Group-Like Uninorms
Uninorms, a.k.a. FL$_e$-algebras over $[0,1]$, play a prominent role both in the theory and the applications of Aggregations and Mathematical 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; these construct group-like FL$_e$-algebras. Then two particular ways of applying the partial-lexicographic product construction will be specified. The first method constructs, starting from $\mathbb R$ and modifying it in some way by $\mathbb Z$'s, what we call basic group-like uninorms, whereas with second method one may modify any group-like uninorm by using $\mathbb Z$ and 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. So 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 by the second method using only basic group-like uninorms. In this way a complete characterization for group-like uninorms which possess finitely many idempotents is given. The paper is illustrated with several 3D plots of group-like uninorms.