Generalized L-Čech closure operators and beyond

Abstract

In this paper, we first give a concrete example to illustrate that the construction of$L$-Čech closure operators given in  [27, Theorem 3.1] fails to hold when $L$ is  a continuous residuated lattice. By strengthening $L$ to be  completely distributive, we propose two new kinds of operators, which are respectively called generalized closure operators and generalized$L$-closure operators, and establish the categorical relation between them. As an application, we prove the existence of a Galois connection between the category of stratified generalized$L$-closure spaces and the category of stratified$L$-convex spaces. As a natural extension of probabilistic uniform spaces,  we further introduce and study$L$-Čech uniform spaces, and also present their connection with $L$-Čech closure spaces.

Keywords

• L-Čech closure operator
• generalized L-closure operator
• L-convex structure
• L-Čech uniformity

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