Giuseppina Barbieri

A Topological Approach to the Study of Fuzzy Measures

A topological approach to the study of fuzzy measures is developed. To do so we need (instead of a clan of fuzzy sets) a more general structure of the domain of the fuzzy measures. This structure is defined by means of some equations. Our general setting allows us to treat simultaneously fuzzy measures, group-valued measures on Boolean rings, and linear operators on Riesz spaces. We deal with extension and decomposition theorems. Also we study connected, totally disconnected, and compact MV-algebras.

The Hahn decomposition theorem for fuzzy measures and applications

We deal with measures on Δ-l-semigroups, in particular on MV-algebras. Examples of such measures are T∞-valuations on clans of fuzzy sets. We first provide the Hahn decomposition theorem for measures on Δ-l-semigroups. This is then used to obtain a representation theorem for such measures, which itself is a basic tool in the proof of Liapounoff type theorems.

Range of finitely additive fuzzy measures

Abstract We extend Liapunov theorem to R n -valued finitely additive measures of fuzzy sets. Moreover, we study the convexity of the range of B-convex-valued measures.

Characterization of T-measures

As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They made the first steps towards a characterization of monotonic real-valued T-measures for a Frank triangular norm T. Later on, Mesiar and the authors of this paper found independently two generalizations, one for vector-valued T-measures with respect to Frank t-norms (in particular for nonmonotonic ones) [3], the other for monotonic real-valued T-measures with respect to general strict t-norms [15]. Here we present a common generalization – a characterization of nonmonotonic T-measures with respect to an arbitrary strict t-norm. Moreover, we prove this for vector-valued T-measures. Using this characterization, we generalize Ljapunov Theorem to this context.

On the Alexandroff decomposition theorem

We prove an Alexandroff decomposition type theorem, which extends a decomposition theorem proved in [de LUCIA, P.—MORALES, P.: Decomposition of group-valued measures in orthoalgebras, Fund. Math. 158 (1998), 109–124].

Decomposition of Pseudo-effect Algebras and the Hammer–Sobczyk Theorem

We prove an algebraic and a topological decomposition theorem for complete pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for group-valued modular measures defined on pseudo-D-lattices and compactness of the range of every ℝn

The extension of measures on D-lattices

\mathbb {R}^{n}

On some properties of k-subadditive lattice group-valued capacities

-valued σ-additive modular measure on a σ-complete pseudo-D-lattice.

Answer to Raczkowski's questions on convergent sequences of integers

Abstract A lattice ordered group valued measure is extended from a D-lattice into a σ-complete D-lattice. A D-lattice is a lattice with a greatest element 1 and a smallest element 0 endowed with an order-compatible operation, called a difference, which satisfies a list of axioms. The result generalizes the classical result known for measures on Boolean algebras.

Lyapunov's Theorem for Measures on D-posets

Abstract We investigate some properties of (uniformly) absolutely continuous, singular and non-concentrated k-subadditive lattice group-valued capacities, and we examine some relations between them. Furthermore, we pose some open problems.