Weak relatively uniform convergences on abelian lattice ordered groups

Abstract

ABSTRACT. The notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G + , then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex -subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.

Weak relatively uniform convergences on MV-algebras

“…Next, we will apply the basic properties of β(M )-convergence presented in [7]. The symbol S(G) will denote the system of all nonempty subsets of A(G) closed with respect to the addition and s(G) will be the system of all convergences β(M ) where M runs over the system S(G).…”

“…[7]) Let M be a nonempty subset of A(G), (x n ) a sequence in G and x ∈ G. We say that the sequence (…”

“…The system s(G) of all wru-convergences on G is partially ordered in a natural way. In [7] there is proved that s(G) is a Brouwerian lattice.…”

“…Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A). A different standpoint is applied in [5]; here, there are studied archimedean lattice ordered groups with a fixed regulator.The notion of ru-convergence in archimedean lattice ordered groups was generalized in [7] in two directions. First, the lattice ordered group G under consideration was assumed to be abelian (this is a weaker condition than the assumption of the archimedean property).…”

“…The notion of ru-convergence in archimedean lattice ordered groups was generalized in [7] in two directions. First, the lattice ordered group G under consideration was assumed to be abelian (this is a weaker condition than the assumption of the archimedean property).…”