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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].

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Cubic Intuitionistic Structures Applied to Ideals of BCI-Algebras


In this paper, the notion of closed cubic intuitionistic ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals in BCI-algebras are introduced, and several related properties are investigated. Relations between cubic intuitionistic subalgebras, closed cubic intuitionistic ideals, cubic intuitionistic q-ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic p-ideal are provided. Characterizations of a cubic intuitionistic a-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic a-ideal is established.

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