The non-commutative, idempotent, associative functions are studied. It is well known that each commutative, idempotent (internal), associative function can be represented by a partial (linear) order. In this work it is shown that each non-commutative, idempotent (internal), associative function can be represented by a (linear) pair-order. Moreover, each internal, associative function can be expressed as an ordinal sum of trivial semigroups and semigroups, where the corresponding semigroup operation is the projection to one of the coordinates. Vice versa, each linear pair-order induces a unique internal, associative function and a condition under which each (non-linear) pair-order defined on a chain induces a unique monotone, idempotent, associative function is also introduced. Several examples of non-commutative, idempotent, associative functions and related pair-orders are shown.