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Independence in D-posets

Independence in D-posets. Chovanec Ferdinand, Drobná Eva Department of Natural Sciences, Armed Forces Academy, Liptovský Mikuláš , Slovakia Nánásiová Oľga Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering Slovak Technical University, Bratislava , Slovakia

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Independence in D-posets

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  1. Independence in D-posets Chovanec Ferdinand, Drobná Eva Department of Natural Sciences, Armed ForcesAcademy, Liptovský Mikuláš, Slovakia Nánásiová Oľga Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering Slovak Technical University, Bratislava, Slovakia Mathematical Structures for Nonstandard Logics Prague, Czech Republic, December 10-11, 2009

  2. Classical approach • Kolmogorov, A. N. Grundbegriffe der Wahrscheikchkeitsrechnung. Springer, Berlin, 1933. • De Finetti • Rényi, A. On a new axiomatic theory of probability. Acta Math Acad Sci Hung 6: 285–335, 1955. • Bayes

  3. Kolmogorov ( Ω , S, P ) (Ω∩E , SE , PE ) E , AS,P(E) > 0

  4. De Finetti, Rényi S0 S f: S × S 0 → [0, 1] 1. f ( E, E ) = 1 for everyES 0 2. f ( . ,E ) σ – additive measure 3. f (A ∩ B, C ) = f ( A, B ∩ C ) f (B, C ) for everyA, BS , C, B ∩ CS 0

  5. Comparison • These approaches give the same result • Independence of random events

  6. Algebraic structures D-posets Orthoalgebras D-lattices Orthomodular Posets Multivalued Algebras Orthomodular Lattices Boolean Algebras

  7. Beltrametti E, Bugajski S(2004) Separating classical and quantum correlations. Int J Theor Phys 43:1793–1801 • Beltrametti E, Cassinelli G(1981)The logic of quantum mechanics. Addison-Wesley, Reading • Cassinelli G, Truini P(1984) Conditional probabilities on orthomodular lattices. Rep Math Phys 20:41–52 • Dvurečenskij A, Pulmannová S(2000) New trends in quantum structures. Kluwer/Ister Science, Dordrecht/Bratislava • Gudder SP(1984) An extension of classical measure theory. Soc Ind Appl Math 26:71–89 • Khrennikov A Yu(2003) Representation of the Kolmogorov model having all distinguishing features of quantum probabilistic model. Phys Lett A 316:279–296 • Nánásiová O(2003) Map for simultaneous measurements for a quantum logic. Int J Theor Phys 42:1889–1903 572 • Nánásiová O(2004) Principle conditioning. Int J Theor Phys 43(7– 8):1757–1768

  8. D-poset Kôpka F, Chovanec F(1994) D-posets, Mathematica Slovaca, 44 (P,, 0P, 1P)bounded poset ⊖partial binary operation –difference on P b ⊖aexists iff a b (D1) a ⊖0P= a for any a P (D2) a  b cimplies c ⊖ b  c ⊖a and (c ⊖ a)⊖(c ⊖ b) = b ⊖ a (P, , 0P, 1P, ⊖)D-poset (P, , , , 0P , 1P, ⊖)D-lattice

  9. dual partial binary operation to a difference – orthogonal sum a b = (a⊖b)for a b where x = 1P⊖x – orthosupplement ⊙partial binary operation – product a ⊙b = a⊖b for ba Chovanec F, Kôpka F(2007) D-posets, handbook of quantum logic and quantum structures: quantum structures. Elsevier B.V.,Amsterdam, pp 367–428 Chovanec F, Rybáriková E(1998) Ideals and filters in D-posets. Int J Theor Phys 37:17–22

  10. Conditional state on a D-poset LetP be a D-poset and P 0 P be its nonempty subset. f:P × P 0 → [0, 1] is said to be a conditional state on P iff (CS1)f(a, a) = 1 for everyaP 0 (CS2)Ifb, bnPforn = 1, 2, ..., and bn b then f(bn , a) f(b, a) (CS3) Ifb, cP, bcthen f(c⊖b, a) = f(c, a) – f(b, a)for everyaP 0 (CS4) IfbP 0 , baand a⊖b P 0 then for everyxP f(x, a) = f(x, b) f(b, a) (CS5) Ifb, a⊖bP 0 then for everyxP f(x, a) = f(x, b)f(b, a) + f(x, a⊖b) f(a⊖b, a)

  11. Example 1 P0= { a, a  , b, b ,1 } 1 b a b a 0

  12. Filter in a D-poset A non-empty subset F of a D-poset P is said to be a filter in Piff (F1) a F, bP, a bbF (F2) a F, bP, b aand(a ⊖ b) F bF (F2*) a F, bF, b aa ⊙ bF F is a proper filter in a D-poset P iff 0PF a F a F

  13. Example 2 F 1 = { b, a  , 1 } 1 b a a b 0

  14. Example 3 F 2 = { b , 1 } 1 b a a b 0

  15. Example4 P0= { b, b , a  , 1 } 1 b a b a 0

  16. Maximal conditional system is a union of all proper filters in a D-poset. ( Ω , S, P ) E S ,P(E) > 0 SE = { A S ; E  A} is a proper filter in S S 0= SE

  17. Independence in D-posets LetP be a D-poset bP , aP 0 and f be a conditional state on P . b is said to be independent of an element a with respect to f iff f(b, a) = f(b, 1P) b ↪a

  18. Boolean algebras B↪A iff A↪B Orthomodular lattices, MV-algebras, D-posets B↪A ⇏ A↪B Chovanec F, Drobná E, Kôpka F, Nánásiová O Conditional states and independence in D-posets. Soft Computing (2010) DOI 10.1007/s00500-009-0487-0

  19. Example5 f(a,b) = 1/2f(a,1P) = 1 ais not ↪ b f(b,a) = 0f(b,1P) = 0 b↪a 1 b a b a 0

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