Two classes of pseudo-triangular norms and fuzzy implications

Two kinds of extensions of triangular norms (t-norms) are proposed, and the relations between these extensions and fuzzy implications are discussed in this paper. First, two classes of pseudo-t-norms (ps-t-norms), called type-A and type-B ps-t-norms, and their respective residual operators are defined. Then, we prove that these residual operators are fuzzy implications and satisfy the left neutral property. For these two classes of pseudo-t-norms, we give a series of equivalent conditions of left-continuity with respect to their first or second variable. By combining the axioms of the two classes of pseudo-t-norms, we simply get the definition of the triangular seminorms. Furthermore, we define two classes of induced operators from fuzzy implications and give the conditions under which they are type-A ps-t-norms, type-B ps-t-norms and t-seminorms. For a fuzzy implication, a series of equivalent conditions of right-continuity with respect to its second variable are established. Finally, another method inducing type-A ps-t-norms, type-B ps-t-norms and t-seminorms by implications is proposed.