Replacing trapezoidal membership functions by triangular membershipfunctions for ⊗-transitivity

摘要

Although trapezoidal and Π-shaped membership functions arefrequently used as generalizations of triangular membership functions infuzzy modeling, they lead to the violation of ⊗-transitivity:μ(x, z)⩾max{0, μ(x, y)+μ(y, z)-1}, which is one of theweakest forms of transitivity. The triangular membership functions donot have this problem. We show that for any fuzzy relation μ(x, y)there is a unique smallest ⊗-transitive relationμ⊗(x,y)⩾μ(x, y). We show that under fairlygeneral conditions, a trapezoidal membership function can be replaced bya triangular membership function. This also leads to a representationtheorem for an arbitrary ⊗-transitive relation μ(x, y), whichmay not be symmetric