Transitive Closures of Ternary Fuzzy Relations

Recently, we have introduced six types of composition of ternary fuzzy relations. These compositions are close in spirit to the composition of binary fuzzy relations. Based on these types of composition, we have introduced several types of transitivity of a ternary fuzzy relation and investigated their basic properties. In this paper, we prove additional properties and characterizations of these types of transitivity of a ternary fuzzy relation. Also, we provide a representation theorem for ternary fuzzy relations satisfying these types of transitivity. Finally, we focus on the problem of closing a ternary fuzzy relation with respect to the proposed types of transitivity.