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In this paper we propose an algorithm to solve group decision making problems using n-dimensional fuzzy sets, namely, sets in which the membership degree of each element to the set is given by an increasing tuple of n elements. The use of these sets has naturally led us to define admissible orders for n-dimensional fuzzy sets, to present a construction method for those orders and to study OWA operators for aggregating the tuples used to represent the membership degrees of the elements. In these conditions, we present an algorithm and apply it to a case study, in which we show that the exploitation phase which appears in many decision making methods can be omitted by just considering linear orders between tuples.

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Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications R O -implications, the strong implications (G, N )-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N . Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x, I(x, y)) ≤ y, for any fuzzy implication I and overlap function O, and, in particular, for R O -implications, (G, N )-implications, QL-implications and D-implications derived from tuples (O, G, N ), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T -conditionality and U -conditionality, for t-norms T and uninorms U , respectively. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ inference processes via the generalized Modus Ponens (GMP) and Modus Tollens (GMT), carried out via the Zadeh's Compositional Rule of Inference [5]. 1 The GMP, which is the focus of this paper, can be implemented by a scheme enabled by the functional inequality called the Law of T -Conditionality, which, for a t-norm T and a fuzzy implication I, is stated by:In fact, (TC) means x * T (x → y) ≤ y, which generalizes the Modus Ponens x ∧ T (x → y) y to the fuzzy context [7, 8].In the literature, the studies on the T -Conditionality have been done just for the three main families of fuzzy implications, namely, R-, (S, N )and QL-implications. 2 R-implications are generalizations to [0, 1] of Boolean implications defined by the identity given, for a universe set X, bywhere A, B ⊆ X, and the intersection is generalized by a t-norm. This class of implications is related to a residuation concept from the intuitionistic logic. On the other hand, (S, N )-implications are generalizations to [0, 1] of the Boolean material implication defined aswhere ∨ and ¬ are...

In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Lukasiewicz implication.

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The final goal of this study is to adapt the concept of fuzzy entropy of De Luca and Termini to deal with Type-2 Fuzzy Sets. We denote this concept Type-2 Fuzzy Entropy-Set. However, the construction of the notion of entropy measure on an infinite set, such us [0, 1], is not effortless. For this reason, we first introduce the concept of quasi-entropy of a Fuzzy Set on the universe [0, 1]. Furthermore, whenever the membership function of the considered Fuzzy Set in the universe [0, 1] is continuous, we prove that the quasi-entropy of that set is a fuzzy entropy in the sense of De Luca y Termini. Finally, we present an illustrative example where we use Type-2 Fuzzy Entropy-Sets instead of fuzzy entropies in a classical fuzzy algorithm.