Decomposition of pseudo-uninorms with continuous underlying functions via ordinal sum

The decomposition of all pseudo-uninorms with continuous underlying functions, defined on the unit interval, via Clifford's ordinal sum is described. It is shown that each such pseudo-uninorm can be decomposed into representable and trivial semigroups, and special semigroups defined on two points, where the corresponding semigroup operation is the projection to one of the coordinates. Linear orders, for which the ordinal sum of such semigroups yields a pseudo-uninorm, are also characterized.