Replacing trapezoidal membership functions by triangular membership functions for ⊗-transitivity

Although trapezoidal and Π-shaped membership functions are frequently used as generalizations of triangular membership functions in fuzzy modeling, they lead to the violation of ⊗-transitivity: μ(x, z)⩾max{0, μ(x, y)+μ(y, z)-1}, which is one of the weakest forms of transitivity. The triangular membership functions do not have this problem. We show that for any fuzzy relation μ(x, y) there is a unique smallest ⊗-transitive relation μ^⊗(x,y)⩾μ(x, y). We show that under fairly general conditions, a trapezoidal membership function can be replaced by a triangular membership function. This also leads to a representation theorem for an arbitrary ⊗-transitive relation μ(x, y), which may not be symmetric