Lattice Uniformities on Effect Algebras

Abstract

Let L be a lattice ordered effect algebra. We prove that the lattice uniformities on L which make uniformly continuous the operations and ⊕ of L are uniquely determined by their system of neighborhoods 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.

Lattice uniformities on pseudo-D-lattices

“…These results extend similar ones estabilshed in [6] for lattice-ordered effect algebras, in [30] for orthomodular lattices, and in [10,22] for MV-algebras. Morevover they give, as a particular case, the order-isomorphism found in [23].…”

“…On the other hand, in order to generate lattice uniformities on orthomodular lattices [1], and hence also D-uniformities on D-lattices [6], one has to resort to suitable modifications of submeasures, which have been called k-submeasures.…”

“…a lattice-ordered pseudo-effect algebra, we are interested in the study of modular measures defined on it. To this end, using the same terminology as in the commutative case (see for instance [6]), we have introduced the concept of a D-uniformity, which our object of investigation in this paper.…”

On Lattices of Uniformities

“…[6]) or, more generally, modular functions on D-lattices 2 (cf. [2,3]), not all lattice uniformities on the given structure L are of interest, but only those which make all operations of the MV-algebra or of the D-lattice, respectively, uniformly continuous. A first natural question is whether this lattice is a sublattice of LU(L) and whether LU(L) is a sublattice of the lattice U(L) of all uniformities on L. The answer is "yes".…”

“…In 2 For the significance of D-lattices we refer to [12]. 3 An analogous result for ring topologies on a ring with unit is contained in [25,Satz 2.2]. 4 Basile and Traynor [7] used the term "monotonely Cauchy" instead of "locally exhaustive".…”