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Structure of \(n\)-uninorms. (English) Zbl 1122.03044
Summary: 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 \(U^{n}\), 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 \(U^{n}\) 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 \(U^n\) follows the famous series called Catalan numbers.

MSC:
03E72 Theory of fuzzy sets, etc.
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