A New Class of Uninorm Aggregation Operations for Fuzzy Theory

Uninorms play a prominent role both in the theory and applications of Aggregations, Fuzzy Theory, and of Mathematical Fuzzy Logic. In this paper the class of group-like uninorms is introduced. First, two variants of a construction method – called partial-lexicographic product – will be recalled; it constructs a large subclass of group-like FL\(_e\)-algebras. Then two specific ways of applying the partial-lexicographic product construction to construct uninorms will be presented. The first one constructs starting from \(\mathbb R\) and modifying it in some way by \(\mathbb Z\)’s, what we call basic group-like uninorms, whereas with the second one may extend group-like uninorms by using \(\mathbb Z\) and a basic uninorm to obtain further group-like uninorms. All group-like uninorms obtained this way have finitely many idempotents. On the other hand, we assert that the only way to construct group-like uninorms which have finitely many idempotents is to apply this extension (by a basic group-like uninorm) consecutively, starting from a basic group-like uninorm. In this way a complete characterization for group-like uninorms which possess finitely many idempotents is given. The obtained uninorm class can be candidate for the aggregation operation of several applications. The paper is illustrated with several 3D plots.

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