Binary tree based construction of n-uninorm aggregation operators

Abstract

The concept of n-uninorm aggregation operators was introduced as generalization of uninorms and nullnorms. The structure of the operator is based on the existence of an n-neutral element for an associative, monotone increasing in both variables and commutative binary operator on [0, 1]. It has been shown that the number of subclasses of n-uninorms is the (n + 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> Catalan number. Stack sortable permutations and binary trees are well studied as combinatorial objects whose total number are also enumerated by Catalan numbers. We give a one-to-one correspondence between n-uninorms, stack sortable permutations and binary trees and use it to give an iterative algorithm to construct the operator on [0, 1] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for any of its Catalan number of subclasses.