Independence in D-posetsChovanec Ferdinand, Drobná Eva

Department of Natural Sciences, Armed Forces Academy, Liptovský Mikuláš, Slovakia

Nánásiová OľgaDepartment 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

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

Kolmogorov

( Ω , S , P )

(Ω ∩ E , SE , PE ) E , A S , P(E) > 0

)(

)()(

EP

EAPAPE

De Finetti, Rényi

S0 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

Comparison

• These approaches give the same result

• Independence of random events

)(

)()(

EP

EAPAPE

)()(

)()( AP

EP

EAPAPE

Algebraic structures

Boolean Algebras

Multivalued Algebras

D-posets

Orthoalgebras

Orthomodular Posets

Orthomodular Lattices

D-lattices

• 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

D-posetKôpka F, Chovanec F (1994) D-posets, Mathematica Slovaca, 44

(P , , 0P , 1P ) bounded poset

⊖ partial binary operation – difference on P b ⊖ a exists iff a b

(D1) a ⊖ 0P = a for any a P (D2) a b c implies c ⊖ b c ⊖ a and (c ⊖ a) ⊖ (c ⊖ b) = b ⊖ a

(P , , 0P , 1P , ⊖) D-poset

(P , , , , 0P , 1P , ⊖) D-lattice

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

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, bn P for n = 1, 2, ..., and bn b then f(bn , 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 bP 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)

Example 1P 0 = { a, a , b, b , 1 }

0

a

b

b a

1s \ t a a b b 1

a 1CS1

a 0=1 1CS1

b 1CS1

b 1CS1

1 1CS3 1CS3 1CS3 1CS3 1CS1

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 0P F a F a F

Example 2F 1 = { b, a , 1 }

s \ t b a 1

a 0 0 0

b 0 0 0

b 1 1 1

a 1 1 1

1 1 1 10

a

b

a

1

b

Example 3F 2 = { b , 1 }

s \ t b 1

a 1/2 1/2

b 1 1

b 0 0

a 1/2 1/2

1 1 10

a

b

a

1

b

Example 4P0 = { b, b , a , 1 }

0

a

b

b

s \ t b a b 1

a 0 0 1/2 0

b 0 0 1 0

b 1 1 0 1

a 1 1 1/2 1

1 1 1 1 1

a

1

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

E

}0)(;{ APA

Independence in D-posets

Let P be a D-poset bP , 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

)()()(iff)()/( BPAPBAPBPABP

B ↪ A A ↪ Biff

Orthomodular lattices, MV-algebras, D-posets

Boolean algebras

B ↪ A A ↪ B⇏

Chovanec F, Drobná E, Kôpka F, Nánásiová OConditional states and independence in D-posets. Soft Computing (2010) DOI 10.1007/s00500-009-0487-0

Example 5

0

a

b

b

s \ t b a b 1

a 0 0 1/2 0

b 0 0 1 0

b 1 1 0 1

a 1 1 1/2 1

1 1 1 1 1

a

1

f(a,b) = 1/2 f(a,1P ) = 1 a is not ↪ b

f(b,a) = 0 f(b,1P ) = 0 b ↪ a