### Abstract:

Most of problems encountered can be modelled mathematically, but these models require
assumptions that are sometimes too restrictive, making application to the real world difficult.
Real-world problems must take into account imprecise, uncertain information. The concept
of fuzzy set was introduced in 1965 by A. Zadeh [3], many authors were interested by this
concept [1, 4, 13]. The main problem in fuzzy mathematics is how to carry out the ordinary
concepts to the fuzzy case.
The partially ordered algebraic systems play an important role in algebra. Some important
concepts in partially ordered systems are ordered groups and lattice ordered groups. These
concepts play a major role in many branches of Algebra.
In 1971, A. Rosenfeld applied the notion of fuzzy set theory on group theory in his book
[2], he introduced the concept of fuzzy subgroup and show that many theorem can be extended
to develop the fuzzy group theory. Next, many authors worked on fuzzy theory and introduced
the concept of fuzzy orders, fuzzy cosets and fuzzy lattice [7, 9, 15].
Convexity play an important role in the study of compatible orders, ordered groups and
especially in lattice-ordered groups. Our main aim in this work is to investigate some properties
and characterizations theorems of the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered
subgroup) of an ordered group (resp. lattice-ordered group). Some more results related to this
topic are also derived.
This memory is organized in three chapter as follows :
In the first chapter, we recall some definitions and well-known about the ordered sets, coset,
groups, and ordered groups. This chapter also focuses on lattice, lattice-ordered group and some
related concept which we will need in the sequel.
In the second chapter, we give some basic notions and generalities about the fuzzy sets, their
characteristic notion and level sets. Also, we define fuzzy subgroup and give some properties.
In the last chapter, we specified our searches about convexity in fuzzy case more precisely
fuzzy convex subgroups and fuzzy convex lattice-ordered subgroups.