-PADV-PM); Marconi, Umberto (I-PADV-PM); Pelant, Jan (CZ-AOS) On supercomplete $\omega_\mu$-metric spaces. Bull. Polish Acad. Sci. Math. 44 (1996), no. 3, 299–310.
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Davey at the 1981 Banff Conference on Ordered Sets raised a similar issue (page 847 of Rival (ed.), Ordered Sets). He wanted a single example of a question about posets that could be solved using distributive lattices.

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Farley and Klippenstine claim that their 2009 solution to a problem from Stanley's 1986 text Enumerative Combinatorics, Volume I is such an example. See page 1112 of Farley and Klippenstine and page 156 of Stanley. Farley used similar techniques to solve a problem of Rosenberg (Farley, "Distributive Lattices of Small Width I: A question of Rosenberg from the 1981 Banff Conference on Ordered Sets," Mathematica Pannonica 24 (2013), 231-242).

In the 1982 Proceedings of the American Mathematical Society, Matatyahu Rubin and Miriam Cohen made a conjecture about partially-ordered topological spaces that Farley resolved using ideas related to distributive lattice duality (page 6 of Farley, "A Problem Attributed to Rado from Mirsky's 1971 Monograph Transversal Theory and a Conjecture from the 1982 Proceedings of the American Mathematical Society", Mathematica Pannonica 24 (2013), 3-14).

Farley also claims to solve a problem of Stanley about ranked posets using Priestley duality for distributive lattices. See page 299 of Farley, "Linear extensions of ranked posets, enumerated by descents. A problem of Stanley from the 1981 Banff Conference on Ordered Sets," Advances in Applied Mathematics 34 (2005), 295-312.

Farley's paper "Does the Endomorphism Poset $P^P$ Determine Whether a Finite Poset $P$ Is Connected? An issue Duffus raised in 1978” (Mathematica Bohemica 148 (2023), 435-446) also uses some ideas about distributive lattices in the proof of Theorem 16 to resolve this issue about posets.

Note: Please do not take my referring to Farley so much as a clue concerning my identity. I might very well be Barbra Streisand.

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