Duality for r.e. sets with applications V. Yu. Shavrukov v.yu.shavrukov@gmail.com Dagstuhl 15441, 2015–10 Duality for r.e. sets with applications Duality for r.e. sets with applications V. Yu. Shavrukov v.yu.shavrukov@gmail.com Dagstuhl 15441, 2015–10 • Good morning. • First of all, I have to offer my apologies to those members of the audience who have heard fragments of my talk on previous occasions. • We are going to talk about the lattice of r.e. sets and how classical Priestley duality for bounded distributive lattice applies to it. • Another important ingredient will be non-standard models arithmetic — we use those to represent individual prime filters on the lattice. • Rather than opt for a comprehensive survey, I am going to briefly cover some prerequisite basics and focus on a particular application. • That application, in my view, supports the thesis that duality can play more than just decorative role with respect to traditional questions in the theory of r.e. sets. • You can see how a point like that can be important on some kind of personal level to a proponent of duality methods.
dual powers ef21 skies 1sky ciao ( E ∗ ) ⋆ E = � r.e. subsets of ω, ∪ , ∩ , ∅ , ω ) — the lattice of r.e. sets E ∗ = E / fin ( E ∗ ) ⋆ = � the set of prim(e filter)s on E ∗ , ⊆ , π � — the Priestley dual larger prime filters smaller prime filters X ⋆ is the picture of X . p ∈ X ⋆ Let X be r.e. p ∋ X ⇐⇒ X ⋆ is ↑ -closed. R ⋆ is � -closed ⇐⇒ R is recursive. ⊆ is ↑ -forestlike on ( E ∗ ) ⋆ . Observation. Reduction Principle = ⇒ Duality for r.e. sets with applications ( E ∗ ) ⋆ E = � r.e. subsets of ω, ∪ , ∩ , ∅ , ω ) — the lattice of r.e. sets The dual E ∗ = E / fin ( E ∗ ) ⋆ = � the set of prim(e filter)s on E ∗ , ⊆ , π � — the Priestley dual ( E ∗ ) ⋆ larger prime filters smaller prime filters X ⋆ is the picture of X . Let X be r.e. p ∋ X ⇐⇒ p ∈ X ⋆ X ⋆ is ↑ -closed. R ⋆ is � -closed ⇐⇒ R is recursive. ⊆ is ↑ -forestlike on ( E ∗ ) ⋆ . Observation. Reduction Principle = ⇒ • The lattice E of r.e. sets consists of all recursively enumerable subsets of ω together with the set theoretical operations. Which makes it a bounded distributive lattice. • When you quotient this by finite differences between r.e. sets, you get the lattice called E ∗ . • ( E ∗ ) ⋆ is then the classical Priestley space of E ∗ . That’s the ordered topological space comprised by the collection of all prime filters on the lattice of r.e. sets which we are going to just call primes or even points , together with the order relation of inclusion and an appropriate topology. • Note that the two asterisks in the notation for the dual are very different: The inner one says goodbye to finite differences while the outer asterisk gets you from the lattice to the dual space. • When you attempt to visualize this space, you probably get a picture like this. • We think of the smaller primes going towards the bottom, and the larger primes, towards the top. • Given an r.e. set X , we can draw the picture of this set and call it X -star . It consists of all primes that contain the set X . We tend to identify it with the r.e. set X . • We have an equivalence between a prime filter p containing X and the same prime filter, now thought as a point in he dual space, lying within the picture of X . This is probably the key slogan of pictorial duality. • Thus one way to think of the dual space is as a canvas for a kind of Venn diagrams. • Observe that the picture of any r.e. set is upward closed, for once an r.e. set belongs to some prime filter, it has to belong to all larger ones. • The picture of a recursive set is the both upward and downward closed, because the complement of a recursive set is r.e. and must therefore also be upward closed. • Finally, let us point out that the inclusion ordering is upwards-forestlike — this is an easy consequence of the Reduction Principle for r.e. sets.
dual powers ef21 skies 1sky ciao Maximal, r-maximal, and hyperhypersimple r.e. sets M ⋆ M is maximal ⇐⇒ M ⋆ is a singleton Q is r-maximal ⇐⇒ min Q ⋆ is a singleton. Q ⋆ H ⋆ H is hhsimple ⇐⇒ H ⋆ ⊆ min ( E ∗ ) ⋆ Duality for r.e. sets with applications Maximal, r-maximal, and hyperhypersimple r.e. sets The dual M ⋆ M is maximal ⇐⇒ M ⋆ is a singleton Maximal, r-maximal, and hyperhypersimple r.e. sets Q is r-maximal ⇐⇒ min Q ⋆ is a singleton. Q ⋆ H ⋆ H is hhsimple ⇐⇒ H ⋆ ⊆ min ( E ∗ ) ⋆ • We are now going to look at pictures of typical representatives of some familiar classes of r.e. sets. • An r.e. set M is maximal if any r.e. superset of M only differs finitely from either ω or M itself. This is equivalent to the complement of the picture of M only having a single element — you cannot split that complement by any r.e. set into two infinite halves — that single point is either in or out. • Maximality can be generalized in two orthogonal directions. • An r.e. set Q is r-maximal if its picture covers all minimal points of the dual space except one, which is then called the heel of that set. • The original definition is that no recursive set splits the complement of Q into two infinite halves — the complement is the white area in the picture. I.o.w., that complement is r-cohesive . You cannot split it by any picture that is both upwards and downwards closed. If there was more than one point at the bottom of the complement, you could easily split those apart. • Finally, an r.e. set H is hyperhypersimple if the complement of its picture consists of minimal points of the dual space only. The complement thus forms a single-element-wide strip at the bottom of the dual space. • Hyperhypersimple sets were introduced by Emil Post, as the next step after simple and hypersimple sets. These are classes of sets with smaller and smaller complements, and I guess we should be thankful that it was not him who invented the maximal sets.
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