Paolo Vitolo

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.

A representation theorem for quasi-metric spaces

Abstract We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable metric space, endowed with the Hausdorff quasi-metric. Therefore a topological space is quasi-metrizable if and only if it can be embedded in a Hausdorff quasi-metric hyperspace.

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].

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.

A generalization of set-difference

A structure called d0-algebra is defined and studied, which leads to a new description of D-lattices and generalized D-lattices by means of only one totally defined operation.

Topologies and uniformities on d0-algebras

Abstract A d0-algebra, which is a generalization of a D-lattice, is an algebraic structure with one operation, called difference, and a distinguished element denoted by 0. In this paper we investigate those uniformities on a d0-algebra which make the difference uniformly continuous. We prove that such uniformities are univocally determined the neighbourhoods of 0, thus extending the corresponding result known for D-lattices.

Comparability of Lower Attouch–Wets Topologies

The comparison of Attouch --Wets topologies generated by two compatible metrics has been characterized by Beer and Di Concilio . In the present paper characterization of the comparison of lower Attouch --Wets topologies is given.

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