  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.
