<i>L</i>-Quasi (Pseudo)-Metric in <i>L</i>-Fuzzy Set Theory

Abstract

The aim of this paper is to focus on the metrization question in L-fuzzy sets. Firstly, we put forward an L-quasi (pseudo)-metric on the completely distributive lattice LX by comparing some existing lattice-valued metrics with the classical metric and show a series of its related properties. Secondly, we present two topologies: ψp and ζp, generated by an L-quasi-metric p with different spherical mappings, and prove ψp=ζp′ if p is further an L-pseudo-metric on LX. Thirdly, we characterize an equivalent form of L-pseudo-metric in terms of a class of mapping clusters and acquire several satisfactory results. Finally, based on this kind of L-metric, we assert that, on LX, a Yang–Shi metric topology is Q−CI, but an Erceg metric topology is not always so.

Keywords

• <i>L</i>-quasi (pseudo)-metric
• co-prime element
• irreducible element
• way below
• R-neighborhood
• <i>T</i>₁-space