Decomposition of idempotent pseudo-uninorms via ordinal sum

The decomposition of idempotent pseudo-uninorms is investigated. We show that each idempotent pseudo-uninorm on the unit interval can be decomposed into an ordinal sum of trivial semigroups and non-commutative idempotent semigroups defined on two elements, where the corresponding semigroup operation is the projection to one of the coordinates. Linear orders yielding idempotent pseudo-uninorms via ordinal sum of this type of semigroups are also investigated. The link between linear orders corresponding to an idempotent pseudo-uninorm and its dual pseudouninorm is shown for two types of duality.