Further study on $(L,M)$-fuzzy topologies and $(L,M)$-fuzzy neighborhood systems

Following the idea of L-fuzzy neighborhood system as introduced by Fu-Gui Shi, and its generalization to (L;M)-fuzzy neighborhood system, the relationship between (L;M)-fuzzy topology and (L;M)-fuzzy neighborhood system will be further studied. As an application of the obtained results, we will describe the initial structures of (L;M)-fuzzy neighborhood subspaces and (L;M)-fuzzy topological product spaces. 1. Introduction and Preliminaries In this paper, based on the idea of (L;M)-fuzzy topological space introduced by T. Kubiak and A. Sostak (6, 7), and the notion of (L;M)-fuzzy neighborhood system as a generalization of L-fuzzy neighborhood system of Fu-Gui Shi (10), the relationship between (L;M)-fuzzy topology and (L;M)-fuzzy neighborhood system will be further studied. As an application of the obtained results, we will describe the initial structures of (L;M)-fuzzy neighborhood subspaces and (L;M)-fuzzy topological product spaces. The following preliminaries will be used throughout this paper, which can be found in (3, 8). A complete lattice L is called completely distributive, if one of the following conditions hold (the second then follows as a consequence (3)): (CD1)