1. 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 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, S E, P E ) E, A  S, P(E) > 0

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

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

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

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, , 0 P, 1 P ) bounded poset ⊖ partial binary operation – difference on P b ⊖ a exists iff a  b (D1) a ⊖ 0 P = a for any a  P (D2) a  b  c implies c ⊖ b  c ⊖ a and (c ⊖ a) ⊖ (c ⊖ b) = b ⊖ a ( P, , 0 P, 1 P, ⊖ ) D-poset ( P, , , , 0 P, 1 P, ⊖ ) D-lattice

9.  dual partial binary operation to a difference – orthogonal sum a  b = ( a  ⊖ b )  for a  b  where x  = 1 P ⊖ 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 Let P 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 every a  P 0 (CS2) If b, b n  P for n = 1, 2,..., and b n b then f(b n, a) f(b, a) (CS3) If b, c  P, b  c then f(c ⊖ b, a) = f(c, a) – f(b, a) for every a  P 0 (CS4) If b  P 0, b  a and a ⊖ b  P 0 then for every x  P f(x, a) = f(x, b) f(b, a) (CS5) If b, a ⊖ b  P 0 then for every x  P f(x, a) = f(x, b) f(b, a) + f(x, a ⊖ b) f(a ⊖ b, a)

11. Example 1 P 0 = { a, a , b, b , 1 } 0 aa bb ba 1 s \ taa  bb  1 a1 CS1 a a  0=1 1 CS1 b b b  11 CS3 1 CS1

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

13. Example 2 F 1 = { b, a , 1 } s \ tba  1 a000 b  000 b111 a  111 1111 0 aa bb a 1 b

14. Example 3 F 2 = { b , 1 } s \ tb  1 a1/2 b  11 b00 a  1/2 111 0 aa bb a 1 b

15. Example 4 P 0 = { b, b , a , 1 } 0 aa bb b s \ tba  b  1 a001/20 b  0010 b1101 a  111/21 11111 a 1

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

17. Independence in D-posets Let P 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, 1 P ) b ↪ a

18. B ↪ AB ↪ A A ↪ BA ↪ B iff Orthomodular lattices, MV-algebras, D-posets Boolean algebras B ↪ AB ↪ AA ↪ BA ↪ 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. Example 5 0 aa bb b s \ tba  b  1 a001/20 b  0010 b1101 a  111/21 11111 a 1 f(a ,b  ) = 1/2 f(a ,1 P ) = 1 a  is not ↪ b  f(b ,a  ) = 0 f(b ,1 P ) = 0 b  ↪ a 