The role of the overlap relation in constructive mathematics

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The role of the overlap relation in constructive mathematics Francesco Ciraulo Department of Mathematics and Computer Science University of PALERMO (Italy) ciraulo@math.unipa.

1 The role of the overlap relation in constructive mathematics Francesco Ciraulo Department of Mathematics and Computer Science University of PALERMO (Italy) ciraulo Constructive Mathematics: Proofs and Computation Fraueninsel, 7-11 June 2010 Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 1/25

2 Overview 1 Overlap algebras & Locales Overlap Algebras and Overt Locales Atoms and Points Regular opens and Dense sublocales Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 2/25

3 Overview 1 Overlap algebras & Locales Overlap Algebras and Overt Locales Atoms and Points Regular opens and Dense sublocales 2 Overlap relation & Dedekind-MacNeille completion The overlap relation for non-complete posets A variation on the Dedekind-MacNeille completion Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 2/25

4 Part I Overlap Algebras & Locales In which the definition of overlap algebra is recalled as well as some well and less well known facts connecting overlap algebras, overt locales and regular open sets. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 3/25

5 Problem: fill in the blank with a suitable ALGEBRAIC STRUCTURE in such a way that THE DIAGRAM COMMUTES. INTUITIONISTIC world CLASSICAL world Powersets wear classical glasses Powersets algebraize algebraize wear classical glasses cba s Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 4/25

6 Problem: fill in the blank with a suitable ALGEBRAIC STRUCTURE in such a way that THE DIAGRAM COMMUTES. INTUITIONISTIC world CLASSICAL world Powersets wear classical glasses Powersets algebraize algebraize Overlap Algebras wear classical glasses cba s Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 4/25

7 Overlap Algebras = complete lattice + >< binary relation s.t.: x >< y x >< y y z y >< x symmetry x >< z monotonicity x >< y x >< ( x >< (x y) refinement i I y i) ( i I )(x >< y i ) splitting [z >< x].. z >< y x y density Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 5/25

8 Overlap Algebras as Overt Locales A characterization Idea: read x >< y as Pos(x y). 1 overt locales o-algebras Overlap algebras = overt locales + [Pos(z x)]. Pos(z y) x y density 2 1 Vice versa, Pos(x) (x >< x). 2 Not to be confused with the notion of a dense sublocale. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 6/25

9 Overlap Algebras as a solution to the starting problem: Powersets classically Powersets algebraize? classically cba s algebraize 1 O-algebras are an algebraic version of powersets: P(S) is an o-algebra (the motivating example). P(S) is atomic {z }. (see below) Every atomic o-algebra is a powerset (Sambin). 2 Classically: Overlap algebras = complete Boolean algebras (Vickers) (where: x >< y x y 0). Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 7/25

10 Atoms and points (I) (1) A minimal NON-ZERO element. What is an ATOM? (2) A minimal POSITIVE element. (1) is too weak: one cannot even prove that a singleton is an atom in a powerset! (2) works well (but one needs Pos or >< to define positive for elements). Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 8/25

11 Atoms and points (II) TFAE (in any overt locale) 1 a is a minimal positive element i.e.: Pos(a) and Pos(x) & (x a) = (x = a) 2 a >< x }{{} Pos(a x) a x (for any x); Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 9/25

12 Atoms and points (III) TFAE (in any o-algebra) 1 a is an atom; 2 {x a }{{ >< x } } is a completely prime filter Pos(a x) (in that case {x a >< x} = {x a x}). In other words: the mapping x Pos(a x) is a frame homomorphism if and only if a is an atom. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 10/25

13 Atoms and points (IV) {atoms of an o algebra} {points of an o algebra} Open questions: {atomic o algebras} {spatial o algebras} What is a spatial o-algebra like? What is the relationship between o-algebras and discrete locales 3? 3 Discrete locale = overt + the diagonal map is open. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 11/25

14 Regular elements (I) Even if Sambin s density does not hold for an overt locale, in general, nevertheless: [Pos(z x)]. Pos(z y) x y density (in any overt locale) there exists a unique nucleus r s.t.: [Pos(z x)]. Pos(z y) x r(y) Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 12/25

15 Regular elements (II) For L overt, let: L r def = {x L x = r(x)} 1 L r is a o-algebra (hence an overt locale (w.r.t. the same Pos of L)) 2 L = {x L x = x} L r (the least dense sublocale of L) (classically L = L r ) 3 L r is the least positively dense }{{} Pos(r(x)) Pos(x) sublocale of L (this notion has been studied by Bas Spitters) 4 L r = L iff L is an o-algebra (in particular, every o-algebra is of the kind L r ) Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 13/25

16 Regular elements (III) REGULAR open sets = the interior of its closure. Topological closure: (1) complement of the interior of the complement. (2) set of adherent points. Regular elements: (1) x = x (2) x = r(x) Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 14/25

17 Part II Overlap Algebras & Dedekind-MacNeille completion In which a new approach to the Dedekind-MacNeille completion is presented (which works for posets with overlap). Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 15/25

18 Posets with overlap Idea: modify the definition of an overlap relation in such a way that it would make sense for arbitrary posets. x >< y x >< (x y) refinement x >< ( i I y i) ( i I )(x >< y i ) splitting x >< y z(x >< z & z x & z y) i I y i exists x >< ( i I y i) ( i I )(x >< y i ) Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 16/25

19 N.B.: usually, the addition of an overlap relation greatly enrich the underlying structure. For instance, any lattice with overlap is automatically distributive. Moreover (classically): bounded lattice + pseudo-complement + overlap = Boolean algebra Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 17/25

20 Example: Heyting algebras with overlap What happens if one adds an overlap relation to a Heyting algebra? Classically: Heyting algebra + overlap relation = Boolean algebra. Not surprisingly: many classical examples of Boolean algebras (which are no longer so intuitionistically) become Heyting algebras with overlap! Example: a suitable version of the classical Boolean algebra of finite-cofinite subsets. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 18/25

21 Towards a new kind of completion The case of a Boolean algebra (classically). The Dedekind-MacNeille completion DMN(S) of a poset (S, ) can be presented as the complete lattice of all formal open subsets associated to a (basic) cover relation on S, namely: a U def ( b S) ( ( u U)(u b) a b ). Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 19/25

22 Towards a new kind of completion The case of a Boolean algebra (classically). The Dedekind-MacNeille completion DMN(S) of a poset (S, ) can be presented as the complete lattice of all formal open subsets associated to a (basic) cover relation on S, namely: a U def ( b S) ( ( u U)(u b) a b ). If S is a Boolean algebra AND we adopt a classical metalanguage, then: a U ( b S) ( ( u U)(u b) a b ) ( b S) ( ( u U)(u b = 0) a b = 0 ) ( b S) ( a b 0 }{{} a><b ( u U)(u b 0) ) }{{} u><b Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 19/25

23 Completion via overlap (I) In any poset with overlap: a DMN U def ( b S) ( a >< b ( u U)(u >< b) ) Accidentally (!?), this is the (basic) cover represented by the basic pair (S, ><, S). Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 20/25

24 Completion via overlap (I) In any poset with overlap: a DMN U def ( b S) ( a >< b ( u U)(u >< b) ) Accidentally (!?), this is the (basic) cover represented by the basic pair (S, ><, S). If S is already complete: a DMN U ( b S)` a >< b ( u U)(u >< b) {z } ( W U)><b {z } a W U Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 20/25

25 Completion via overlap (II) DMN(S) : usual Dedekind-MacNeille completion. DMN >< (S) : completion via overlap. When S is a Boolean algebra: 1 DMN(S) is a complete Boolean algebra; (actually, the least cba which contains S) 2 DMN >< (S) is an overlap algebra; (actually, the least o-algebra which contains S) 4 4 (w.r.t. a suitable notion of morphism) Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 21/25

26 Completion via overlap (III) For any poset with overlap: 1 DMN >< (S) is an overlap algebra; 2 there exists an embedding S DMN(S) which preserves all existing joins (and meets); 3 DMN >< (S) embeds in any other o-algebra satisfying 2. Classically: the completion via overlap of a poset with overlap is always a cba! Actually, DMC(S) DMN ><(S). Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 22/25

27 Future work Spatial o-algebras & Discrete locales Completions via overlap Inductive-Coinductive generation 5 5 For inductively generated formal topologies, a positivity relation, hence the positivity predicate Pos, can be defined co-inductively. This suggests that the overlap relation >< can probably be defined by co-induction every time is defined by induction. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 23/25

28 References G. Sambin, The Basic Picture: Structures for Constructive Topology, Oxford University Press, to appear. F. C., G. Sambin, The overlap algebra of regular opens, J. Pure Appl. Algebra 214 (11), pp (November 2010). F. C., Regular opens in Formal Topology and a representation theorem for overlap algebras, submitted. F. C., M. E. Maietti, P. Toto, Constructive version of Boolean algebra, submitted. Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 24/25

29 Thank you! Francesco Ciraulo - Palermo (IT) - The overlap relation in constructive mathematics - (CMPC 2010) 25/25

Constructive version of Boolean algebra

Constructive version of Boolean algebra Francesco Ciraulo, Maria Emilia Maietti, Paola Toto Abstract The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean