A notion of independence for probability MV-algebras
by
Ioana Leustean
Technische Universität Darmstadt, Germany and University of Bucharest, Romania

An appropiate definition for the notion of stochastically independent s-subalgebras is an open problem mentioned by Riecan and Mundici in [7]. In the following, we propose a concept of independent structures for probability MV-algebras, generalizing the classical theory for probabilities defined on Boolean algebras [1, 3].

In our approach, a probability MV-algebra is a pair (A, s) where A is an MV-algebra and s:A→ [0, 1] is a (finitely additive) state. A s-probability MV-algebra is a probability MV-algebra (A, s) such that A is s-complete and s is a s-continuous state. A state s is called extreme if it is also an MV-algebra homomorphism into the standard MV-algebra [0, 1].

For any two MV-algebras A and B, the tensor product A⊗_oB was defined in [2]. The main difference between A⊗_o B and the MV-algebraic tensor product defined by Mundici in [6] is that we no more assume that the bilinear functions involved are bimorphisms. As a consequence, the tensor product A⊗_oB is uniquely defined, up to an isomorphism, by the following universal property:

Our first result is the following.

Theorem 1. If (A, s_A) and (B, s_B) are probability MV-algebras then there exists a unique extreme state s_{A, B}:A⊗_o B→ [0, 1] such that s_{A, B}(a⊗_o b)=s_A(a)·s_B(b) (where · is the real product) for all a ∈ A, b ∈ B.

The above result suggets the following definition.

Definition 2. Let (A, s_A), (B, s_B) and (T, s_T) be (s-probability) probability MV-algebras. We say that (A, s_A) and (B, s_B) are (T, s_T)-independent if there exists a bilinear function b:A×B→ T satisfying s_T(b(a, b))=s_A(a)·s_B(b) for all a ∈ A, b ∈ B.

It is easy to see that if A and B are Boolean subalgebras of a Boolean algebra T and P:T→ [0, 1] is a boolean probability, then we get the usual concept: since b(a, b)=a∧b is a bilinear map, A and B are called P-independent if P(a∧b)=P(a)·P(b) for any a ∈ A, b ∈ B.

From Theorem 1 and Definition 2 we get the following result.

Theorem 4. For any two probability MV-algebras (A, s_A) and (B, s_B) there exists a probability MV-algebra (T, s_T) and a bilinear function b_{A, B}:A×B→ T such that the following proprieties hold:
(a) (A, s_A) and (B, s_B) are (T, s_T)-independent,
(b) s_T is an extreme state.
If b_{A, B}:A×B→ T is the bilinear map which gives the independence, then
(c) b_{A, B}(A×B) generates T as an MV-algebra,
(d) for any probability MV-algebra (C, m) and for any bilinear function g:A×B→ C such that m is an extreme state and m(g(a, b))=s_A(a)·s_B(b) for any a ∈ A, b ∈ B, there exists a unique MV-algebra homomorphism f:T→ C such that mf=s_T and f(b_{A, B}(a, b))=g(a, b) for any a ∈ A, b ∈ B.

We remark that the structure T is A⊗_o B, s_T is the probability given by Theorem 1 and b_{A, B}(a, b)=a⊗_o b for any a ∈ A, b ∈ B. By (d), the structure ((T, s_T), b_{A, B}) is uniquely defined up to an isomorphism. The above theorem can be generalized to arbitrary families of probability MV-algebras, following the classical theory [1, 3].

In order to obtain a similar result for s-probability MV-algebras, we need a preliminary construction: the (metric) completion of an MV-algebra with respect to a state. We refer to [4] and [1] for the similar construction in lattice ordered groups and Boolean algebras.

Definition 5. If (A, s) is a probability MV-algebra, we define r_s(a, b)=s(d(a, b)) for any a, b ∈ A. It follows that r_s is a pseudo-metric on A. Moreover, s and the MV-algebra operations are uniformly continuous w.r.t r_s. Let (A^#, r^#_s) be the pseudo-metric completion of (A, r_s) and F:A→ A^# the natural map [5].

Proposition 6. If (A, s) is a probability MV-algebra and s is extreme, then the following properties hold:
(a) A^# is a s-complete MV-algebra,
(b) there exists a s-continuous state s^#:A^#→ [0, 1] such that s^#F = s.

Let (A, s_A) and (B, s_B) be s-probability MV-algebras, T=A⊗_o B and s_T is s_{A, B} from Theorem 1. If we define T^# and s^#_T as above, then (T^#, s^#_T) is a s-probability MV-algebra by Proposition 6.

Theorem 7. Under the above hypothesis, the s-probability MV-algebras (A, s_A) and (B, s_B) are (T^# , s^#_T)-independent.

As a consequence, for any two s-probability MV-algebras (A, s_A) and (B, s_B) there exists a s-probability MV-algebra (T, s_T) such that (A, s_A) and (B, s_B) are (T, s_T)-independent.

Acknowledgement This work was supported by a research fellowship of the Alexander von Humboldt Foundation.

References
[1] I. Cuculescu, O. Onicescu, Probability theory on Boolean algebras of events, Editura Academiei Republicii Socialiste Romania, 1976.
[2] P. Flondor, I. Leu stean, Tensor products of MV-algebras, Soft Computing 7 (2003), 446-457.
[3] D.H. Fremlin, Measure Theory, available from the author's site at the University of Sussex, 1995.
[4] K.R. Goodearl, D.E. Handelman, Matric completions of partially ordered abelian groups, Indiana University Mathematics Journal 29 (1980), 861-895.
[5] J.L. Kelly, General Topology. The university series in higher mathematics, D. Van Nostrand Company, 1955.
[6] D. Mundici, Tensor products and the Loomis-Sikorski theorem for MV-algebras, Advances in Applied Mathematics 22 (1999), 227-248.
[7] B. Riecan, D. Mundici Probability in MV-algebras, in: E. Pap (Editor), Handbook of Measure Theory, North-Holland, Amsterdam, 2002, 869-909.

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Date received: February 1, 2008