Study of Two Families of Generalized Yager's Implications for Describing the Structure of Generalized (h,e)-Implications

Article

Study of Two Families of Generalized Yager’s Implications for Describing the Structure of Generalized ( h, e ) -Implications

Raquel Fernandez-Peralta^1,2,* , Sebastia Massanet^1,2 and Arnau Mir^1,2

Citation: Fernandez-Peralta, R.;

Massanet, S.; Mir, A. Study of Two Families of Generalized Yager’s Implications for Describing the Structure of Generalized

(h,e)-Implications.Mathematics2021, 9, 1490. https://doi.org/10.3390/

math9131490

Academic Editor: Basil Papadopoulos

Received: 25 May 2021 Accepted: 21 June 2021 Published: 24 June 2021

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4.0/).

1 Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma, Spain; s.massanet@uib.es (S.M.);

arnau.mir@uib.es (A.M.)

2 Health Research Institute of the Balearic Islands (IdISBa), 07010 Palma, Spain

* Correspondence: r.fernandez@uib.es

Abstract: In this study, we analyze the family of generalized(h,e)-implications. We determine when this family fulfills some of the main additional properties of fuzzy implication functions and we obtain a representation theorem that describes the structure of a generalized(_h,e)-implication in terms of two families of fuzzy implication functions. These two families can be interpreted as particular cases of the(f,g)_and(g,f)-implications, which are two families of fuzzy implication functions that generalize the well-knownf andg-generated implications proposed by Yager through a generalization of the internal factorsxand¹_x, respectively. The behavior and additional properties of these two families are also studied in detail.

Keywords:fuzzy implication function; generalized(h,e)-implications; generalized Yager’s implica- tions; characterization; horizontal threshold method

1. Introduction

In fuzzy logic, fuzzy implication functions play a key role as operators that generalize the classical implications in crisp logic. They are functionsI:[0, 1]²→[0, 1]fulfilling some boundary properties in order to coincide with the classical implication in{0, 1}²and some monotonicity properties. These operators are important for both theory and applications.

For instance, in the same way classical implications are used in inference schemas such as modus ponens, modus tollens, etc., fuzzy implication functions play a similar role in the generalization of these schemas, which use fuzzy statements whose value is in [0, 1]

instead of being in{0, 1}. In addition, fuzzy implication functions also play a different role in other applications such as fuzzy mathematical morphology, fuzzy control or data analysis [1]. Depending on the context and the proper rule and its behavior, various fuzzy implication functions with different properties can be adequate. This fact has motivated the study and definition of many families of fuzzy implication functions, and more than 100 families have been defined until now. For further information, consult the surveys [1–3]

and the books [4,5].

Although apparently there is a huge amount of fuzzy implication functions available, these families can present intersection or even coincide with others already known [6].

For this reason, it is of the utmost importance to study the additional properties that the operators of certain families satisfy and to provide an axiomatic characterization of the new operators in the literature in order to find its possible relation with respect to those already known. In this respect, the characterization of several families of fuzzy implication functions have already been achieved:(S,N)-implications with a continuous negation [7], R-implications obtained from left-continuous t-norms [5,8,9], someQL-implications [10], Yager’s implications [11],h-implications [12], probabilistic and survivalS-implications [6];

among others [13,14].

Mathematics2021,9, 1490. https://doi.org/10.3390/math9131490 https://www.mdpi.com/journal/mathematics

In this paper, we are interested in the study of the family of(h,e)-implications. These fuzzy implication functions were defined for the first time in [15] under the motivation of generalizingh-implications to a new family of functions satisfying the propertyI(e,y) =y for ally∈[0, 1]and somee∈(0, 1). The family of(h,e)-implications has presented very interesting properties such as the controlled increase in the second variable by the parameter eor that they constitute a whole family of fuzzy implication functions fulfilling the exchange principle, but not the law of importation for any t-normT[16]. Additionally, they have been applied on image processing for edge detection (see [17]), obtaining good results with respect to other families of fuzzy implication functions. Although the properties of (h,e)-implications were analyzed for the first time in [15], in [18] it was pointed out that a more general definition was possible. Therefore, our first contribution is to adapt all the existing results to this more general definition and to provide all the proofs.

In [12], it was proved thath-implications were characterized by the fact that their structure is determined by two Yager’s implications through the threshold horizontal method [12], more specifically, the structure of anh-implication is like an adequately scaled f-implication whenevery≤eand like an adequately scaledg-implication whenevery>e.

In this paper, as a second and main contribution, we provide a similar result for(h,e)- implications but, in our case, the structure of(h,e)-implications is described in terms of two new families of fuzzy implication functions that are generalizations of Yager’s implications, called(f,g)_and(g,f)-implications [19].

To date, there have been several proposals of generalizations of Yager’s implications by considering different approaches: generalizing the inner factorsxand ¹_x, in the expression of f- andg-generated implications, respectively, [19–22], considering a different internal function from the product [23–25]; among others [26,27]. The families of(f,g)and(g,f)- implications are of the first kind since they consider the internal factorxas a continuous and strictly decreasing function g : [_{0, 1}] → [_0,+_∞]_with g(₀) = _{0 and} ¹_x as a strictly decreasing function f : [0, 1] → [0,+_∞] with f(0) = +∞. This approach is different from all the other generalization proposals, and then the interest in these two families is twofold: to describe the structure of(h,e)-implications and to study these families in order to compare its properties with other generalizations considered in the literature. Although these two families were preliminary studied in [19], the results were provided without any proof and some of them were partially erroneous. Our third contribution is to provide the proofs of all the results and to rectify the mistakes in some statements.

The paper is structured as follows. In Section2, we list essential results on fuzzy implication functions. In Section3, the family of generalized(h,e)-implications and its main properties are recalled, and a proof of the representation theorem is given. In Section4, the families of(f,g)and(g,f)-implications are presented and their properties are studied.

Finally, in Section5, we gather concluding remarks and we outline some future work.

2. Preliminaries

In this section, we recollect some concepts and results that are useful throughout the paper. First, we recall the definition of fuzzy negation, t-norm and t-conorm, which are generalizations of the classical negation, conjunction and disjunction, respectively, to fuzzy logic.

Definition 1([8]). A decreasing function N : [0, 1] → [0, 1] is called afuzzy negation if N(₀) =₁and N(₁) =0.

Definition 2([28]). Atriangular normort-normis a binary function T:[0, 1]²→[0, 1]that is commutative, associative, increasing in both variables and 1 is its neutral element. Atriangular conormort-conormis a binary function S : [0, 1]² → [0, 1]that is commutative, associative, increasing in both variables and 0 is its neutral element.

Next, we introduce the definition of fuzzy implication function.

Definition 3([5,8]). A binary operator I : [0, 1]² → [0, 1]is said to be afuzzy implication functionif it satisfies:

(I1) I(x,z)≥I(y,z) when x≤y, for all z∈[0, 1]. (I2) I(x,y)≤I(x,z) when y≤z, for all x∈[0, 1]. (I3) I(0, 0) =I(1, 1) =1 and I(1, 0) =0.

From this definition, it is straightforward to prove that a fuzzy implication function is such thatI(x, 1) = 1 andI(0,x) =1 for allx ∈[0, 1]. However, the 0-horizontal section I(x, 0)and 1-vertical sectionI(1,x)forx∈(0, 1)cannot be deduced from the definition.

Since the definition of fuzzy implication function is rather general, additional proper- ties on these operators are considered. Although there exist many additional properties, we recall here the ones that have been more studied.

• Theidentity principle

(_IP) I(x,x) =1, x∈[0, 1].

• Theordering property

(OP) I(x,y) =1⇔x≤y, x,y∈[0, 1].

• Theexchange principle

(EP) I(x,I(y,z)) = I(y,I(x,z)), x,y,z∈[0, 1].

• Thelaw of importationwith respect to a t-normT

(LIT) I(T(x,y),z) =I(x,I(y,z)), x,y,z∈[0, 1].

• Theleft neutrality principle

(NP) I(1,y) =y, y∈[0, 1].

• Theleft neutrality principle with e∈(0, 1)

(NPe) I(e,y) =y, y∈[0, 1].

• Thecontrapositive symmetrywith respect to a fuzzy negationN, (_CP(_N)) I(x,y) =I(N(y)_,N(x))_, x,y∈[_{0, 1}]_.

From the valuesI(x, 0)of a fuzzy implication functionI, we obtain a fuzzy negation called the natural negation ofI.

Definition 4([5]). Let I be a fuzzy implication function. The function NIdefined by NI(x) =I(x, 0) for all x∈[0, 1]is called thenatural negationof I.

The wide definition of fuzzy implication function allows the existence of many fam- ilies with different additional properties. Depending on the construction method of a family, we can distinguish between three classes: those obtained by the combination of other logical operators such as fuzzy negations, t-norms or t-conorms; those generated by univaluated functions in the interval [0, 1]; and those generated from other fuzzy implica- tion functions. In [29], the reader can find an overview of construction methods of fuzzy implication functions.

Among those fuzzy implication functions whose definition is based on the use genera- tor functions, we can highlight Yager’s implications, calledf andg-generated implications.

These fuzzy implication functions are generated from additive generators of continuous Archimedean t-norms and t-conorms, respectively.

Definition 5([5], Definition 3.1.1). Let f : [0, 1] → [0,+∞] be a strictly decreasing and continuous function with f(₁) =0. The function I_f :[_{0, 1}]²→[_{0, 1}]defined by

I_f(x,y) = f⁻¹(x· f(y)), x,y∈[0, 1],

with the understanding0·(+∞) =0, is called an f-generated implication. The function f itself is called an f-generator.

Definition 6 ([5], Definition 3.2.3). Let g : [0, 1] → [0,+_∞] be a strictly increasing and continuous function with g(0) =0. The function Ig:[0, 1]²→[0, 1]defined by

Ig(x,y) =g⁽⁻¹⁾ 1

xg(y)

, x,y∈[0, 1],

with the understanding¹₀ = +_∞and+_∞·0= +_∞is called ag-generated implication, where the function g⁽⁻¹⁾is the pseudo-inverse of g given by

g⁽⁻¹⁾(x) =

g⁻¹(x) if x∈[0,g(1)], 1 if x∈[g(1),+_∞].

In this case, g is called the g-generator of the fuzzy implication function I defined as above.

Another family of fuzzy implication functions of this kind is the h-implications, introduced in [15] following the idea behind the definition of Yager’s implications, but considering additive generators of representable uninorms.

Definition 7([15], Definition 7). Fix an e∈(0, 1)and let h:[0, 1]→[−∞,+∞]be a strictly increasing and continuous function with h(0) =−∞, h(e) =0and h(1) = +∞. The function I^h:[_{0, 1}]²→[_{0, 1}]defined by

I^h(x,y) =







1 if x=0,

h⁻¹(x·h(y)) if x>0and y≤e, h⁻¹(¹_x·h(y)) if x>₀and y>e.

is called an h-implication. The function h itself is called an h-generator(with respect to e) of the fuzzy implication function I^hdefined as above.

These three families of fuzzy implication functions were characterized in [11,12].

Moreover, it was proved that the characterization ofh-implications could be derived from the characterization of Yager’s implications since the structure of anh-implication is given by anf and ag-generated implication through the horizontal threshold generation method.

The horizontal threshold generation method is a construction method that generates a fuzzy implication function from two given ones and it consists in an appropriate scaling of the second variable of the two fuzzy implication functions.

Theorem 1([12], Theorem 3). Let I₁, I₂be two fuzzy implication functions and e∈(0, 1). Then, the binary function I_I1−I₂ :[0, 1]²→[0, 1], called the e-horizontal threshold generated implication from I1and I2, defined as

I_I1−I₂(x,y) =







1 if x=0,

e·I₁ x,^y_e

if x>₀and y≤e, e+ (1−e)·I₂

x,^y−e_1−e

if x>0and y>e, is a fuzzy implication function.

3. Generalized(h,e)-Implications

In [15], a new class of fuzzy implications was presented, the family of(h,e)-implications.

The motivation behind its definition was modifying theh-implications towards fulfilling the property(NPe). Indeed, the family ofh-implications presented an unexpected behav- ior. Most of the families of fuzzy implication functions generated from uninorms such as RU-implications [30], which are generalizations of R-implications, tend to satisfy(NPe) instead of(NP). Beingh-generators the additive generators of representable uninorms, it would be expected that this family of fuzzy implications functions would satisfy(_NPe) instead of(_NP). This is achieved with a slight modification in the definition leading to the so-called(h,e)-implications.

Although(h,e)-implications were first defined in [15], in [18] it was pointed out that a more general definition was possible. The latter is the one we recall here.

Definition 8([18], Definition 11). Fix an e ∈ (0, 1)and let h : [0, 1] → [−∞,+∞] be a strictly increasing and continuous function with h(e) = 0and h(1) = +∞. The function I^hg,e :[0, 1]²→[0, 1]defined by

I^hg,e(x,y) =







1 if x=0,

h^{(−1) x}_eh(y) if x>0,y≤e, h⁻¹ _x^eh(y) if x>0,y>e, where the function h⁽⁻¹⁾is the pseudo-inverse of h given by

h⁽⁻¹⁾(x) =

h⁻¹(x) if x∈[h(₀)_,+_∞)_, 0 if x∈(−_∞,h(0)),

is called ageneralized(h,e)-implication. The function h itself is called a generalized h-generator (with respect to e) of the implication function I^hg,edefined as above.

Although the above definition was proposed in [18] and the properties that were already studied for(h,e)-implications in [15] were reconsidered for this more general defi- nition, the results in [18] were announced without any proof. Therefore, hereafter we recall those results, but provide the corresponding proof. Having said this, the next proposition ensures that generalized(h,e)-implications are indeed fuzzy implication functions.

Proposition 1([18], Proposition 9). If h is a generalized h-generator with respect to a fixed e∈(0, 1), then I^hg,eis a fuzzy implication function.

Proof.

- Letx₁,x₂,y∈[0, 1]withx₁<x₂. Sincehis strictly increasing, we haveh(x₁)<h(x₂) and it holds thath⁽⁻¹⁾is an increasing function. Now, we have to distinguish three cases:

- Ifx₁=0 thenI^hg,e(0,y) =1≥ I^hg,e(x2,y).

- Ifx₁6=0 andy≤ethenh(y)≤0 and^x_e¹h(y)≥ ^x_e²h(y). Consequently, I^hg,e(x1,y) =h⁽⁻¹⁾x₁

e h(y)≥h⁽⁻¹⁾x₂

e h(y)= I^hg,e(x2,y). - Ifx₁6=0 andy>ethenh(y)>0 and_x^e

1h(y)> _x^e

2h(y). Therefore, I^hg,e(x1,y) =h⁽⁻¹⁾

e x1h(y)

≥h⁽⁻¹⁾ e

x2h(y)

= I^hg,e(x2,y).

- Let x,y₁,y₂ ∈ [0, 1] withy₁ < y₂. Then, similarly to the previous item, we have h(y₁)<h(y2)and we have to consider four different cases:

- Ifx =_{0 then}I^hg,e(_0,y₁) =₁= I^hg,e(_0,y₂)_.

- Ifx 6=0 andy₁<y₂≤ewe have thath(y₁)<h(y₂)≤h(e) =0 and I^hg,e(x,y₁) =h⁽⁻¹⁾x

eh(y₁)≤h⁽⁻¹⁾x

eh(y₂)= I^hg,e(x,y₁)_.

- Ifx 6=0 andy₁≤e<y2, we have thath(y₁)≤0<h(y2)so ^x_eh(y₁)≤0< _x^eh(y2) and we get that

I^hg,e(x,y1) =h⁽⁻¹⁾x

eh(y1)≤h⁽⁻¹⁾(0) =e<h⁻¹e

xh(y2)= I^hg,e(x,y2). - Ifx 6=0 ande<y1<y2then we have that 0<h(y1)<h(y2). Thus,

I^hg,e(x,y1) =h⁻¹e

xh(y1)≤h⁻¹e

xh(y2)=I^hg,e(x,y2). - Finally,I^hg,esatisfies the boundary conditions since

- I^hg,e(0, 0) =1 by construction.

- I^hg,e(1, 1) =h⁻¹

1

eh(1)=h⁻¹(+_∞) =1.

- I^hg,e(1, 0) =h⁽⁻¹⁾

1

eh(0)=0.

Moreover, like in the case ofh-implications, the generator of a generalized(h,e)- implication is unique up to a positive multiplicative constant piecewise.

Proposition 2([18], Proposition 10). Let h₁,h₂ : [0, 1] → [−∞,+_∞] be two generalized h- generators with respect to a fixed e∈(0, 1). Then, the following statements are equivalent:

(i) I^h1,g,e =I^h2,g,e.

(ii) There exist constants k,c∈(0,+∞)such that h₂(x) =

k·h1(x) if x∈[0,e), c·h1(x) if x∈[e, 1]. Proof. The proof is identical to the proof of Theorem 17 in [15].

Hereunder, we recall some of the basic properties of generalized(h,e)-implications.

First of all, the next result studies the natural negation. Notice that in contrast with(h,e)- implications, the presence ofh⁽⁻¹⁾in the more general definition implies that the behavior of the natural negation depends on the value ofhin zero.

Proposition 3([18], Proposition 11). Let h be a generalized h-generator. Then,

- If h(0) =−∞, then the natural negation N_Ihg,e is the Gödel negation or least negation N_D1, given by

N_D1(x) =

1 if x=0, 0 otherwise.

- If h(0)>−∞, then the natural negation N_Ihg,e is given by

N_Ihg,e(x) =I^hg,e(x, 0) =







1 if x=_0,

h^{−1 x}_eh(₀) if x≤e,

0 if x>e.

Proof. Lethbe a generalizedh-generator. Then,

- Ifh(0) =−_{∞, then}h⁽⁻¹⁾=h⁻¹and for everyx∈[0, 1]we get N_Ihg,e(x) =I^hg,e(x, 0) =

1 if x=0, h⁻¹(−_∞) if x∈(0, 1], =

1 ifx=0, 0 ifx∈(0, 1]. - Ifh(0)>−_∞then we have

N_Ihg,e(x) = I^hg,e(x, 0) =







1 if x=0,

h^{−1 x}_eh(0) if ^x_eh(0)≥h(0), 0 if ^x_eh(0)<h(0),

=







1 ifx=0,

h^{−1 x}_eh(0) ifx≤e,

0 ifx>e.

The following proposition studies when generalized (h,e)-implications fulfill the additional properties of fuzzy implication functions considered in this paper.

Theorem 2([18], Theorem 22). Let h be a generalized h-generator and e∈(0, 1). The following properties hold:

(i) I^hg,e(x,y)≤e if and only if(x>₀and y≤e). Moreover, I^hg,e(x,e) =e for all x>0.

(ii) I^hg,esatisfies(EP)if and only if h(0) =−∞.

(iii) I^hg,e(x,y) = 1if and only if x = 0or y = 1. Thus, I^hg,e does not satisfy either(OP) or(IP).

(iv) I^hg,eis continuous, except at the points(0,y)with y≤e.

(v) I^hg,esatisfies(NPe), but does not satisfy(NP).

(vi) I^hg,edoes not satisfy(LIT)with respect to any t-norm T.

(vii) I^hg,edoes not satisfy(CP(N))with any fuzzy negation N.

Proof.

(i) It is clear that if x > 0 and y ≤ e, then I^hg,e(x,y) = h^{(−1) x}_eh(y) ≤ e since h(y)≤0. Otherwise, if x = 0, I^hg,e(0,y) = 1 for all y ∈ [0, 1] and ifx > 0 and y > e, I^hg,e(x,y) = h^{−1 e}_xh(y) > e because h(y) > 0. Moreover, we have that I^hg,e(x,e) =h^{(−1) x}_eh(e)=h⁽⁻¹⁾(0) =efor allx>0.

(ii) Assume that I^hg,e satisfies(EP). Now, let us considerh(0)> −∞and we will get a contradiction. On the one hand, if 0<x0<e², we have

I^hg,e(x₀,I^hg,e(1, 0)) = I^hg,e(x₀, 0) =h⁽⁻¹⁾x0

e h(0)=h⁻¹x0

e h(0).

On the other hand, let us computeI^hg,e(1,I^hg,e(x₀, 0)). First, we have x0<e⇒ ^x0

e h(0)>h(0)⇒I^hg,e(x0, 0) =h⁻¹x0

e h(0). Now, sincex0<e²and by item (i)I^hg,e(x0, 0)≤ewe get that

I^hg,e(_1,I^hg,e(_{x0, 0})) = _I^hg,e_1,h^−1x0

e h(₀)=_h^(−1)x0

e²h(₀)=_h^−1x0 e²h(₀)_. Since we have thath⁻¹is strictly increasing in[h(0),+∞], we get

h⁻¹x₀

e²h(0)<h⁻¹x₀ e h(0).

Hence, I^hg,e(1,I^hg,e(x0, 0)) < I^hg,e(x0,I^hg,e(1, 0)), in contradiction with the fact that I^hg,esatisfies(EP).

For the reverse implication, ifh(0) =−∞, we know that in this caseh⁽⁻¹⁾=h⁻¹and N_Ihg,e = ND₁. For anyx,y,z∈[0, 1], let us distinguish five cases:

- Ifx =0, then for ally,z∈[0, 1]we have

I^hg,e(0,I^hg,e(y,z)) =1=I^hg,e(y, 1) =I^hg,e(y,I^hg,e(0,z)). – Ify=0 for allx,z∈[0, 1]we have

I^hg,e(x,I^hg,e(0,z)) = I^hg,e(x, 1) =1=I^hg,e(0,I^hg,e(x,z)). - Ifx 6=0,y6=0 andz=0, we obtain

I^hg,e(x,I^hg,e(y, 0)) = I^hg,e(x,N_Ihg,e(y)) =I^hg,e(x, 0) =N_Ihg,e(x) =0

= N_Ihg,e(y) =I^hg,e(y, 0) =I^hg,e(y,N_Ihg,e(x))

= I^hg,e(y,I^hg,e(x, 0)).

- Ifx 6= 0,y 6=0 andz≤ e, by the item (i),I^hg,e(y,z)≤ eandI^hg,e(x,z)≤ eand consequently

I^hg,e(x,I^hg,e(y,z)) =I^hg,e

x,h⁻¹y

eh(z)=h⁻¹xy e²h(z). Similarly,

I^hg,e(y,I^hg,e(x,z)) =I^hg,e

y,h⁻¹x

eh(z)=h⁻¹xy e²h(z).

- Finally, ifx 6=0,y 6=0 ande< z≤1, then again by (i) we have I^hg,e(y,z)> e andI^hg,e(x,z)>eand thus

I^hg,e(x,I^hg,e(y,z)) =I^hg,e

x,h⁻¹ e

yh(z)

=h⁻¹ e²

xyh(z)

,

I^hg,e(y,I^hg,e(x,z)) =I^hg,e

y,h⁻¹e

xh(z)=h⁻¹ e²

xyh(z)

.

(iii) It is obvious that if x = 0 ory = 1, I^hg,e(x,y) = 1 sinceI^hg,eis a fuzzy implication function. Ify≤e, by item (i),I^hg,e(x,y)≤e<1 and ife<y<1, 0<h(y)<+_∞and we have

I^hg,e(x,y) =h⁻¹e

xh(y)<h⁻¹(+∞) =1.

(iv) By definition, the implication I^hg,e is continuous for all(x,y) ∈ (0, 1]×[0,e)and for all (x,y) ∈ (0, 1]×(e, 1]. Further, the vertical sections with a fixed x > 0 are continuous since

I^hg,e(x,e) =h⁽⁻¹⁾x

eh(e)=h⁽⁻¹⁾(0) =e, and

y→elim⁻I^hg,e(x,y) = lim

y→e⁻h⁽⁻¹⁾x

eh(y)=h⁽⁻¹⁾(0) =e,

y→elim⁺I^hg,e(x,y) = lim

y→e⁺h⁻¹e

xh(y)=h⁻¹(0) =e.

On the other hand, the horizontal sections withy>eare continuous sinceI^hg,e(0,y) =1 and

x→0lim⁺I^hg,e(x,y) = lim

x→0⁺h⁻¹e

xh(y)=h⁻¹(+∞) =1.

However, fixed 0<_y≤e,−∞<_h(_y)≤0 and we know thatI^hg,e(_0,y) =_{1, but}

x→0lim⁺I^hg,e(x,y) = lim

x→0⁺h⁽⁻¹⁾x

eh(y)=h⁽⁻¹⁾(0) =e,

thusI^hg,ehorizontal sections withy≤eare continuous except at the points(0,y)with 0 < y ≤ e. Finally, by Proposition3,I^hg,eis also not continuous at the point(0, 0). Now, applying ([5] Theorem A.0.4) adequately, we can prove thatI^hg,eis continuous, except at the points(0,y)withy≤e.

(v) For ally∈[0, 1]we have that I^hg,e(e,y) =

h⁽⁻¹⁾(h(y)) if y≤e, h⁻¹(h(y)) if y>e, =y.

Thus,I^hg,esatisfies(NPe). On the other hand, for ally>e, we have I^hg,e(1,y) =y⇔h⁻¹(eh(y)) =y⇔eh(y) =h(y)⇔e=1.

Thus,I^hg,edoes not satisfy(NP).

(vi) Suppose thatI^hg,efulfills(LIT)with respect to a t-normT, then we know that it also fulfills(EP), and by item (ii),h(₀) =−∞. Now, takingx =y=_{1 and}e<z<_{1, since} T(1, 1) =1 we find that

I^hg,e(T(1, 1),z) =I^hg,e(1,I^hg,e(1,z))⇔h⁻¹(eh(z)) =h⁻¹(e²h(z))⇔e=1.

Thus,I^hg,edoes not satisfy(_LIT)with respect to any t-normT.

(vii) Suppose thatI^hg,esatisfies(CP(N))with a fuzzy negationN. So, we haveI^hg,e(x,y) = I^hg,e(N(y),N(x))for allx,y∈[0, 1]. Takingx=1 andy=e, we know by item (i) that I^hg,e(1,e) =eand then

e=I^hg,e(1,e) =I^hg,e(N(e),N(1)) =I^hg,e(N(e), 0) =N_Ihg,e◦N(e). Ifh(0) =−_∞thenN_Ihg,e =N_D1 and we obtain a contradiction. Ifh(0)>−_∞then

e=N_Ihg,e◦N(e) =







1 ifN(e) =0

h⁻¹_N(e)

e h(0) ifN(e)≤e, 0 ifN(e)>e,

and the only feasible case isN(e)∈(0,e]and thenN(e)h(0) =h(e)·e=0, which is also a contradiction.

Perhaps one of the main properties of(h,e)-implications is given by (i) in the previous theorem. It reflects that these operators have a controlled increase with respect to the second variable produced by the insertion of the parametere, as we can graphically see in Figure1. Observe that the fuzzy implication functions generated by the horizontal threshold method in Theorem 1had a similar property, so it is intuitive to think that generalized(h,e)-implications are related in some way with this method. Furthermore, notice that the family of generalized(h,e)-implications provides an example of fuzzy implication functions that do not satisfy(LIT)with respect to any t-normTand yet satisfy (EP)whenh(0) =−∞, providing another argument of the fact that(LIT)is stronger than (EP)[16].

Example 1. Let us consider the generalized h-generator

h(x) =

( ln ^x_e

if x≤e,

−ln

1−x 1−e

if x>e, with e∈(0, 1). The corresponding(h,e)-implication is the following one

I^h1,e(x,y) =











1 if x=0,

e ^y_e^x_e

if x>0,y≤e, 1−(1−e)·^1−y_1−e

e

x if x>0,y>e.

(1)

The plot of this(h,e)-implication for different values of the parameter e can be seen in Figure1.

(a)e=0.25. (b)e=0.5. (c)e=0.75.

Figure 1.Plot of the(h,e)-implication given by Equation (1) for different values ofe.

Representation Theorem

Although there is no axiomatic characterization of(h,e)-implications, in [19] a rep- resentation theorem for this family was presented without the corresponding proof. In this article, we provide a proof to that result and we adjust it to the case of generalized (h,e)-implications.

Theorem 3. Let I :[0, 1]²→[0, 1]be a binary function and e∈(0, 1). Then, I is a generalized (h,e)-implication with respect to e if and only if there exist an f -generator and a g-generator with g(1) = +∞such that I is given by

I(x,y) =







1 if x=0,

e· f^{(−1) x}_e ·f ^y_e

if x>0,y≤e, e+ (1−e)·g⁻¹

e

x·g_y−e

1−e

if x>0,y>e.

(2)

Moreover, in this case generators h, f and g are related in the following way:

f(x) =−h(e·x) for all x∈[0, 1], g(x) =h(e+ (1−e)·x) for all x∈[0, 1],

h(x) =

−f ^x_e

if x≤e, g ^x−e_1−e

if x>e.

Proof. Let I be a generalized(h,e)-implication with respect to e. We know thathis a continuous and strictly increasing function withh(e) = 0 andh(1) = +∞. First of all, note that f(x) =−h(ex)andg(x) =h(e+ (1−e)x)are f andg-generators, respectively, sincef is a continuous and strictly decreasing function with f(1) =−h(e) =0 andgis a continuous and strictly increasing function withg(0) = h(e) =0. Note that sinceh⁻¹is well-defined on[h(₀)_,+_∞)_withh(₀)<0 then we have for allx∈[_0,+_∞)_that

f⁽⁻¹⁾(x) = ^h

(−1)(−x)

e , g⁻¹(x) = ^h

−1(x)−e 1−e . We will split the proof in two cases:

- Ifx>0 andy≤e, then e f⁽⁻¹⁾x

efy e

=e f⁽⁻¹⁾

−^x

eh(y)=h⁽⁻¹⁾x

eh(y)=I(x,y). - Ifx>0 andy>ethen

e+ (1−e)·g⁻¹ e

xg y−e

1−e

= e+ (1−e)· ^h

−1

e xh

e+ (1−e)^y−e_1−e−e 1−e

= h⁻¹e

xh(y)= I(x,y).

For the reverse implication, let us consider f andg-generators such that I is given by Equation (2). Consider

h(x) =

−f ^x_e

if x≤e, g ^x−e_1−e

if x>e.

This function is continuous, strictly increasing,h(_e) =−f(₁) =_{0 andh}(₁) =_g(₁) = +_∞.

Now, let us prove thatI=I^hg,e. Notice that h⁽⁻¹⁾(x) =

h⁻¹(x) if x∈[h(0),+_∞), 0 if x∈(−_∞,h(0)),

=







0 if x ∈(−∞,−f(0)),

e· f⁻¹(−x) if x∈[−f(0), 0], e+ (1−e)·g⁻¹(x) if x∈(0,+∞). Then, studying again two cases we have that

- Ifx>0 andy≤ethen I^hg,e(x,y) =h⁽⁻¹⁾x

eh(y)=h⁽⁻¹⁾

−^x efy

e

=e· f⁽⁻¹⁾x efy

e

=I(x,y). - Ifx>0 andy>ethen

I^hg,e(x,y) = h⁻¹e

xh(y)=h⁻¹ e

xg y−e

1−e

=e+ (1−e)·g⁻¹ e

xg y−e

1−e

= I(x,y).

The next example provides the construction of an(h,e)-implication by using the threshold horizontal method given an f-generator andg-generator withg(1) = +_∞.

Example 2. Take, for instance, e= ¹₂and the subsequent f and g-generators f(x) =−ln

x 2−x

, g(x) =ln 1+x

1−x

.

Then, it is easy to check that the following functions are fuzzy implication functions I₁(x,y) = f⁽⁻¹⁾x

ef(y)= ^2y

2x

(2−y)^2x+y^2x, I2(_x,y) = _g^−1e

xg(_y)= (1+y)^2x1 −(1−y)^2x1 (1+y)^2x1 + (1−y)^{2x1 .}

Then, an(h,e)-implication is constructed from I1and I2by using the threshold horizontal method as Theorem3shows. Concretely, the h-generator corresponds to

h(x) =ln x

1−x

. We can see the construction method graphically in Figure2.

(a)I1 (b)I2 (c)I^hg,e

Figure 2.Plot of an(h,e)-implication withe= ¹₂ constructed via the horizontal threshold method jointly with its generators.

Although Theorem3gives a useful description of the family of generalized(h,e)- implications, it is not an axiomatic characterization of this family, i.e., a characterization in terms of their own properties. For providing such results, a deeper study of this family is needed.

Let us recall that the characterization ofh-implications presented in [12] was written in terms of the threshold horizontal method, in particularh-implications are characterized by the fact that they are generated by an f-implication and ag-implication through the horizontal threshold method. In this case, the axiomatic characterization was not provided, but it can be easily obtained by using the characterizations of Yager’s implications presented in [11]. For the case of generalized(h,e)-implications, it is straightforward to prove that if we consider an f-generator, the functionI_f,e:[0, 1]²→[0, 1]defined by

I_f,e(x,y) = f⁽⁻¹⁾x

ef(y), x,y∈[0, 1],

with the understanding+_∞·0=0 is a fuzzy implication function and if we consider a g-generator withg(1) = +∞, the functionIg,e :[0, 1]²→[0, 1]defined by

Ig,e(x,y) =g⁽⁻¹⁾e

xg(y), x,y∈[0, 1],

with the understanding ¹₀ = +∞and+∞·0= +∞is also a fuzzy implication function.

Notice that Theorem3discloses that generalized(h,e)-implications are also characterized by the fact that they can be generated through the horizontal threshold method by the two new families of fuzzy implication functions just introduced. Therefore, in order to obtain a characterization of(h,e)-implications, we have to study and characterize the two families of fuzzy implication functions defined.

In particular,I_f,eandIg,eare fuzzy implication functions that belong to two families which are generalizations of the well-known Yager’s implications. In the next section, we deeply study these two families.

4. Generalized Yager’s Implications

In [19], two new families of fuzzy implication functions, called the(f,g)and(g,f)- generated implications, were defined as a generalization of the well-known Yager’s f and g-generated implications, respectively. In the definition of the f-generated implications, one can consider the functionxas a particular case of a family of strictly increasing and continuous functions defined as g : [_{0, 1}] → [_0,+_∞] _{such that} g(₀) = 0. The same happens to the role of ¹_x as a concrete case of a continuous, strictly decreasing function f : [0, 1] → [0,+_∞]such that f(0) = +∞. In this section, we recall the definitions and properties of these two families of fuzzy implication functions published in [19], providing the corresponding proofs and rectifying some wrongly stated results.

4.1. Generalization of f -Generated Implications

First, we will study a generalization of the f-generated implications, generalizing the functionxin its definition as a strictly increasing functiong :[_{0, 1}] → [_0,+_∞]_with g(0) =0.

Definition 9. Let f : [0, 1] →[0,+_∞]be a strictly decreasing and continuous function with f(1) =0and g:[0, 1]→[0,+_∞]be a continuous and strictly increasing function with g(0) =0.

The function If,g:[0, 1]²→[0, 1]defined by

I_f,g(x,y) = f⁽⁻¹⁾(g(x)f(y)), x,y∈[0, 1], (3) with the understanding0·(+∞) =0, is called an(f,g)-generated operation.

Remark 1. An initial difference between the family of f -generated implications and its general- ization is that we need to consider the pseudo-inverse of f . This is because when f(0) < +_∞, g(x)· f(y)may be bigger than the initial value f(0). Nevertheless, notice that Equation (3) can also be written in the following form without explicitly using the pseudo-inverse of f :

I(x,y) = f⁻¹ min

f(x)g(y),f(0), x,y∈[0, 1]. (4) An(_f,g)-generated operation may not fulfill all the conditions in Definition3, and then it is not always a fuzzy implication function.

Theorem 4. An (f,g)-operation I_f,gis a fuzzy implication function if and only if one of the following conditions hold:

(i) f(0) = +_∞.

(ii) f(0)<+_∞and g(1)≥1.

Proof. First, we will consider thatI_f,gis a fuzzy implication function with f(0)<+_{∞. In} this case,

I_f,g(1, 0) = f⁽⁻¹⁾(g(1)f(0)) =







f⁻¹(g(1)f(0)) if g(1)<1,

0 if g(1)≥1.

Since I_f,g is a fuzzy implication function, then it holds that I_f,g(1, 0) = 0 and hence, g(1)≥1.

Now, let us consider an(f,g)-operation satisfying (i) or (ii). The fact thatI_f,gis a fuzzy implication function can be seen from the following:

- Letx₁,x2,y∈[0, 1]withx₁≤x2. Sincegis strictly increasing, we have thatg(x₁)≤ g(x2). Now, sincef is strictly decreasing, f⁽⁻¹⁾is decreasing, and we find that

I_f,g(x₁,y) = f⁽⁻¹⁾(g(x₁)f(y))≥ f⁽⁻¹⁾(g(x2)f(y)) =I_f,g(x2,y), andIf,gsatisfies(I1).

- Considerx,y1,y2∈[0, 1]withy1≤y2then, again by the strictly decreasing nature of f,f⁽⁻¹⁾is decreasing, and hence, we have

f(_y1)≥ f(_y2) ⇒ g(_x)·f(_y1)≥g(_x)·f(_y2)

⇒ f⁽⁻¹⁾(g(x)· f(y1))≤ f⁽⁻¹⁾(g(x)· f(y2))

⇒ I_f,g(x,y1)≤ I_f,g(x,y2), andIf,gsatisfies(I2).

- If,g(0, 0) = f⁽⁻¹⁾(g(0)f(0)) = f⁽⁻¹⁾(0) =1.

- If,g(1, 1) = f⁽⁻¹⁾(g(1)f(1)) = f⁽⁻¹⁾(0) =1.

- If,g(1, 0) = f⁽⁻¹⁾(g(1)f(0))and we have two cases. If f(0) = +∞thenIf,g(1, 0) = f⁻¹(+∞) = 0. Otherwise, if f(0) < +∞andg(1) ≥ 1 then, f(0)g(1) ≥ f(0)and If,g(1, 0) =0.

Remark 2. In [21], a similar approach to provide a generalization of Yager’s f -implications was considered. In this case, the authors consider the fuzzy implication function given by I_f,g(x,y) = f⁽⁻¹⁾(g(x)f(y)) where f is an f -generator and g : [0, 1] → [0, 1] is an increasing function satisfying g(0) =0and g(1) =1. In this case, they consider functions g, which are not necessarily continuous, but with g(1) = 1. Our approach restricts to the case when g is continuous, but allows any value in(0,+∞)of g(1)whenever f(0) = +∞and any value g(1) ≥ 1whenever f(₀)<+∞. Clearly, the two families intersect when we consider a continuous, strictly decreasing function g with g(1) =1. Moreover, by Remark 2.1 in [21], in this particular case the resulting (f,g)-implications are in factφ-conjugated of f -generated implications with f generator given by

f◦g⁻¹andϕ=g.

When an(f,g)-operation fulfills Definition3, we will use the nomenclature(f,g)- implication and we will call an admissible pair of generators to the pair of functions(f,g). The next result shows that it is enough to consider the pairs (f,g) of admissible generators such that f(0) = +_∞orf(0) =1.

Proposition 4. Let I_f,gbe a fuzzy implication function with f(0) < +∞, then there exists a function f1with f1(0) =1such that(f1,g)is an admissible pair of generators and I_f,g=I_f1,g.

Proof. LetI_f,gbe a fuzzy implication function with f(0)<+_∞and consider f₁(x) = ^f_f(0)^(x). Then,(f₁,g)is an admissible pair of generators withf₁(₀) = ^f_f⁽⁰⁾₍₀₎ =1 and since f₁⁻¹(x) =

f⁻¹(x f(₀))_then

If₁,g(x,y) = f₁⁽⁻¹⁾(g(x)f1(y)) = f₁⁽⁻¹⁾

g(x)^f(y) f(0)

= f₁⁻¹

min

g(x)^f(y) f(0)^,f1(0)

= f⁻¹(min{g(x)f(y),f(0)}) =I_f,g(x,y).

Then next proposition shows that the(f,g)-generated implications have non-trivial zero region for some choice of generators.

Proposition 5. Let(f,g)be an admissible pair of generators. Then, the following statements hold:

(i) If g(1)< f(0) = +_{∞, then If,g}(x,y) =0if and only if y=0<x.

(ii) If g(1) = f(0) = +_{∞, then If,g}(x,y) =0if and only if y<x=1or y=0<x.

(iii) If f(0)<+_{∞, then If,g}(x,y) =0if and only if g(x)≥1and y≤ f⁻¹_f(0)

g(x)

.

Proof.

(i) Let us assume that g(1) < f(0) = +_∞then f⁽⁻¹⁾ = f⁻¹. Hence, for every x,y∈[0, 1], we find that

If,g(x,y) = f⁻¹(g(x)f(y)) =0⇔g(x)f(y) = f(0) = +∞.

However, we know thatg(x)≤g(1)<+∞, then the only possibility isf(y) = +_∞ andg(x)6=0. Consequently,y=0<x.

(ii) Again we have that f⁽⁻¹⁾= f⁻¹and then,

I_f,g(x,y) =₀⇔g(x)f(y) = +_∞.

Therefore,g(x) = +_∞and f(y) > 0 or, g(x) > 0 and f(y) = +∞. Hence, the results follows.

(iii) Considerx,y∈[0, 1]then

I_f,g(x,y) =₀⇔ f⁽⁻¹⁾(g(x)f(y)) =₀⇔g(x)f(y)∈[f(₀)_,+_∞)_.

Now, sincefis strictly decreasing,f(y)≤ f(0)for ally∈[0, 1]and then necessarily g(x)≥1. Finally,

g(x)f(y)≥ f(0)⇔ f(y)≥ ^f(0)

g(x) ⇔y≤ f⁻¹ f(0)

g(x)

.

On the other hand, the next proposition shows that the region where the (f,g)- generated implications take the value 1 is independent of their generators.

Proposition 6. Let(f,g)be an admissible pair of generators. Then I_f,g(x,y) =1if and only if x=0or y=1.

Proof. Let(f,g)be an admissible pair of generators andx,y∈[0, 1]. Then, I_f,g(x,y) =₁ ⇔ f⁽⁻¹⁾(g(x)f(y)) =₁⇔g(x)f(y) =₀

⇔ g(x) =0 or f(y) =0⇔x =0 ory=1.

The determination of the one region obtained in the previous proposition is a property of fuzzy implication functions deeply studied in [31] where it is explained that the property I(x,y) = 1 ⇔ x = 0 ory = 1 is very important for the definition of strong equality indices. Consequently,(f,g)-implications could be used to generate strong equality indices.

Furthermore, in [11], this property plays a crucial role in the characterization off-generated implications.

From the previous proposition, the following result is straightforward.

Corollary 1. Let(f,g)be an admissible pair of generators. Then, the(f,g)-implication I_f,gdoes not satisfy either(IP)or(OP).

The next proposition studies under which conditions(NP)is satisfied by the(f,g)- generated implications. This property is satisfied by many of the most well-known families and therefore, to determine when(f,g)-implications fulfill(NP)is a necessary step for forthcoming studies on the intersections of this family with other existing families.

Proposition 7. Let(f,g)be an admissible pair of generators. Then, I_f,gsatisfies(NP)if and only if g(1) =1.

Proof. Consider(f,g)an admissible pair of generators. It is clear from the definition of fuzzy implication function that I_f,g(1, 0) = 0 and I_f,g(1, 1) = 1. Otherwise, since f is strictly decreasing and continuous with f(1) =0, we can choosey ∈ (0, 1)such that g(1)f(y)< f(0)and then

I_f,g(_1,y) = f⁽⁻¹⁾(g(₁)f(y)) =y⇔g(₁)f(y) = f(y)⇔g(₁) =_1.

The following result studies the natural negation of these fuzzy implication functions.

The properties of the natural negation play an important role in many characterization results of fuzzy implication functions.

Proposition 8. Let(f,g)be an admissible pair of generators. Then, the following properties hold:

(i) If f(0) = +∞, then the natural negation NI_f,g is the Gödel or least negation N_D1. (ii) If f(0)<+∞, then the natural negation NI_f,g is given by

N_If,g(x,y) =

f⁻¹(g(x)f(0)) if g(x)<1,

0 if g(x)≥1.

(iii) The natural negation NI_f,g is continuous and strictly increasing if and only if f(0)<+_∞ and g(1) =1.

Proof.

(i) If f(0) = +_{∞, then}f⁽⁻¹⁾= f⁻¹and for everyx∈[0, 1]we get NI_f,g(x) = f⁻¹(g(x)f(0)) =

f⁻¹(+_∞) if g(x)6=0, f⁻¹(0) if g(x) =0, =

0 if x>0, 1 if x=0,

= ND₁(x). (ii) If f(0)<+_∞then we have

NI_f,g(x) = f⁽⁻¹⁾(g(x)f(0)) =

f⁻¹(g(x)f(0)) if g(x)f(0)< f(0), 0 if g(x)f(0)≥ f(0),

=

f⁻¹(g(x)f(0)) if g(x)<1,

0 if g(x)≥1.

(iii) If f(0) < +_∞ and g(1) = 1, it is straightforward from point (ii) that N_If,g is a continuous function since it is the composition of real continuous functions.

Considerx1<x2, by the strictly increasing nature ofg, we have thatg(x1)f(0)<

g(x2)f(0). Now, since f is strictly decreasing, f⁻¹is strictly decreasing in[0,f(0)], and we get that

N_If,g(x₁) = f⁻¹(g(x₁)f(0))> f⁻¹(g(x₂)f(0)) =N_If,g(x₂).

Hence,NI_f,g is strictly decreasing. Reciprocally, items (i) and (ii) prove that the obtained natural negations are not strictly decreasing when f(0) = +∞or when

f(0)<+∞andg(1)>1.

At this point, we analyze the discontinuity points of the(f,g)-generated implications, which can be(0, 0)or(1, 1).

Proposition 9. Let(f,g)be an admissible pair of generators. Then If,gis continuous everywhere except at point(0, 0)when f(0)= +∞or at point(1, 1)when g(1) = +∞.

Proof. Let(f,g)be an admissible pair of generators, then by definitionI_f,gis continuous at each(x,y)∈[0, 1]²by being the composition of real continuous functions except for the cases when (g(x) =0 and f(y) = +_{∞) or (g}(x) = +_∞or f(y)=0), since in these situations we have considered the convention 0·(+_∞) =0. These two situations correspond to the following two cases:

• Ifx =y=0 andf(0)= +∞, then because of (i) in Proposition8, the natural negation ofI_f,gis not continuous atx=0, and thereforeI_f,gis non-continuous at(0, 0).

• Ifx=y=1 andg(1) = +∞, then I_f,g(1,y) = f⁽⁻¹⁾(+_∞·f(y)) =

f⁻¹(0) if y=1, 0 if y<1, =

1 if y=1, 0 if y<1.

Hence, we have that

y→1lim⁻I_f,g(1,y) =06=1=I_f,g(1, 1), andI_f,gis not continuous at(1, 1).

Consequently, there are members of the family that are continuous in the whole domain, and therefore, feasible to be applied in several fields.

Finally, we present two results that determine completely when the(f,g)-generated implications satisfy(EP)or(LIT).

Proposition 10. Let(f,g)be an admissible pair of generators. Then, the following statements are equivalent:

(i) I_f,gsatisfies(EP).

(ii) f(0) = +∞or ( f(0)<+∞and g(1) =1).

Proof. Assume thatIf,gsatisfies(EP). Now, let us consider f(0)<+∞andg(1)>1 and get a contradiction. On the one hand, we have that

I_f,g(x,I_f,g(1, 0)) = I_f,g(x, 0) = f⁽⁻¹⁾(g(x)f(0)).