Fuzzification of the category of (L, M)-fuzzy convex spaces

Abstract

In this paper, a fuzzy approach to the category of $(L, M)$-fuzzy convex spaces (denoted by \textbf{LM-FCon}) is introduced.
To be specific, the objects and morphisms of
\textbf{LM-FCon} are given some degree value, denoted by $\omega(X,\mathscr{C})$ and $\mu(f)$ respectively, which extends the \textbf{LM-FCon} to the fuzzy case.
What is more, we show that the set $\mathfrak{C}_{\alpha}(L,M,X) = \{\mathscr{C}:L^X \rightarrow M
\mid \omega(X,\mathscr{C}) \geqslant \alpha\}$ is a bounded complete lattice and discuss the relationships among $\mathfrak{C}_{\alpha}(L,M,X)$, $L$-convex structures and $(L,M)$-fuzzy convex structures. Finally, subspace, product space, join space and quotient space of $\mathfrak{C}_{\alpha}(L,M,X)$ are studied
and related properties are obtained.