In this talk, we will focus on novel ordinal construction methods for non-commutative associative functions defined on a real interval.
After a brief introduction, where we will recall ordinal and z-ordinal sum construction methods, we will focus on extension of these methods for non-commutative case. We will show that z-ordinal sum (ordinal sum) construction can be represented by an associated commutative, associative, idempotent (internal) function defined on the corresponding index set, and based on this fact we will then define a non-commutative ordinal sum construction based on an associated non-commutative, associative, idempotent function defined on the corresponding index set. We will show that similarly as each commutative, associative, idempotent function can be represented by a partial order, each non-commutative, associative, idempotent function can be represented by a pair-order.