Do any Stone-like dualities have some self-dualities hidden inside them?

Question

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures of some $D^S$ (a product of $D$ with itself many times), so they both carry some topology induced from the Tychonoff topology on the product, although this topology is usually discarded on one of the sides.

I understand relatively well when in the Pontryagin case the dual of a compact abelian group sits inside a power of $D$ as a discrete subgroup, so the topology honestly disappears. But this is relatively rare, right? A discrete subgroup of a compact group must be finite, so any infinite abelian group acquires some canonical non-discrete topology in this way, what is this topology? In any case, on one hand the category of finite abelian groups becomes self-dual and on the other hand the category of locally compact abelian groups becomes self-dual. After that it is easy to name additional structure on the level of individual objects which makes them self-dual.

There is a vast labyrinth of dualities for vector spaces which I understand much worse but somehow realize that in the end of the analog of the above paragraph one arrives at something like Hilbert spaces. But I have no idea what is the analog of Hilbert spaces for some of the versions, e. g. for the locally linearly compact vector spaces in the sense of Lefschetz. Some relatively recent work on the latter by Kapranov is very interesting in this respect but I do not really understand it well.

And I am at complete loss with things like Stone, Priestley and Esakia dualities for Boolean algebras, distributive lattices and Heyting algebras. What might be self-dual objects there? What is the topology on a distributive lattice $L$ induced from the embedding $L\hookrightarrow2^{\textrm{prime filters of }L}$? Or is it more appropriate to consider the topology induced from the Vietoris space of the dual? Or are these actually the same? On the other side, -- what are those Priestley spaces $(X,\leqslant)$ which happen to be distributive lattices with respect to $\leqslant$? In short,

In what sense may one speak of self-dual objects in context of Priestley and other Stone-like dualities? Is there any self-dual category at all either including or included in both Priestley spaces and distributive lattices?

There is some kind of analog of locally compact abelian group thing in this situation - the self-dual category of complete join-semilattices, but here also I do not understand why has topology disappeared altogether. There is an "induced-Stone" topology on each complete join-semilattice, why is it not in the game here? Also, what are self-dual objects in this category?

For rings, a similar question leads to looking at affine ring schemes, Hopf ring schemes and alike. Are there any real-life examples of self-dual Hopf-ring schemes?

What about Gelfand duality in this context?

Does anybody have further examples of dualities where one may either restrict or enlarge to a self-dual category, and then self-dual objects give some known and meaningful mathematical objects?

(added slightly later) Somehow the adele construction looks like relevant. It embeds a (number or function) field $F$ into a self-dual locally compact ring $\mathbb A_F$ as a discrete subfield. Then self-duality of this object seems to play important rôle in class field theory and arithmetic algebraic geometry. However I cannot quite figure out where does this self-duality take place. On the surface of it, $\mathbb A_F$ is just self-dual as an abelian group wrt addition. The multiplicative part (including the group of ideles) is somehow separate...

Answer 1

any infinite abelian group acquires some canonical non-discrete topology in this way, what is this topology?

The maximal precompact topology (used to define almost-periodic functions). See section 9.9 "Bohr topology on discrete groups" pp. 633-662 in "Topological Groups and Related Structures" by Alexander Arhangel'skii, Mikhail Tkachenko.

something like Hilbert spaces

also pre-Hilbert, or generally spaces with a (possibly even with non - symmetric orthogonality) nondegenerate sesquilinear form. (Not only in the locally convex case, or more generally for locally compact fields of scalars, but also in the Lefschetz case, i.e. duality when the skew field of scalars is discrete and in the vector spaces there is a basis at 0 of finite-codimensional subspaces).

This can be seen from many equivalent points of view: M. Pertich, Categories of algebraic systems, Springer LNM 554. (One has many crypto-equivalent definitions of structures only slightly weaker than dual pairs of vector spaces; the isomorphisms of such structures naturally correspond to proportionality classes of semi-linear isomorphisms among vector spaces). Among the many equivalent points of view, I prefer the point of view of complete irreducible DAC lattices [see F. Maeda, S. Maeda, theory of symmetric lattices], where self-duality of an object corresponds to lattice anti automorphism i.e., in classical geometric terms, a correlation of projective geometries.

(The first trick described by barcelos works in full generality, giving the "exchange" involution, corresponding to the exchange involution on a direct product ring $R\times R^{op}$ and analogoulsy for lattices; in ring and lattice theory is is considered a "degenerate" kind of involution)

Notable, somehow extreme and strange, cases: Ornstein's dual pairs (following the founding works of Mackey and Kaplansky) with some modular (but not projective or dually) subcases; Keller's orthomodular (but not Hilbert) spaces.

There is a common generalisation (of vector spaces dualities and Pontryagin - van Kampen duality) to modules over (suitable, topological) rings Flood; Orsatti, Menini, Bazzoni, \dots]. Google scholar search for "duality ring modules Pontryagin"

Morita dualities (for suitable artinian, or only perfect, rings) can be seen in this "linearly compact" context. These dualities are induced by a bimodule, which has a role like the unit circle for locally compact abelian groups or the (skew) field of scalars for vector spaces.

One can consider also dualities between more general abelian categories of modules; in this context I like the WQF (weakly quasi Frobenius) a.k.a. IF (injectives are flat) rings, i.e. the coherent rings such that the usual dual (Hom to the ring of scalars) gives a duality of the abelian categories of finitely presented right and left modules. This can be seen equivalently in terms of Hutchinson (modular) lattices of the abelian categories; WQF rings such that intervals in this lattice are complemented (resp. noetherian or artininan or finite length) are exactly the von Neumann regular rings (resp. quasi-Frobenius rings).

[Note that in these last cases no topology is used, since all considered modules are finitely presented: in these (and the following) algebraic contexts, topology only usefully appears when also infinite direct products, or dually sums, make sense; then these topologies are 0-dimensional, being subspaces of infinite direct products of discrete spaces (the algebraic object that induces the duality, and that need not be self-dual, has a discrete topology in purely algebraic contexts). Analogously, in the theory of field extensions, a topology usefully appears in the Galois group only for infinite extensions, and the group is pro-finite]

What is the topology on a distributive lattice $L$ induced from the embedding $L\hookrightarrow 2^{\textrm{prime filters of }L}$? Or is it more appropriate to consider the topology induced from the Vietoris space of the dual? Or are these actually the same?

Homomorphisms to $\{0,1\}$ (prime ideals are their kernels) i.e. dispersion free (finitely additive probability) measures i.e. extreme points of the compact convex set of (finitely additive probability) measures are the same for a distributive lattice and its freely generated Boolean algebra. This explains why in Priestley representation inside a compact T$_2$ space (the above extreme points, closed subset of the compact convex set and generating it in the Krein - Millman way) one needs additional structure (the specialisation order, coming from the original T$_0$ spectral space of Stone, which is crypto - equivalent to the totally order disconnected T$_2$ compact space: the Stone advantage is that no extra-topological structure is needed, the disadvantage is that is is not T$_2$ in the non Boolean case).

In the Boolean case, a variable clopen weakly converges (in this duality with measures, or even only the delta measures) to a fixed clopen iff it has eventually the same points (in the Stone space); it is the same as the convergence induced on clopen sets by the Vietoris topology on the space of closed sets of the compact T$_2$ space (may "hyperspace" topologies are known and useful for topological spaces, but for compact metrisable spaces they all coincide). For sequences, you see the traditional LimInf and LimSup of sequences of sets, and generally one of the traditional intrinsic topologies in a ordered set (see below).

the self-dual category of complete join-semilattices, but here also I do not understand why has topology disappeared altogether.

Various convergences (and topologies) are natural and useful in complete lattices: see Birkoff's book and the compendium of continuous lattices and its successor book (also partially useful in the context of these dualities). [Warning: except the completely distributive case (complete sub lattices of direct products of complete chains), some of the important natural convergences are not topological.] They do not disappear, but they are implicit and more than one.

things like Stone, Priestley and Esakia dualities

In general: natural dualities (Davey). Google scholar search with "Pontryagin Priestley duality" gives useful introductions to the subject, such as http://www.researchgate.net/publication/242014244_Duality_Theory_on_Ten_Dollars_a_Day/file/60b7d51cba47a1d37e.pdf

In what sense may one speak of self-dual objects in context of Priestley and other Stone-like dualities? Is there any self-dual category at all either including or included in both Priestley spaces and distributive lattices?

in many cases, possibly almost none. See below.

What about Gelfand duality in this context?

Yes, it is included [in terms of probability measures these are the Radon measures, the Dirac's deltas being the extreme points, the compact convex set being the unit ball in the weak topology of the duality. This way of looking at the duality extends to the non commutative case: looking at homomorphisms to the ring of scalars is insufficient in the general case, and using representations in Hilbert spaces gives various notions of "points" (equivalence classes of suitable representations): the above ones, via GNS representation; the primitive spectrum; the closed prime spectrum]

The Gelfand case (which includes Stone duality) is a typical case that shows how little sense can sometimes have a hunt for self-duality in this context: on one side of the duality one has (suitable, complete, semiprimitive, commutative) normed algebras; on the other side one has locally compact spaces, and a normed space is locally compact iff finite dimensional, which for complex commutative semiprimitive algebras means finite direct product of copies of the complex field, so its dual is a finite discrete space, so it cannot be a complex algebra (unless 0).

what are those Priestley spaces $(X,\leqslant)$ which happen to be distributive lattices with respect to $\leqslant$?

Here I consider only the Priestley space of finite distributive lattices (but algebraic dually algebraic distributive lattices can be treated in the same way). Here Priestley duality, once restated as spectral duality (the original form by Stone), reduces to the Birkhoff transform: the equivalence between finite posets, finite T$_0$ (topological, even T$_D$ and sober) spaces, finite distributive lattices. [The poset is the poset of join-irreducible elements of the lattice; the order on the poset is the specialisation order of the topology and the subposet order from the lattice; the lattice is the lattice of closed sets of the spectral topology, the points are the dense points of the irreducible closed sets]. So you have (finite) distributive lattices which are the lattices of all order ideals in a (finite) distributive lattice. No Boolean algebras except $\{0,1\}$ (and $0=1$) are of this form, since (complete atomic) Boolean algebras are the distributive lattices that in Birkhoff transform correspond to posets that are antichains. Generally, no self-dual object is obtained except finite chains, since chains are exactly the distributive lattices where each element is join-irreducible.

Finite chains are so special than it is perhaps better to look at them not simply as distributive lattices, but as more structured objects: as truth values generalising $0,1$ one looks at them as Post algebras, then Chang MV algebras (equivalently, by Mundici's equivalence, as unit intervals in lattice ordered abelian groups with fixed strong unit). As lattices, the natural "noncommutative" generalisation of finite chains are semiprimary lattices (someone instead likes Elliott's theorem that two AF $C^*$-algebras with a natural lattice order on dimensions [the Murray - von Neumann classes of projections] are isomorphic iff the associated MV algebras of dimensions are isomorphic. However, this theorem is not a equivalence: no natural composition preserving bijection is given for isomorphisms on the two sides). These very special modular lattices of finite length were studied by Baer, Inaba, J\'onsson - Monk culminating in their coordinatization theorem (in the primary decomposable case).

The typical examples of primary decomposable lattices are: lattice of subgroups of finite abelian groups (including the theory of elementary divisors of a matrix with integer entries); lattice of invariant subspaces for a single semi-linear endomorphisms of a finite dimensional vector space (including rational and Jordan form of matrices over fields); more generally, lattices of all submodules of finitely generated modules over artinian principal ideal rings (including all torsion finitely generated modules over principal ideal domains and Dedekind domains, since any proper homomorphic image of these is an artinian principal ideal ring). In terms of the Hutchinson lattice as above, the artinian rings such as the Hutchinson lattice of the abelian category of finitely generated modules has semiprimary (resp. primary decomposable) intervals are exactly the artinian serial rings (resp. artinian principal ideal rings); the abelian categories of right and left finitely generated modules are then in Morita duality (but a dualizing bimodule is the ring of scalars only in the quasi Frobenius case, which includes the principal ideals case but not every serial case). [There is also a theory for finitely presented modules over non artinian serial rings, including all discrete valuation domains, and even generic valuation domains; also related: every proper homomorphic image of a Pr\"ufer domain is a WQF ring]

If there are cases of "natural dualities" where a self-duality is known and useful, Davey et. al. should have them. Perhaps start with Theorem 3.15 (The Strong Self-Duality Theorem) in the "ten dollars" survey cited above, then consider endodualisability

But it seems to me that the most useful path to find self-dualities is in the "analytic" Pontryagin vein (incidentally, Hopf algebras appear when dualizing locally compact noncompact nonabelian groups), not in the "algebraic" Stone vein.

Answer 2

Not sure if this is relevant to your question but there are two natural and well known ways to get self-dual spaces in the category of locally convex spaces, namely objects of the form $E \times E'$ (the cartesian product of a lcs with its dual---this works for reflexive spaces) and the tensor product $E \otimes E'$ (which works for nice nuclear spaces, say Frechet (can't get the accent right) or $DF$---due to the nuclearity one doesn't have to specify which tensor product is being used). The latter space can be identified with the operator space $L(E)$ with the strong topology in this situation.

Answer 3

In the Stone duality / Boolean algebra case, presumably the self-dual object is the two-element Boolean algebra. I guess we want to place the discrete topology on this. Then every Boolean algebra maps to an infinite product of these, and there is an induced topology by pullback. We can describe the open balls as inverse image of points in maps to finite Boolean algebras, since every finite Boolean algebra embeds in a finite product of the two-element Boolean algebra.

So this topology is a kind of "profinite" topology on the algebra. It's nontrivial, but I don't know any use for it. That is not very strong evidence for anything, though, as I don't know very much about Boolean algebras.