Measures and Topologies on MV-algebras

My talk is based on article [1]. Our aim is to give a unified topological approach to several results about a certain type of fuzzy measures, namely T∞−valuations (2) on clans of fuzzy sets (1) (in the sense of Butnariu and Klement [2]). We will transfer the method of [3], used to study FN-topologies (4) and measures on Boolean rings, to the study of T∞−valuations. So we need, for the domain of the measures, instead of a clan of fuzzy sets a more general structure which is equationally defined and therefore closed with respect to quotients and uniform completions. A structure which satisfies these requirements is that of an MV-algebra. In this talk I will first mention some algebraic facts about MV-algebras. In particular, it is shown that any σ-complete MV-algebra L is isomorphic to a clan of fuzzy sets, more precisely: Let Ω and A, respectively, be the Stone space and the Stone algebra of the center C(L) of L and l∞(Ω) the space of real-valued bounded functions on Ω; then L can be embedded in the closed linear subspace L∞(A) of (l∞(Ω), ‖ ‖∞) generated by the characteristic functions χA, A ∈ A. As an immediate consequence one obtains a representation of a measure (3) on L as an integral with respect to a measure on A. Another consequence is the following decomposition theorem: Any complete MV-algebra is isomorphic to a product L0× ∏ α∈A Lnα where L0 is an MV-algebra such that C(L0) is atomless, nα ∈ (N ∪ {∞}) \ {1}, Ln is a chain of n elements if n ∈ N and L∞ is the real interval [0, 1]. Then we study uniform MV-algebras, i.e. MV-algebras L endowed with a uniformity making the operations of L uniformly continuous. It turns out that the uniformity of any uniform MValgebra can be generated by a family of submeasures. A submeasure η on L is a monotone [0,∞]−valued function on L such that η(0) = 0 and η(x + y) ≤ η(x) + η(y) for all x, y ∈ L; then η induces a semimetric on L by the formula d(x, y) := η(x ∨ y − x ∧ y) and so (L, d) is a uniform MV-algebra. As a consequence of the algebraic decomposition of complete MV-algebras mentioned above and the known characterization of compact Boolean algebras one obtains e.g. that a uniform MV-algebra is compact if and only if it is (topologically and algebraically) isomorphic to a product ∏ α∈A Lnα , 2 ≤ nα ≤ ∞. A particular role play exhaustive uniform MV-algebras, i.e. uniform MV-algebras such that any monotone sequence is Cauchy. We establish a relationship between these uniformities on an MV-algebra L and the