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

Document Type : Research Paper


Department of Mathematics, University of Qom, Qom, Iran


In this paper, we generalize all of the fuzzy structures which we have discussed in \cite{MM} to $L$-fuzzy set theory, where
$L= <L,\leq,\bigwedge,\bigvee, '>$ 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.