Lattice-valued convergence ring and its uniform convergence structure

Abstract

Considering L a frame, we introduce the notion of stratified L-neighborhood topological ring, produce some characterization theorems including its Luniformizability. With the help of the notions of stratified convergence structures attributed to Gunther J¨ager [10], we introduce and study various subcategories of stratified L-convergence rings. In doing so, we bring into light, among others, the notions of stratified L-uniform group and stratified L-uniform convergence group in an attempt to show that every stratified L-convergence ring carries in a natural way the stratified L-uniform convergence structure of J¨ager and Burton [15], and the category of stratified L-uniform groups and uniformly continuous group homomorphisms, SLUnifGrp is isomorphic to the category of principal stratified L-uniform convergence groups and uniformly continuous group homomorphisms, SL-PUConvGrp. Introducing the notion of stratified L-Cauchy ring, we show that the category of stratified L-Cauchy rings and Cauchy-continuous ring homomorphisms, SL-ChyRng is topological over the category of rings, Rng with respect to the forgetful functor, and that every stratified L-Cauchy ring is a stratified L-convergence ring. We observe that if L is a Boolean algebra, then each stratified L-uniform convergence ring serves as a natural example of a stratified L-Cauchy ring. We give necessary and sufficient conditions for a stratified L-convergence structure and a ring structure to be a stratified L-convergence ring.

Keywords: Frame; L-topology; L-neighborhood topological system; stratified L-neighborhood topological ring; stratified L-convergence ring; stratified L-Cauchy ring; L-uniform group; L-uniform convergence structure; category

Quaestiones Mathematicae 33(2010), 21–51