# $(L,M)$-fuzzy convex structures

## TL;DR

This paper introduces a new framework called $(L,M)$-fuzzy convex structures that generalizes existing convex structures using lattice-valued fuzzy sets, and explores their categorical properties.

## Contribution

It defines $(L,M)$-fuzzy convex structures, extends convexity preserving functions, and establishes categorical relationships including an adjunction between related structures.

## Key findings

- Defined $(L,M)$-fuzzy convex structures and their properties.
- Extended convexity preserving functions to lattice-valued cases.
- Established an adjunction between categories of 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.