Title: $(L,M)$-fuzzy convex structures,
Abstract: In this paper, the notion of $(L,M)$-fuzzy convex structures is introduced.
It is a generalization of $L$-convex structures and $M$-fuzzifying convex
structures. In our definition of $(L,M)$-fuzzy convex structures, each
$L$-fuzzy subset can be regarded as an $L$-convex set to some degree. The
notion of convexity preserving functions is also generalized to lattice-valued
case. Moreover, under the framework of $(L,M)$-fuzzy convex structures, the
concepts of quotient structures, substructures and products are presented and
their fundamental properties are discussed. Finally, we create a functor
$\omega$ from $\mathbf{MYCS}$ to $\mathbf{LMCS}$ and show that there exists an
adjunction between $\mathbf{MYCS}$ and $\mathbf{LMCS}$, where $\mathbf{MYCS}$
and $\mathbf{LMCS}$ denote the category of $M$-fuzzifying convex structures,
and the category of $(L,M)$-fuzzy convex structures, respectively.