On a certain type of unary operators

In our study we give a general form for modifiers that includes negation, different types of hedge and the sharpness operators. We will show that the four operators have a common form in the Pliant system and they will be called modifier operators. By changing the parameter value of a modifier we get the modalities, negation and the sharpness operators.


I. INTRODUCTION
The concept of hedge and modifiers appears at the very beginning of fuzzy set theory. They are related to an attempt to model meanings like "very", "more or less", "somewhat", "rather" and "quite". A hedge modifies the shape of the fuzzy set, inducing a change in the membership function. Thus a hedge transforms one fuzzy set into another fuzzy set. Here we will deal with strictly monotonously increasing and decreasing membership function.

A. Historical background
In the early 1970s, Zadeh [1] introduced a class of powering modifiers. He proposed computing with words as an extension of fuzzy sets and logic theory (Zadeh [2]). As pointed out by Zadeh [3]- [5], linguistic variables and terms are closer to human thinking (which emphasise importance more than certainty) and are used in everyday life. For this reason, words and linguistic terms can be used to model human thinking systems. Zadeh [1] said that a proposition such as "The sea is very rough" can be interpreted as "It is very true that the sea is rough." A number of studies [6], [7] have been conducted that discuss fuzzy logic and fuzzy reasoning with linguistic truth values. Basic notions of linguistic variables were formalized in different works by Zadeh in the mid 1970s [3]- [5]. These papers sought to provide a mathematical model for linguistic variables.

II. THE PLIANT OPERATOR SYSTEM
Here, we will be concerned with strict operators (strict tnorms and t-conorms).
Using the general representation theorem, we have for the strict t-norm (conjunctive operator) and the strict t-conorm (disjunctive operator). Those familiar with fuzzy logic theory will find that the terminology used here is slightly different from that used in standard texts [8]- [12].
In the Pliant system, we look for a class of operators with infinitely many negation operators.
Proof: See [13]. The general form of the multiplicative Pliant system is where ( ) is the generator function of the strict t-norm operator and : [0, 1] → [0, ∞] is a continuous and strictly decreasing function.
The corresponding negation operator is where * is the fix point of the negation, i.e. ( * ) = * and when we fix a certain 0 threshold then takes this value, i.e. ( ) = 0 . A characterization of this operator class can be found in [13]. We can introduce the aggregative operator (uninorm) consistent with the conjunctive and disjunctive operators and negations [14]. Here, we use the multiplicative form of a solution of the associative functional equation.
. (8) where ( ) is the generator function of either conjunctive or disjunctive operator. In the Pliant system the aggregative operator is unique [14]. The pan operators of the conjunctive and disjunctive operators have the same form as the Pliant system.

A. Modifiers based on connectives
We will start with the definition of the substantiating, and the weakening modifier. These modifiers are compositional modifiers.
Definition 3: The substantiating operator of grade induced by the conjunctive operator is and the corresponding dual modifier i.e. the weakening modifier is Using the representation theorem we get: We generalize the operator that is a positive real valued number. Instead of we will characterize the modifiers by 0 and values. First choose 0 and we define the unary operators in terms of , i.e. 0 is the fixed threshold.
The unary operators can be characterized by and . Now ( ) and ( ) can be expressed in terms of 0 , and . So we have: , .

Special cases
Product case: As a special case of the strict monotonous the operator if ( , ) = , we get Using the 0 and values and Eq.(16) is the Zadeh case, but ( ) ( ) differs from the previously defined weakening operator. We call this modifier system the product modifier system.  (14) and (15) have the same form and we can drop the and indices.

B. On the equivalence of conjunctive and disjunctive modifiers The mathematical model
We will introduce the modifiers induced by connectives: where , (0, ∞).
It is interesting to ask under what in conditions the following is valid for a certain choice and . Theorem 5: and from this is a strictly continuously decreasing function. Let = and we wish to find a that depends on . We have to solve the functional equation.

IV. THE SHARPNESS OPERATOR
As we saw previously, modifiers can be introduced by repeating the arguments of conjunctive and disjunctive operators -times. The next step is that is extended to any real number.
We will introduce the sharpness operator by repeating the arguments of the aggregation operator. Because in the Pliant system we have [14] ( 1) The negation operator: 2) The hedge operator (necessity and possibility operator): 3) The sharpness operator: These three types of operators can be represented in a common form.
Definition 10: The general form of the modifier operators is ( ) Theorem 11: The negation (29), the hedge (30) and the sharpness (31) are special cases of the modifier operators 32. Proof: is the sharpness operator Remark 12: The sharpness operator in the fuzzy concept is called the dispersion value transformation.

VI. KAPPA FUNCTION BASED ON DIFFERENTIAL EQUATION
We saw previously that ( ) is closely related to the generator function namely it is an isomorphic mapping of the abstract space of object to the real line. If we have different types of mapping, then we will have different operators. If we change the isomorphic function we get conjunctive, disjunctive or aggregation operators. See Figure 1. We will characterize ( ) by this function.
Let us introduce the relative effectiveness of ( ) by defining the following where ∕ = 0. This is the general form of the modifier operator. Proof: Let be a strictly monotonous transformation of ( ).
or writing this equation, we can have i.e. the proportion of the speed of the transformed value ( ) and the speed of are the same as the proportion of the transformed value and the value multiplied by a constant .

VIII. ODDS AND KAPPA FUNCTION
Let us denote the odds of the input variable by and by the odds of the output (transformed) value, i.e.
Let ( ) be the corresponding function between the input and output odds = ( ).

IX. SOME PROPERTIES OF THE MODIFIER
In the following we will summarize some of the basic properties of the modifier.   , has the same form as we can find in [16] where we show that this is the general form of a certain class of membership function. In this article we presented the general form of the negation, weakening, strenghtening and sharpness operators. We extended the Pliant operator system with a kappa function that describes various unary operators.