Basic Formal Properties of Triangular Norms and Conorms

In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2].

After defining corresponding Mizar mode using four attributes, we introduced the following t-norms:

• minimum t-norm minnorm (Def. 6),

• product t-norm prodnorm (Def. 8),

• Łukasiewicz t-norm Lukasiewicz_norm (Def. 10),

• drastic t-norm drastic_norm (Def. 11),

• nilpotent minimum nilmin_norm (Def. 12),

• Hamacher product Hamacher_norm (Def. 13),

and corresponding t-conorms:

• maximum t-conorm maxnorm (Def. 7),

• probabilistic sum probsum_conorm (Def. 9),

• bounded sum BoundedSum_conorm (Def. 19),

• drastic t-conorm drastic_conorm (Def. 14),

• nilpotent maximum nilmax_conorm (Def. 18),

• Hamacher t-conorm Hamacher_conorm (Def. 17).

Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm).

This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].