Commutative, associative and non-decreasing functions continuous around diagonal

We characterize all functions that can be obtained as a $z$-ordinal sum of semigroups related to continuous t-norms, t-conorms, representable uninorms and idempotent semigroups. We show that this class of functions is bigger than the class of $n$-uninorms with continuous underlying functions. Vice versa, we show that the characterization of $n$-uninorms with continuous underlying functions via $z$-ordinal sum can be extendedto any  commutative, associative and non-decreasing binary function on the unit interval, which has continuous Archimedean components and is continuous on the diagonal.