Generalization of QL-operators based on general overlap and general grouping functions Cecilia Botelho * and Alessandra Galvao * Lab. of Ubiquitous and Parallel Systems (LUPS/CDTEC) * Federal University of Pelotas (UFPEL) Pelotas, Brazil {cscbotelho, argalvao}@inf.ufpel.edu.br Helida Santos †‡ C3 - † Universidade Federal do Rio Grande (FURG) Rio Grande, Brazil ISC - ‡ Universidad Publica de Navarra (UPNA) Pamplona, Spain helida@furg.br Jocivania Pinheiro § Dep. de Ciˆ encias Exatas, Matem´ atica e Estat´ ıstica (DCME) § Universidade Federal Rural do Semi- ´ Arido (UFERSA) Mossor´ o, Brazil vaniamat@ufersa.edu.br Benjam´ ın Bedregal ¶ Dep. de Matem´ atica e Inform´ atica Aplicada (DIMAp) ¶ Universidade Federal do Rio Grande do Norte (UFRN) Natal, Brazil bedregal@dimap.ufrn.br Adenauer Yamin * and Renata Reiser * Lab. of Ubiquitous and Parallel Systems (LUPS/CDTEC) * Federal University of Pelotas (UFPEL) Pelotas, Brazil {adenauer, reiser}@inf.ufpel.edu.br Abstract—Firstly, this work discusses the main conditions guarantying that general overlap (grouping) functions can be obtained from n-dimensional overlap (grouping) functions. Fo- cusing on QL-implications, which are usually generated by strong negations together with t-norms and t-conorms, we consider a non-restrictive construction, by relaxing not only the associativity and the corresponding neutral elements (NE) but also the reverse construction of other properties. Thus, the main properties of the QL-implication class are studied, considering a tuple (G,N,O) generated from grouping and overlap functions together with the greatest fuzzy negation. In addition, in order to provide more flexibility, we define a subclass of QL-implications generated from general overlap and general grouping functions. Some examples are introduced, illustrating the constructive methods to generate such operators. Index Terms—General Overlap Function, General Grouping Function, QL-implication, Fuzzy Implication, Aggregation func- tion I. I NTRODUCTION In the scope of fuzzy logic, implication functions are a vital element of its constitution. Defining and using implication functions to represent different scenarios in fuzzy inference systems is still an open challenge, exploring different classes of implications [1]–[6]. Firstly, this paper provides the essential conditions guar- antying that general overlap/grouping functions can be con- structed from the equivalent n-dimensional operators. This This work was partially supported by CAPES, UFERSA, PQ/CNPq (309160/2019-7; 311429/2020-3), PqG/FAPERGS (21/2551-0002057-1) and FAPERGS/CNPq PRONEX (16/2551-0000488-9). study also includes a study on extensions of fuzzy implication operators obtained via general overlap and grouping functions, addressing mainly the QL-operators and QL-implications [3]. The main proposal of this article is to explore constructive methods to generate implications through the concepts of general overlap and general grouping functions, relaxing some properties. The methods studied provide more flexibility to the underlying fuzzy inference system structure. This paper is organized as follows. Section 2 brings the main theoretical concepts. In Section 3, we discuss how to obtain general overlap functions from n-dimensional overlap functions, and likewise, in Sect. 4, we present general grouping functions from n-dimensional grouping functions. Sect. 5 addresses the QL-operations and their properties. Moreover, in Sect. 6 we discuss the conditions for generating QL- implications from a tuple (G,N, O). We conclude in Sect. 7 with the final remarks and future works. II. PRELIMINARIES A. Fuzzy Negation Definition 2.1: A function N : [0, 1] → [0, 1] is fuzzy negation if it: (N1) is decreasing; (N2) satisfies N (0) = 1 and N (1) = 0 (boundary conditions). A fuzzy negation N is said to be strong if it is involutive (N3) N (N (x)) = x, ∀x ∈ [0, 1], as the strict negation N S (x)=1 − x. However, a counterexample of the involution © 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other work.