Additive generators based on generalized arithmetic operators in interval-valued fuzzy and Atanassov's intuitionistic fuzzy set theory

In this paper we investigate additive generators in Atanassov's intuitionistic fuzzy and interval-valued fuzzy set theory. Starting from generalized arithmetic operators satisfying some axioms we define additive generators and we characterize continuous generators which map exact elements to exact elements in terms of generators on the unit interval. We give necessary and sufficient condition under which a generator actually generates a t-nporm and we show that the generated t-norm belongs to particular classes of t-norms depending on the arithmetic operators involved in the defintion of the generator

Quasi-arithmetic means and OWA functions in interval-valued and Atanassov's intuitionistic fuzzy set theory

In this paper we propose an extension of the well-known OWA functions introduced by Yager to interval-valued (IVFS) and Atanassov’s intuitionistic (AIFS) fuzzy set theory. We first extend the arithmetic and the quasi-arithmetic mean using the arithmetic operators in IVFS and AIFS theory and investigate under which conditions these means are idempotent. Since on the unit interval the construction of the OWA function involves reordering the input values, we propose a way of transforming the input values in IVFS and AIFS theory to a new list of input values which are now ordered

The quest for rings on bipolar scales

We consider the interval $]{-1},1[$ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t.~$0$ are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.