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),

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].