Modiﬁcations of Łukasiewicz’s intuitionistic fuzzy implication

: In [6], G. Klir and B. Yuan named after J. Łukasiewicz the implication p → q = min(1 , p + q ) . In a series of papers, 198 different intuitionistic fuzzy implications have been introduced, and their basic properties have been studied. Here we introduce six new implications which are modiﬁcations of Łukasiewicz’s intuitionistic fuzzy implication, and we describe and prove some of their properties.

Here, following the scheme from [2,4,5], we will construct eight new implications, related to the Łukasiewicz's intuitionistic fuzzy implication but we will observe that two of them coincide and one is trivial as it permanently yields in result a constant. Therefore, of these eight six are the new implications which will present and study in details.
Let everywhere below for IFPs x and y: For the IFPs, in [3], different operations and relations have been defined. For our aims, we will remind the definitions of only three of these relations and one operation:
Theorem 1. For every two IFPs x and y Proof. Let the two IFPs x and y be given. Then from the inequalities the validity of Theorem 1 follows.
In contrast to [4], in the general case, the relations in the form are not valid for 1 ≤ i, j ≤ 4.
Indeed, for i = 1 we obtain: which is not equal to any ¬(x → j y) with 1 ≤ j ≤ 4.
Similarly to [4], if for the scheme ( * ) we use M 1 , M 2 ∈ { + , × }, we will receive the following four new intuitionistic fuzzy implications: Obviously, implication → 7 is trivial, because its result is always a constant, while implications → 5 and → 8 coincide and hence, we can only work with the first of them, → 5 . Now, we formulate Theorem 2, which can be proved in the same manner as Theorem 1, hence the proof is skipped. Theorem 2. For every two IFPs x and y x → 5 y ≥ x → 6 y.
We directly check the validity of the following equalities.  .
Following [2] and using, e.g. [7], we mention that Therefore, the new implications generate the following negations: We see immediately that for each IFP x It is important to mention that the constructed here implications are new ones, while negations ¬ 1 and ¬ 2 coincide with negation ¬ 8 from [2], negations ¬ 3 and ¬ 4 -with negation ¬ 4 from [2], negations ¬ 6 -with negation ¬ 35 from [2] and only negation ¬ 5 is a new one.
In [6], G. Klir and B. Yuan give nine axioms for fuzzy implication. In [2], K. Atanassov pre-formulated them for the case of intuitionistic fuzziness. Let O * = 0, 1 , The IFP is a tautology iff a = 1 and b = 0, while it is an Intuitionistic Fuzzy Tautology (IFT) iff a ≥ b.