Modiﬁcations of the Goguen’s intuitionistic fuzzy implication

: On the basis of the Goguen’s intuitionistic fuzzy implication, seven of its modiﬁcations are constructed. Some of their basic properties are studied. The negations, generated by these implications are introduced and some of their properties are also described.


Introduction
In [6], an intuitionistic fuzzy form of the Joseph Amadee Goguen's (28 June 1941 -3 July 2006) implication is given and some of its basic properties are studied. Its name is related to the following fuzzy implication (see, e.g., [9]): Here, following and combining ideas from [4,6,8,11], we are going to construct 7 new implications, related to the Goguen's intuitionistic fuzzy implication.

Preliminary results
In the beginning, the necessary concepts from Intuitionistic Fuzzy Pairs theory will be given, following [5,7].
In some definitions we are using functions sg and sg defined by: Let everywhere intuituinistic fuzzy truth values of variables x and y be: In [7], these pairs are called Intuituinistic Fuzzy Pairs (IFPs). The IFP a, b is: • a tautology if and only if (iff) a = 1 and b = 0; • an intuitionistic fuzzy tautology For two IFPs x and y: x ≤ y iff a ≤ c and b ≥ d.
Let everywhere below we suppose that for each p > 0 the following condition holds: The Goguen's intuitionistic fuzzy implications was defined in [6] for the IFPs x and y as: Let us define the empty IFP, the totally uncertain IFP, and the unit IFP (see [7,5]) by: Over a fixed IFP x a number of standard and extended modal operators have been defined (see [5]). Here, we use only four of them:

Main results
In [8], if ⊃ is a fixed implication, the following new modal implication can be introduced: where x and y are propositional variables, i.e., having IFP-representation.
Having the Goguen's intuitionistic fuzzy implication (→ G ) and following [4], we will construct eight new implications, but as we will see below, one of them (the last one) is trivial and it will be omitted.
As we mentioned above, the result of implication → * 8 is a constant and by this reason, we eliminate it.
First, we check that the definitions are correct. For implication → * 1 , we obtain. First, let If a > c, then sg(a − c) = 1, sg(a − c) = 0 and hence If a ≤ c, then sg(a − c) = 0, sg(a − c) = 1 and hence i.e., the implication → * 1 is correct. By the same manner, we check of the correctness of the rest implications.
Second, for the basic properties of implications we obtain that: are tautologies. Third, let us mention by ⊃ i the expression x → i y. Then we formulate and prove the next theorem.
Theorem 1. The following relations hold: Proof. For the check that ⊃ 2 ≥⊃ 1 , we put: If a > 1 − d(≥ c), then All other relations are proved in the same manner.
Therefore, the intuitionistic condition x → ¬¬x is satisfied as an IFT, as well as, as a tautology.
On the other hand, and it is not a tautology, because, e.g., for 0 < a < 1 = a, 1 − a that is not an IFT when a < 0.5 and therefore, it is not a tautology. By the same manner we can check the relations for → 6 and → 7 and can formulate the following theorem.