Basic Formal Properties of Triangular Norms and Conorms
In the article we present in the Mizar system ,  the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets . The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality . 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 , . 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  started in  and ; 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 , the approach which was chosen allows however for merging both theories , .