The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

Mutually exchangeable fuzzy implications

“…Let ðI; IÞ satisfies (ME), i.e., I satisfies (EP). Then from Lemma 5.4 in [29], it follows that ðI; I ½n Þ satisfies (ME) for all n 2 N.…”

“…Fact 1: From the results obtained in [29], it is clear that (ME) occupies a central position of importance in the preservation of many properties, functional equations and families w.r.t. the~-composition.…”

“…the~-composition. In fact, it was shown in [29] that for a pair of fuzzy implications satisfying a property (functional equation or coming a from a family of fuzzy implications) and (ME), their~-composition also satisfies the same, thus generating fuzzy implications that preserve the properties, functional equations and families of fuzzy implications through the~-composition. Fact 2: The functional equations involving aggregation operators have many applications in fuzzy logic and have been well established by many researchers, see for instance, [2,7,[10][11][12][13]15,16], including in this journal also, see [8,9,27,31].…”

“…(ii) However, when I 2 I F the powers I ½n of I are given by ðI f Þ ½n ðx; yÞ ¼ I f ðx n ; yÞ, for all x; y 2 ½0; 1 and n 2 N. Moreover, from Corollary 6.33 in [29], it follows that I ½n 2 I F for all I 2 I F and n 2 N. …”

“…In [29], Vemuri and Jayaram have proposed yet another generalization, namely mutual exchangeability (ME) of (1) of fuzzy implications as follows: Definition 1.3 ([29], Definition 3.9). A pair (I; J) of fuzzy implications is said to be mutually exchangeable if for all x; y; z 2 ½0; 1, Iðx; Jðy; zÞÞ ¼ Jðy; Iðx; zÞÞ: ðMEÞ Fuzzy implications satisfying (ME) are called as the mutually exchangeable fuzzy implications.…”