Lattice-Valued Convergence Ring and its Uniform Convergence Structure

ISSN： 1607-3606

Source： Quaestiones Mathematicae, Vol.33, Iss.1, 2010-03, pp. : 21-51

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Abstract

Considering L a frame, we introduce the notion of stratified L-neighborhood topological ring, produce some characterization theorems including its L-uniformizability. With the help of the notions of stratified convergence structures attributed to Gunther Jäger [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 Jäger and Burton [15], and the category of stratified L-uniform groups and uniformly continuous group homomorphisms, SL-UnifGrp 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.