Affiliations: College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan province, P.R. China | Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, Sichuan province, P.R. China
Note: [] Corresponding author. Yi Liu, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan province 641112, P.R. China. E-mail: liuyiyl@126.com; xuyang@home.swjtu.edu.cn (Xu Yang).
Abstract: The aim of this paper is further to develop the fuzzy filter theory and fuzzy congruence relation on a residuated lattice. First, we introduce the concepts of interval-valued $\mathcal{T}$-fuzzy (implicative, positive implicative, MV, regular) filters with respect to a t-norm $\mathcal{T}$ on D[0,1] and investigate their properties, and some equivalent characterizations of these generalized fuzzy filters are derived. Next, we introduce the concept of interval-valued $\mathcal{T}$-fuzzy congruence relation on a residuated lattice, and the relation between interval-valued $\mathcal{T}$-fuzzy congruences and interval-valued $\mathcal{T}$-fuzzy filters are investigated. Finally, we construct a new residuated lattice which induced by interval-valued $\mathcal{T}$-fuzzy congruences, the homomorphism theorem is given.
