The purpose of this article is to explore the group of the operators, which can be used for aggregation of the fuzzy sets. There were scrutinized operators to be used for the intersection and union such as the triangular norm and triangular co-norm in the article. All of those operators are defined as the binary operations, where there was indicated that the norms and co-norms are two-valued functions of [0, 1]. Furthermore, there were discussed properties of norm and co-norm functions such as symmetry, associativity, neutral entry and null entry properties, monotonicity properties. Moreover, the consideration of the properties of norm and co-norm functions have routed investigation of the norm and co-norm operators to go further to prove theorem, which converts the union into intersection and vice versa in terms to all possible norm and co-norm functions based on the fuzzy logic sets.

The paper studies the geometry of the Riemann curvature tensor of generalized Kenmotsu manifolds. In this paper, several identities satisfied by the curvature tensor of generalized Kenmotsu manifolds are obtained. Two identities are distinguished from the obtained identities, called the first and second additional identities of curvature of the GK-manifold. Based on additional identities, two subclasses of GK-manifolds are distinguished, and a local characterization of the distinguished classes is also obtained. It is proved that the distinguished two subclasses of GK-manifolds coincide and have dimension 5. In addition, it is proved that the class of distinguished manifolds coincides with the class of almost contact metric manifolds obtained from a cosymplectic manifold by a canonical concircular transformation of a cosymplectic structure of dimension 5.

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The paper introduces the Kirichenko structural tensors of generalized Kenmotsu manifolds, called the first, second, and third structural tensors. The properties of structural tensors of generalized Kenmotsu manifolds are studied, analytical expressions of structural tensors are obtained, and covariant differentials of structural tensors are calculated.