$C^infty$ $L$-fuzzy manifolds with $L$-gradation of openness and $C^infty$ $LG$-fuzzy mappings of them

Abstract

In this paper, we generalize all of the fuzzy structures which we have discussed in cite{MM} to $L$-fuzzy set theory, where $L= $ denotes a complete distributive lattice with at least two elements. We define the concept of an $LG$-fuzzy topological space $(X, mathfrak{T} )$ which $X$ is itself an $L$-fuzzy subset of a crisp set M and $mathfrak{T}$ is an $L$-gradation of openness of $L$-fuzzy subsets of $M$ which are less than or equal to $ X $. Then we define $C^infty$ $L$-fuzzy manifolds with $L$-gradation of openness and $C^infty$ $LG$-fuzzy mappings of them such as $LG$-fuzzy immersions and $LG$-fuzzy imbeddings. We fuzzify the concept of the product manifolds with $L$-gradation of openness and define $LG$-fuzzy quotient manifolds when we have an equivalence relation on $M$ and investigate the conditions of the existence of the quotient manifolds. We also introduce $LG$-fuzzy immersed, imbedded and regular submanifolds.