Anna Avallone

Congruences and Ideals of Effect Algebras

This paper is devoted to a general investigation of congruences and ideals in effect algebras. One of our main results is the existence of an order isomorphism between Riesz congruences and Riesz ideals. We also answer an open question of Dvurečenskij and Pulmannová by showing that an ideal is a Riesz ideal if and only if it is closed under generalized Sasaki projections.

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.

Liapunov theorem for modular functions

We extend Liapunov Theorem to modular functions on complemented lattices.

Modular functions: Uniform boundedness and compactness

We prove the Nikodym Boundedness, Brooks-Jewett and Vitali-Hahn-Saks theorems for modular functions on orthomodular lattices with SIP and on particular complemented or sectionally complemented lattices, and the equivalence, for any complemented or sectionally complemented lattice, between the Brooks-Jewett and Vitali-Hahn-Saks theorems for group-valued modular functions. As consequence, we obtain characterizations of relative, sequential and weak compactness in spaces of modular functions.

Separating points of measures on effect algebras

We prove the existence of separating points for every countable family of nonatomic σ-additive modular measures on a σ-complete lattice ordered effect algebra.

Lebesgue decomposition and bartle—dunford—schwartz theorem in pseudo-D-lattices

Abstract Let L be a pseudo-D-lattice. We prove that the exhaustive lattice uniformities on L which makes the operations of L uniformly continuous form a Boolean algebra isomorphic to the centre of a suitable complete pseudo-D-lattice associated to L . As a consequence, we obtain decomposition theorems—such as Lebesgue and Hewitt—Yosida decompositions—and control theorems—such as Bartle—Dunford—Schwartz and Rybakov theorems—for modular measures on L .

Lattice uniformities on pseudo-D-lattices

Let L be a pseudo-D-lattice. We prove that the lattice uniformities on L which make uniformly continuous the operations of L are uniquely determined by their system of neighbourhoods of 0 and form a distributive lattice.Moreover we prove that every such uniformity is generated by a family of weakly subadditive [0,+∞]-valued functions on L.

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].

On the Liapunov-Richter theorem in the finitely additive setting

Abstract The range of a finitely additive closed-valued nonatomic correspondence is convex if the domain is a σ-algebra.

Extension of measures on pseudo-D-lattices

Abstract We prove a Carathéodory type extension theorem for σ-additive exhaustive modular measures on σ-continuous pseudo-D-lattices.