# Aggregation on lattices isomorphic to the lattice of closed subintervals of the real unit interval

2022

In numerous generalizations of the original theory of fuzzy sets proposed by Zadeh, the considered membership degrees are often taken from lattices isomorphic to the lattice of closed subintervals of the unit interval . This is, for example, the case of intuitionistic values, Pythagorean values or -rung orthopair values. The mentioned isomorphisms allow to transfer the results obtained for the lattice directly to the other mentioned lattices. In particular, basic connectives in Zadeh's fuzzy set theory, i.e., special functions on the lattice , can be extended to the interval-valued connectives, i.e., to special functions on the lattice , and then to the connectives on the lattices of intuitionistic values, of Pythagorean values, and also on the lattice of -rung orthopair values. We give several examples of such connectives, in particular, of those which are related to strict t-norms. For all these connectives we show their link to an additive generator of the considered strict t-norm . Based on our approach, many results discussed in numerous papers can be treated in a unique framework, and the same is valid for possible newly proposed types of connectives based on strict t-norms. Due to this approach, a lot of tedious proofs of the properties of the proposed connectives could be avoided, which gives researchers more space for presented applications.

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