MV-algebras and fuzzy topologies: Stone duality extended

Transcription

1 MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011

2 Outline 1 MV-algebras and their reducts 2 3 4

3 Outline MV-algebras and their reducts MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras 1 MV-algebras and their reducts 2 3 4

4 MV-algebras MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Definition An MV-algebra A,,, 0 is an algebra of type (2,1,0) such that A,, 0 is a commutative monoid, (x ) = x, x 0 = 0, (x y) y = (y x) x. The MV-algebra [0, 1] [0, 1],,, 0, with x y := min{x + y, 1} and x := 1 x, is an MV-algebra, called standard. It generates the variety of MV-algebras both as a variety and as a quasi-variety.

5 Further operations and properties MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Operations x y if and only if x y = 1, 1 = 0, x y = (x y ), defines a structure of bounded lattice. Properties, and distribute over any existing join., and distribute over any existing meet. De Morgan laws hold both for weak and strong conjunction and disjunction: x y = (x y ) and x y = (x y ), x y = (x y ) and x y = (x y ).

6 MV and Boolean algebras MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Boole MV Boolean algebras form a subvariety of the variety of MV-algebras. They are the MV-algebras satisfying the equation x x = x. The Boolean center Let A be an MV-algebra. a A is called idempotent or Boolean if a a = a. a a = a iff a a = a. a is Boolean iff a is. B(A) = {a A a a = a} is a Boolean algebra, called the Boolean center of A. It is, in fact, the largest Boolean subalgebra of A.

7 Semirings and quantales MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Definition A semiring is a structure S, +,, 0 such that S, +, 0 is a commutative monoid, S, is a semigroup, distributes over + from either side. Definition A quantale Q,,, is a sup-lattice equipped with a monoid operation which distributes over arbitrary joins.

8 Reducts of MV-algebras MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras [Di Nola Gerla B., 2005] For any MV-algebra A, A,,, 0, 1 and A,,, 1, 0 are (commutative, unital, additively idempotent) semirings, isomorphic under the negation. So, if A is complete, A,,, 0, 1 and A,,, 1, 0 are isomorphic (commutative, unital) quantales. Moreover, also A,,, 0 and A,,, 1 are isomorphic semirings and, if A is complete, A,,, 0 and A,,, 1 are isomorphic quantales.

9 Outline MV-algebras and their reducts Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras 1 MV-algebras and their reducts 2 3 4

10 Ideals and congruences in MV Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Definition A subset I of an MV-algebra A is called an ideal if 0 I ; I is downward closed; a b I for all a, b I. Proposition Let A be an MV-algebra. For any MV-algebra congruence on A, [0] is an ideal of A. Conversely, for any ideal I, the relation I defined by a I b iff d(a, b) := (a b ) (b a ) I is the unique congruence on A whose zero-class is I.

11 Maximal ideals Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Max A The set of all maximal ideals of A is denoted by Max A. The radical of A is defined as the intersection of all maximal ideals: Rad A := Max A. Proposition If M is a proper ideal of A then the following are equivalent: M is maximal; for any a A, if a / M then there exists n ω such that (a ) n M.

12 Semisimple algebras Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Definition (from Universal Algebra) An algebra A is called semisimple if it is subdirect product of simple algebras. Proposition An MV-algebra A is semisimple if and only if Rad A = {0}. MV ss The class of semisimple MV-algebras form a full subcategory of MV that we shall denote by MV ss. It is worth noticing that, although MV ss is NOT a variety (it is closed under S and P, but not under H), it contains [0, 1], Boole, and free, projective, σ-complete and complete MV-algebras.

13 Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Semisimple MV-algebras are algebras of fuzzy sets Theorem [Belluce, 1986] A is isomorphic to a subalgebra of [0, 1] Max A, for any A MV ss. Sketch of the proof. For any M Max A, A/M is simple. [Chang, 1959]: Any simple MV-algebra is an archimedean chain, hence it is isomorphic to a (unique) subalgebra of [0, 1]. So there exists a unique embedding ι M : A/M [0, 1]. Let ϕ M : A A/M be the natural projection. a A, let â : M Max A ι M (ϕ M (a)) [0, 1]. The map ι : a A â [0, 1] Max A is an MV-algebra embedding.

14 Hyperarchimedean algebras Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Definition Let A be an MV-algebra. An element a A is archimedean if it satisfies the following equivalent conditions: 1 there exists a positive integer n such that na B(A); 2 there exists a positive integer n such that a na = 1; 3 there exists a positive integer n such that na = (n + 1)a. Definition An MV-algebra A is called hyperarchimedean if all of its elements are archimedean.

15 Outline MV-algebras and their reducts The category MV Top The shadow topology 1 MV-algebras and their reducts 2 3 4

16 Open sets MV-algebras and their reducts The category MV Top The shadow topology X, Ω topological space {0, 1} X,,,, 0, 1 is a complete Boolean algebra. Ω,, 0 is a sup-sublattice of {0, 1} X,, 0, Ω,, 1 is a meet-subsemilattice of {0, 1} X,, 1. X, Ω MV-topological space [0, 1] X,,,,,, 0, 1 is a complete MV-algebra. Ω,,, 0 is a subquantale of [0, 1] X,,, 0, Ω,,, 1 is a subsemiring of [0, 1] X,,, 1.

17 Continuous maps The category MV Top The shadow topology Preimage of a function Let X, Y be sets and f : X Y a map. If we identify the subsets of X and Y with their membership functions, the preimage of f is f : χ {0, 1} Y χ f {0, 1} X. Analogously, the fuzzy preimage of f is defined by f : χ [0, 1] Y χ f [0, 1] X. MV-continuity So, if X, Ω X and Y, Ω Y are MV-spaces, f : X Y is said to be MV-continuous if f [Ω Y ] Ω X.

18 Examples and bases The category MV Top The shadow topology X, {0, 1} and X, [0, 1] X are MV-topological spaces. Any topology is an MV-topology. Let d : X [0, + [ be a metric on X and α a fuzzy point of X with support { x. For any r R +, the open ball B r (α) is α(x) if d(x, y) < r B r (α)(y) := 0 if d(x, y) r. The family of fuzzy subsets of X that are joins of open balls is an MV-topology on X that is said to be induced by d. Definition T = X, Ω MV Top. B Ω is called a base for T if, for all o Ω, o = i I b i, with {b i } i I B.

19 The shadow topology The category MV Top The shadow topology Definition For any MV-space T = X, Ω, let B(Ω) := Ω {0, 1} X. Sh T = X, B(Ω) is a topology in the classical sense, called the shadow of T. Sh is a functor Top is a full subcategory of MV Top. The mapping Sh : MV Top Top is a functor. It is, in fact, the left-inverse of the inclusion Top MV Top. The shadow of the MV-topology induced by a metric d is the topology induced by d.

20 Outline MV-algebras and their reducts Stone MV-spaces Stone duality extended Some compositions 1 MV-algebras and their reducts 2 3 4

21 Compactness Stone MV-spaces Stone duality extended Some compositions A more complex situation Due to the presence of two intersection and two union operations, compactness and each separation axiom can have at least two different MV-versions. Compact spaces An MV-space X, Ω is said to be compact if any open covering of X contains an additive covering, i.e., for any Ω Ω such that Ω = 1, there exists a finite subset {o 1,..., o n } of Ω such that o 1 o n = 1; strongly compact if any open covering of X contains a finite covering.

22 Separation MV-algebras and their reducts Stone MV-spaces Stone duality extended Some compositions T 2 axioms An MV-space T = X, Ω is called an Hausdorff (or separated, or T 2 ) space if, for any x y X, there exist o x, o y Ω such that: (i) o x (x) = o y (y) = 1, (ii) o x (y) = o y (x) = 0, (iii) o x o y = 0. T is said to be strongly separated if, for any x y X, there exist o x, o y Ω satisfying (i) and (iv) o x o y = 0. T 2 definition do not need fuzzy points.

23 Stone MV-spaces Stone MV-spaces Stone duality extended Some compositions Remark Strong separation implies separation and they both collapse to classical T 2 in the case of crisp topologies. The same holds for compactness. Clopens and zero-dimensionality Let T = X, Ω be an MV-space and Ξ = Ω be the family of closed fuzzy subsets. We denote by Clop T the family Ω Ξ of clopen fuzzy subsets of X. Clop T MV ss, for any MV-space T. T is called zero-dimensional if Clop T is a base for it. Definition A Stone MV-space is an MV-space which is compact, separated and zero-dimensional.

24 The MV-space Max A, Ω A Stone MV-spaces Stone duality extended Some compositions Remark The category MV Stone of Stone MV-spaces, with MV-continuous maps as morphisms, is a full subcategory of MV Top. The Maximal MV-spectrum Let A be a semisimple MV-algebra. By Belluce representation theorem, there exists a canonical embedding ι : A [0, 1] Max A. Then ι[a] generates, as a base, an MV-topology on Max A. The family of open sets of such a space is denoted by Ω A. So, for any semisimple MV-algebra A, Max A, Ω A denotes the MV-topological space on Max A having (an isomorphic copy of) A as a base.

25 Stone MV-spaces Stone duality extended Some compositions A (proper) extension of Stone duality Theorem 1 The mappings Φ : T MV Top Clop T MV ss Ψ : A MV ss Max A, Ω A MV Top define two contravariant functors. 2 They yield a duality between MV ss and MV Stone, that is for every semisimple MV-algebra A, ΨA is a Stone MV-space and A is isomorphic to the clopen algebra of such a space; conversely, every Stone MV-space T = X, Ω is homeomorphic to ΨΦT. 3 The restriction of such a duality to Boolean algebras and Stone spaces coincide with the classical Stone duality. 4 Φ Sh = B Φ and Ψ B = Sh Ψ.

26 MV-algebras and their reducts Graphically Stone MV-spaces Stone duality extended Some compositions MV ss Ψ MV Stone op Φ B Sh Boole Φ Ψ Stone op Horizontal arrows: equivalences Vertical arrows: inclusions of full subcategories and their left-inverses Corollary Strongly separated Stone MV-spaces are dual to hyperarchimedean MV-algebras.

27 Mundici equivalence Stone MV-spaces Stone duality extended Some compositions Unital Abelian l-groups Let ulg Ab be the category whose objects are Abelian lattice-ordered groups with a distinguished strong order unit and whose morphisms are unit-preserving l-group homomorphisms. Theorem [Mundici, 1986] The categories ulg Ab and MV are equivalent. Γ : ulg Ab MV is defined as follows: for any Abelian ul-group G, +,,,, 0, u, Γ(G) = [0, u],,, 0, where x y := (x + y) u, x = u x. We shall denote by Γ 1 the inverse of Γ.

28 ΦΓ and Γ 1 Ψ Stone MV-spaces Stone duality extended Some compositions Restrictions of Γ and Γ 1 MV ss is equivalent to the subcategory ulgr Ab of ulgab whose objects are, up to isomorphisms, ul-subgroups of a group of bounded real-valued functions from a set X, with 1 as unit. In such a restriction Boolean algebras correspond to subgroups of bounded integer-valued functions (ulg Ab Z ). Corollary Then ΦΓ and Γ 1 Ψ form a duality between ulgr Ab and MV Stone. Such a duality obviously restricts to ulgz Ab and Stone.

29 n-valued MV-algebras Stone MV-spaces Stone duality extended Some compositions MV n and BR n Let n > 1 be a natural number. In [Di Nola Lettieri, 2000], the authors defined the category BR n : objects are pairs (B, R), where B is a Boolean algebra and R is an n-ary relation on B satisfying certain conditions; a morphism from (B, R) to (B, R ) is a Boolean algebra homomorphism f : B B such that (a 1,..., a n ) R implies (f (a 1 ),..., f (a n )) R. Now, let MV n denote the subvariety V(S n ) of MV generated by the (n + 1)-element chain S n = {0, 1 n,..., n 1 n, 1}.

30 Future work Stone MV-spaces Stone duality extended Some compositions Theorem [Di Nola Lettieri, 2000] The categories MV n and BR n are equivalent. MV n MV ss, for all n > 1. MV Stone and Stone From an MV-topological viewpoint, MV n is dual to the category of Stone MV-spaces of fuzzy sets with S n -valued membership functions. Next step will be to characterize a suitable category of Stone spaces with additional conditions which is dual to BR n and, therefore, to MV n.

31 Further possible developments Stone MV-spaces Stone duality extended Some compositions A point-free approach MV-frames (reducts of complete BL-algebras?). Spatial and sober MV-frames. MV-topoi. Applications Mathematical Morphology in digital image analysis. Geosystems. Other ideas or suggestions are welcome...

32 Stone MV-spaces Stone duality extended Some compositions Thank you!

33 Stone MV-spaces Stone duality extended Some compositions References [Belluce, 1986], Semisimple algebras of infinite valued logic and bold fuzzy set theory, Can. J. Math., 38/6, , [Chang, 1959], A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc., 93, 74 90, [Di Nola Gerla B., 2005], Algebras of Lukasiewicz s logic and their semiring reducts, Contemp. Math., 377, , Amer. Math. Soc., [Di Nola Lettieri, 2000], One chain generated varieties of MV-algebras, Journal of Algebra, 225, , [Mundici, 1986] Interpretation of AF C -algebras in Lukasiewicz sentential calculus, J. Functional Analysis, 65, 15 63, [Russo], An extension of Stone duality to fuzzy topologies and MV-algebras, submitted, arxiv: v2 [math.lo].