Structure of n-uninorms

In this paper we study binary operators on [0,1] which are associative, monotone non-decreasing in both variables and commutative (AMC) with neutral element. In this work, we generalize the concept of neutral element and this generalization gives rise to a new class of AMC binary operators on [0,1] called n-uninorms. n-Uninorms are denoted as Un, where n comes from the generalization of the neutral element. We study the structure of n-uninorms. The structure resembles an ordinal sum structure made up of n uninorms. We characterize some special cases of them based on some continuity considerations and show that t-norms, t-conorms, uninorms and nullnorms (t-operators) are special cases of n-uninorms. We also show that given n there are n+1 classes of operators in Un and each of them has many subclasses. We also study the Frank equation involving n-uninorms and show that we need to consider only n-uninorms for the study. Finally, we show that the total number of subclasses of operators in Un follows the famous series called Catalan Numbers.