Group-like Uninorms

Uninorms play a prominent role both in the theory and the applications of Aggregations and Fuzzy Logic. In this paper the class of group-like uninorms is introduced and characterized. First, two variants of a general construction -- called partial-lexicographic product -- will be recalled from \cite{Jenei_Hahn}; these construct odd involutive FL$_e$-algebras. Then two particular ways of applying the partial-lexicographic product construction will be specified. The first method constructs, starting from $\mathbb R$ (the additive group of the reals) and modifying it in some way by $\mathbb Z$'s (the additive group of the integers), what we call basic group-like uninorms, whereas with the second method one can modify any group-like uninorm by a basic group-like uninorm to obtain another group-like uninorm. All group-like uninorms obtained this way have finitely many idempotent elements. On the other hand, we prove that given any group-like uninorm which has finitely many idempotent elements, it can be constructed by consecutive applications of the second construction (finitely many times) using only basic group-like uninorms as building blocks. Hence any basic group-like uninorm can be built using the first method, and any group-like uninorm which has finitely many idempotent elements can be built using the second method from only basic group-like uninorms. In this way a complete characterization for group-like uninorms which possess finitely many idempotent elements is given: ultimately, all such uninorms can be built from $\mathbb R$ and $\mathbb Z$. This characterization provides, for potential applications in several fields of fuzzy theory or aggregation theory, the whole spectrum of choice of those group-like uninorms which possess finitely many idempotent elements.