Keywords

1 Introduction

The distributivity of (classical) fuzzy implications over different fuzzy logical connectives, like t-norms, t-conorms or uninorms has been studied in the recent past by many authors (see [1–3, 7, 9, 13, 15, 30, 31, 33]). Distributivity equations have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems. Given an input “\(\widetilde{x} \) is \(A'\)”, the role of an inference mechanism is to obtain a fuzzy output \(B'\) that satisfies some desirable properties. The most important inference schemas are fuzzy relational inference and similarity based reasoning. In the first case, the inferred output \(B'\) is obtained either as

  1. (i)

    \(\sup - T\) composition, where T is a t-norm, as in the compositional rule of inference (CRI) of Zadeh (see [35]), or

  2. (ii)

    \(\inf -I\) composition, where I is a fuzzy implication, as in the Bandler-Kohout subproduct (BKS) (see [16]),

of \(A'\) and given rules. Since all the rules of an inference engine are exercised during every inference cycle, the number of rules directly affects the computational duration of the overall application.

To reduce the complexity of fuzzy “IF-THEN” rules, Combs and Andrews [18] proposed an equivalent transformation of the CRI to mitigate the computational cost. In fact, they demanded the following classical tautology

$$\begin{aligned} (p\wedge q)\rightarrow r=(p\rightarrow r)\vee (q\rightarrow r), \end{aligned}$$

written in fuzzy logic language, i.e., using t-norms, t-conorms and fuzzy implications. Subsequently, there were many discussions (see [17, 21, 29]), most of them pointed out the need for a theoretical investigation required for employing such equations. Later, the similar method but for similarity based reasoning was presented by Jayaram [27]. For an overview of the most important these methods see [12, Chapter 8].

In [4–6] (for the full article see [10]), [8, 11] we discussed the following distributivity equations

$$\begin{aligned} \mathcal {I}(x,\mathcal {T}_1(y,z))&= \mathcal {T}_2(\mathcal {I}(x,y),\mathcal {I}(x,z)), \\ \mathcal {I}(\mathcal {S}(x,y),z)&= \mathcal {T}(\mathcal {I}(x,z),\mathcal {I}(y,z)), \end{aligned}$$

for t-representable (decomposable) t-norms and t-conorms (in interval-valued fuzzy sets theory) generated from continuous Archimedean operations. In these articles we obtained the solutions for each of the following functional equations, respectively:

$$\begin{aligned} f(u_1+v_1,u_2+v_2)&= f(u_1,u_2)+f(v_1,v_2), \end{aligned}$$
(A)
$$\begin{aligned} g(\min (u_1+v_1,a),\min (u_2+v_2,a))&= g(u_1,u_2)+g(v_1,v_2), \end{aligned}$$
(B)
$$\begin{aligned} h(\min (u_1+v_1,a),\min (u_2+v_2,a))&= \min (h(u_1,u_2)+h(v_1,v_2),b), \end{aligned}$$
(C)
$$\begin{aligned} k(u_1+v_1,u_2+v_2)&= \min (k(u_1,u_2)+k(v_1,v_2),b), \end{aligned}$$
(D)

where \(a,b>0\) are fixed real numbers, \(f:L^{\infty }\rightarrow [0,\infty ]\), \(g:L^{a}\rightarrow [0,\infty ]\), \(h:L^{a}\rightarrow [0,b]\), \(k:L^{\infty }\rightarrow [0,b]\) are unknown functions and

$$\begin{aligned} L^{\infty }&=\{(u_1,u_2)\in [0,\infty ]^2 \; | \; u_1\ge u_2\}, \\ L^{a}&=\{(u_1,u_2)\in [0,a]^2 \; | \; u_1\ge u_2\}. \end{aligned}$$

More precisely, the solutions of Eq. (A) are presented in [4, Proposition 3.2], the solutions of Eq. (B) are presented in [5, Proposition 4.2], the solutions of Eq. (C) are presented in [10, Proposition 5.2] and the solutions of Eq. (D) are presented in [8, Proposition 3.2].

These investigations have been extended to the following functional equation

$$\begin{aligned} \mathcal {I}(x,\mathcal {U}_1(y,z)) = \mathcal {U}_2(\mathcal {I}(x,y),\mathcal {I}(x,z)), \end{aligned}$$
(D-UU1)

when \(\mathcal {U}_1\), \(\mathcal {U}_2\) are decomposable uninorms on \({\mathcal {L}}^I\) generated from two conjunctive representable uninorms and \(\mathcal {I}\) is an unknown function (see [14]). In that article we presented the solutions of the following functional equation

$$\begin{aligned} f(u_1+v_1,u_2+v_2) = f(u_1,u_2) + f(v_1,v_2), \qquad \qquad (u_1,u_2),(v_1,v_2)\in L^{\overline{\infty }}, \end{aligned}$$
(F)

where \(L^{\overline{\infty }}=\{(x_1,x_2)\in [-\infty ,\infty ]^2 \; | \; x_1\le x_2\}\), with the assumption \((-\infty )+\infty =\infty +(-\infty )=-\infty \) in both sets of domain (formally in both projections) and codomain of a function f.

In this article we continue this research for two distributivity laws: Eq. (D-UU1) and the following equation

$$\begin{aligned} \mathcal {I}(\mathcal {U}_1(x,y),z)&= \mathcal {U}_2(\mathcal {I}(x,z),\mathcal {I}(y,z)),\qquad \qquad x,y,z\in L^I, \end{aligned}$$
(D-UU2)

where \(\mathcal {U}_1\), \(\mathcal {U}_2\) are given decomposable uninorms, and function \(\mathcal {I}\) is unknown, in particular an implication. We show that in this case solving both Eqs. (D-UU1) and (D-UU2) reduces to finding solutions of Eq. (F) for some combinations of the following assumptions

figure a

in both sets of domain (formally in both projections) and codomain of a function f.

2 Interval-Valued Fuzzy Sets, Implications and Uninorms

One possible extension of fuzzy sets theory is interval-valued fuzzy sets theory introduced by Sambuc [32] (see also [26, 36]), in which to each element of the universe a closed subinterval of the unit interval is assigned – it can be used as an approximation of the unknown membership degree. Let us define

$$\begin{aligned}&\qquad \quad L^I=\{(x_1,x_2)\in [0,1]^2 \; | \; x_1\le x_2\}, \\&(x_1,x_2) \le _{L^I} (y_1,y_2) \Longleftrightarrow x_1\le y_1 \text { and } x_2\le y_2. \end{aligned}$$

In the sequel, if \(x\in L^I\), then we denote it by \(x = [x_1, x_2]\). In fact, \({\mathcal {L}}^I=(L^I,\le _{L^I})\) is a complete lattice with units \(0_{{\mathcal {L}}^I}=[0,0]\) and \(1_{{\mathcal {L}}^I}=[1,1]\).

Definition 2.1

An interval-valued fuzzy set on X is a mapping \(A:X\rightarrow L^I\).

We assume that the reader is familiar with the classical results concerning basic fuzzy logic connectives, but we briefly mention some of the results employed in the rest of the work.

One possible definition of an implication on \({\mathcal {L}}^I\) is the following one (cf. [12, 20, 24]).

Definition 2.2

Let \({\mathcal {L}}=(L,\le _L)\) be a complete lattice. A function \(\mathcal {I}:L^2\rightarrow L\) is called a fuzzy implication on \({\mathcal {L}}\) if it is decreasing with respect to the first variable, increasing with respect to the second variable and fulfills the following conditions:

$$\begin{aligned} \mathcal {I}(0_{{\mathcal {L}}},0_{{\mathcal {L}}})=\mathcal {I}(1_{{\mathcal {L}}},1_{{\mathcal {L}}}) =\mathcal {I}(0_{{\mathcal {L}}},1_{{\mathcal {L}}})= 1_{{\mathcal {L}}}, \qquad \mathcal {I}(1_{{\mathcal {L}}},0_{{\mathcal {L}}})=0_{{\mathcal {L}}}. \end{aligned}$$
(1)

Uninorms (in the unit interval) were introduced by Yager and Rybalov in 1996 (see [34]) as a generalization of triangular norms and conorms. For the recent overview of this family of operations see [23, 28].

Definition 2.3

Let \({\mathcal {L}}=(L,\le _L)\) be a complete lattice. An associative, commutative and increasing operation \(\mathcal {U}:L^2\rightarrow L\) is called a uninorm on \({\mathcal {L}}\), if there exists \(e\in L\) such that \(\mathcal {U}(e,x) = \mathcal {U}(x,e) = x\), for all \(x\in L\).

Remark 2.4

  1. (i)

    The neutral element e corresponding to a uninorm \(\mathcal {U}\) is unique. Moreover, if \(e=0_{{\mathcal {L}}}\), then \(\mathcal {U}\) is a t-conorm and if \(e=1_{{\mathcal {L}}}\), then \(\mathcal {U}\) is a t-norm.

  2. (ii)

    For a uninorm \(\mathcal {U}\) on any \({\mathcal {L}}\) we get \(\mathcal {U}(0_{{\mathcal {L}}},0_{{\mathcal {L}}})=0_{{\mathcal {L}}}\) and \(\mathcal {U}(1_{{\mathcal {L}}},1_{{\mathcal {L}}})=1_{{\mathcal {L}}}\).

  3. (iii)

    For a uninorm U on \(([0,1],\le )\) we get \(U(0,1) \in \{0,1\}\).

  4. (vi)

    For a uninorm \(\mathcal {U}\) on \({\mathcal {L}}^I\) with the neural element \(e\in L^I\setminus \{0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I}\}\) we get \(\mathcal {U}(0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I}) \in \{0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I}\}\) or \(\mathcal {U}(0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I}) \Vert e\), i.e., \(\mathcal {U}(0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I})\) is not comparable with e (cf. [19, 22]).

  5. (v)

    In general, for any lattice \({\mathcal {L}}\), if \(\mathcal {U}(0_{{\mathcal {L}}},1_{{\mathcal {L}}}) = 0_{\mathcal {L}}\), then it is called conjunctive and if \(\mathcal {U}(0_{{\mathcal {L}}},1_{{\mathcal {L}}}) = 1_{{\mathcal {L}}}\), then it is called disjunctive.

In the literature one can find several classes of uninorms (see [25, 28]). Uninorms that can be represented as in point (ii) of Theorem 2.5 are called representable uninorms.

Theorem 2.5

([25, Theorem 3]). For a function \(U:[0,1]^2\rightarrow [0,1]\) the following statements are equivalent:

  1. (i)

    U is a strictly increasing and continuous on \(]0,1[^2\) uninorm with the neutral element \(e\in ]0,1[\) such that U is self-dual, except in points (0, 1) and (1, 0), with respect to a strong negation N with the fixed point e, i.e.,

    $$\begin{aligned} U(x,y)=N(U(N(x),N(y))), \qquad x,y\in [0,1]^2\setminus \{(0,1),(1,0)\}. \end{aligned}$$
  2. (ii)

    U has a continuous additive generator, i.e., there exists a continuous and strictly increasing function \(h:[0,1] \rightarrow [-\infty ,\infty ]\), such that \(h(0) = -\infty \), \(h(e) = 0\) for \(e\in ]0,1[\) and \(h(1) = \infty \), which is uniquely determined up to a positive multiplicative constant, such that for all \(x,y\in [0,1]\) either

$$\begin{aligned} U(x,y) = {\left\{ \begin{array}{ll} 0 &{} \text {if } (x,y) \in \{(0,1),(1,0)\}, \\ h^{-1}(h(x) + h(y)), &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

when U is conjunctive, or

$$\begin{aligned} U(x,y) = {\left\{ \begin{array}{ll} 1 &{} \text {if } (x,y) \in \{(0,1),(1,0)\},\\ h^{-1}(h(x) + h(y)), &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

when U is disjunctive.

Remark 2.6

(cf. [3]). If a representable uninorm U is conjunctive, then \(U(x,y)=h^{-1}(h(x) + h(y))\) holds for all \(x,y\in [0,1]\) with the assumption

$$\begin{aligned} (-\infty )+\infty =\infty +(-\infty )=-\infty . \end{aligned}$$
(A-)

If a representable uninorm U is disjunctive, then \(U(x,y)=h^{-1}(h(x) + h(y))\) holds for all \(x,y\in [0,1]\) with the assumption

$$\begin{aligned} (-\infty )+\infty =\infty +(-\infty )=\infty . \end{aligned}$$
(A+)

Now we shall consider the following special class of uninorms on \({\mathcal {L}}^I\).

Definition 2.7

(see [19, 22]). A uninorm \(\mathcal {U}\) on \({\mathcal {L}}^I\) is called decomposable (or t-representable) if there exist uninorms \(U_1\), \(U_2\) on \(([0,1],\le )\) such that

$$\begin{aligned} \mathcal {U}([x_1,x_2],[y_1,y_2]) = [U_1(x_1,y_1),U_2(x_2,y_2)], \qquad [x_1,x_2],[y_1,y_2]\in L^I, \end{aligned}$$

and \(U_1\le U_2\). In this case we will write \(\mathcal {U}=(U_1,U_2)\).

It should be noted that not all uninorms on \({\mathcal {L}}^I\) are decomposable (see [22]).

Lemma 2.8

([22, Lemma 8]). If \(\mathcal {U}\) on \({\mathcal {L}}^I\) is a decomposable uninorm, then \(\mathcal {U}(0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I})=0_{{\mathcal {L}}^I}\) or \(\mathcal {U}(0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I})=1_{{\mathcal {L}}^I}\) or \(\mathcal {U}(0_{{\mathcal {L}}^I},1_{{\mathcal {L}}^I})=[0,1]\).

Therefore it is not possible that for decomposable uninorm \(\mathcal {U}=(U_1,U_2)\) on \({\mathcal {L}}^I\) we have that \(U_1\) is disjunctive and \(U_2\) is conjunctive.

Lemma 2.9

(cf. [22, Theorems 5 and 6]). If \(\mathcal {U}=(U_1,U_2)\) on \({\mathcal {L}}^I\) is a decomposable uninorm with the neutral element \(e=[e_1,e_2]\), then \(e_1=e_2\) is the neutral element of \(U_1\) and \(U_2\).

Lemma 2.10

([14, Lemma 3.9]). Let a function \(\mathcal {I}:({\mathcal {L}}^I)^2\rightarrow {\mathcal {L}}^I\) satisfy (1) and Eq. (D-UU1) with some uninorms \(\mathcal {U}_1\), \(\mathcal {U}_2\) defined on \(\mathcal {L}^I\). Then \(\mathcal {U}_1\) is conjunctive if and only if \(\mathcal {U}_2\) is conjunctive.

Lemma 2.11

Let a function \(\mathcal {I}:({\mathcal {L}}^I)^2\rightarrow {\mathcal {L}}^I\) satisfy (1) and Eq. (D-UU2) with some uninorms \(\mathcal {U}_1\), \(\mathcal {U}_2\) defined on \(\mathcal {L}^I\). Then \(\mathcal {U}_1\) is conjunctive if and only if \(\mathcal {U}_2\) is disjunctive and \(\mathcal {U}_1\) is disjunctive if and only if \(\mathcal {U}_2\) is conjunctive.

Proof

Putting \(x=z=0_{\mathcal {L}^I}\) and \(y=1_{\mathcal {L}^I}\) in (D-UU2) we have

$$\begin{aligned} \mathcal {I}(\mathcal {U}_1(0_{\mathcal {L}^I},1_{\mathcal {L}^I}),0_{\mathcal {L}^I})=\mathcal {U}_2(\mathcal {I}(0_{\mathcal {L}^I},0_{\mathcal {L}^I}),\mathcal {I}(1_{\mathcal {L}^I},0_{\mathcal {L}^I})). \end{aligned}$$

If \(\mathcal {U}_1\) is conjunctive, then \(\mathcal {U}_1(1_{\mathcal {L}^I},0_{\mathcal {L}^I})=0_{\mathcal {L}^I}\) and from (1) we obtain \(1_{\mathcal {L}^I}=\mathcal {U}_2(1_{\mathcal {L}^I},0_{\mathcal {L}^I})\), thus \(\mathcal {U}_2\) is disjunctive. If on the other hand \(\mathcal {U}_1\) is disjunctive, then \(\mathcal {U}_1(1_{\mathcal {L}^I},0_{\mathcal {L}^I})=1_{\mathcal {L}^I}\) and by (1) we have \(0_{\mathcal {L}^I}=\mathcal {U}_2(1_{\mathcal {L}^I},0_{\mathcal {L}^I})\), so \(\mathcal {U}_2\) is conjunctive.    \(\square \)

The above results allow us to investigate Eqs. (D-UU1) and (D-UU1) only for some decomposable uninorms.

3 Method for Solving Distributivity Eqs. (D-UU1) and (D-UU2) for Decomposable Uninorms

In this section we derive the Eq. (F) from distributivity Eqs. (D-UU1) and (D-UU2). Let \(\mathcal {U}_1=(U_1,U_2)\), \(\mathcal {U}_2=(U_3,U_4)\) be decomposable uninorms on \(\mathcal {L}^I\). Assume that the projection mappings on \(\mathcal {L}^I\) are defined as the following:

$$\begin{aligned} pr_1([x_1,x_2])=x_1, \qquad \qquad pr_2([x_1,x_2])=x_2, \qquad \qquad \text {for }[x_1,x_2]\in L^I. \end{aligned}$$

Eqs. (D-UU1) and (D-UU2) have the following form:

$$\begin{aligned} \mathcal {I}([x_1,x_2],&[U_1(y_1,z_1),U_2(y_2,z_2)]) \\ =&[U_3(pr_1 (\mathcal {I}([x_1,x_2],[y_1,y_2])),pr_1 (\mathcal {I}([x_1,x_2],[z_1,z_2]))), \\&~U_4(pr_2 (\mathcal {I}([x_1,x_2],[y_1,y_2])),pr_2 (\mathcal {I}([x_1,x_2],[z_1,z_2])))], \end{aligned}$$
$$\begin{aligned} \mathcal {I}([U_1(x_1,y_1),&U_2(x_2,y_2)],[z_1,z_2]) \\ =&[U_3(pr_1 (\mathcal {I}([x_1,x_2],[z_1,z_2])),pr_1 (\mathcal {I}([y_1,y_2],[z_1,z_2]))), \\&~U_4(pr_2 (\mathcal {I}([x_1,x_2],[z_1,z_2])),pr_2 (\mathcal {I}([y_1,y_2],[z_1,z_2])))], \end{aligned}$$

for \([x_1,x_2],[y_1,y_2],[z_1,z_2]\in L^I\). As a consequence we obtain the following four equations, which are satisfied for all \([x_1,x_2],[y_1,y_2],[z_1,z_2]\in L^I\),

$$\begin{aligned} pr_1(\mathcal {I}([x_1,x_2],&[U_1(y_1,z_1),U_2(y_2,z_2)])) \\&=U_3(pr_1 (\mathcal {I}([x_1,x_2],[y_1,y_2])),pr_1 (\mathcal {I}([x_1,x_2],[z_1,z_2]))), \\ pr_2(\mathcal {I}([x_1,x_2],&[U_1(y_1,z_1),U_2(y_2,z_2)])) \\&=U_4(pr_2 (\mathcal {I}([x_1,x_2],[y_1,y_2])),pr_2 (\mathcal {I}([x_1,x_2],[z_1,z_2]))), \end{aligned}$$
$$\begin{aligned} pr_1(\mathcal {I}([U_1(x_1,y_1),&U_2(x_2,y_2)],[z_1,z_2])) \\&=U_3(pr_1 (\mathcal {I}([x_1,x_2],[z_1,z_2])),pr_1 (\mathcal {I}([y_1,y_2],[z_1,z_2]))), \\ pr_2(\mathcal {I}([U_1(x_1,y_1),&U_2(x_2,y_2)],[z_1,z_2])) \\&=U_4(pr_2 (\mathcal {I}([x_1,x_2],[z_1,z_2])),pr_2 (\mathcal {I}([y_1,y_2],[z_1,z_2]))). \end{aligned}$$

Next, let us fix arbitrarily \([x_1,x_2],[z_1,z_2]\in L^I\) and define four functions \(k^1_{[x_1,x_2]}, k^2_{[x_1,x_2]}, l^{[z_1,z_2]}_1, l^{[z_1,z_2]}_2:\mathcal {L}^I \rightarrow \mathcal {L}^I\) by

figure b

where \(\circ \) denotes the standard composition of functions. Thus we have shown that if \(\mathcal {U}_1\) and \(\mathcal {U}_2\) on \(\mathcal {L}^I\) are decomposable, then Eqs. (D-UU1) and (D-UU2) are equivalent, respectively, to the following systems of equations:

$$\begin{aligned} \begin{aligned} k^1_{[x_1,x_2]}([U_1(y_1,z_1),U_2(y_2,z_2)])&= U_3(k^1_{[x_1,x_2]}([y_1,y_2]),k^1_{[x_1,x_2]}([z_1,z_2])), \\ k^2_{[x_1,x_2]}([U_1(y_1,z_1),U_2(y_2,z_2)])&= U_4(k^2_{[x_1,x_2]}([y_1,y_2]),k^2_{[x_1,x_2]}([z_1,z_2])), \end{aligned} \end{aligned}$$
(DUU-1')
$$\begin{aligned} \begin{aligned} l^{[z_1,z_2]}_1([U_1(x_1,y_1),U_2(x_2,y_2)])&= U_3(l^{[z_1,z_2]}_1([x_1,x_2]),l^{[z_1,z_2]}_1([y_1,y_2])), \\ l^{[z_1,z_2]}_2([U_1(x_1,y_1),U_2(x_2,y_2)])&= U_4(l^{[z_1,z_2]}_2([x_1,x_2]),l^{[z_1,z_2]}_2([y_1,y_2])). \end{aligned} \end{aligned}$$
(DUU-2')

Let us look closer to Eq. (DUU-1’). Assume that \(U_1=U_2\) and \(U_3=U_4\) are representable uninorms generated from \(h_1\) and \(h_3\), respectively. Next, by Lemma 2.10, let us assume that both \(U_1\), \(U_3\) are conjunctive or disjunctive. From Remark 2.6, if both uninorms are conjunctive, then we assume the assumption (\(A-\)) on the codomains of \(h_1\) and \(h_3\), while if both uninorms are disjunctive, then we the assumption (\(A+\)) on the codomains of \(h_1\) and \(h_3\).

Using the representation for representable uninorms i.e., Theorem 2.5, we can transform our problem to the following equation (for a simplicity we deal only with \(k^1\) now)

$$\begin{aligned} k^1_{[x_1,x_2]}&([h_1^{-1}(h_1(y_1) + h_1(z_1)),h_1^{-1}(h_1(y_2) + h_1(z_2))]) \\&=h_3^{-1}(h_3(k^1_{[x_1,x_2]}([y_1,y_2])) + h_3(k^1_{[x_1,x_2]}([z_1,z_2]))), \end{aligned}$$

where \([x_1,x_2],[y_1,y_2], [z_1,z_2]\in L^I\). Let us put \(h_1(y_1)=u_1\), \(h_1(y_2)=u_2\), \(h_1(z_1)=v_1\) and \(h_1(z_2)=v_2\). It is obvious that \(u_1,u_2,v_1,v_2\in [-\infty ,\infty ]\) and \(u_1\le u_2\), \(v_1\le v_2\), since \(y_1\le y_2\), \(z_1\le z_2\), and generator \(h_1\) is strictly increasing. If we define

$$\begin{aligned} f^1_{[x_1,x_2]}(u,v):= h_3\circ k^1_{[x_1,x_2]}([h_1^{-1}(u),h_1^{-1}(v)]), \qquad \qquad u,v\in [-\infty ,\infty ], \; u\le v, \end{aligned}$$

then we get the following functional equation

$$\begin{aligned} f^1_{[x_1,x_2]}&(u_1+v_1,u_2+v_2) = f^1_{[x_1,x_2]}(u_1,u_2)+f^1_{[x_1,x_2]}(v_1,v_2), \end{aligned}$$
(2)

where \((u_1,u_2),(v_1,v_2)\in L^{\overline{\infty }}\) and \(f^1_{[x_1,x_2]}:L^{\overline{\infty }}\rightarrow [-\infty ,\infty ]\) is an unknown function. By \(L^{\overline{\infty }}\) we denoted the set \(\{(x_1,x_2)\in [-\infty ,\infty ]^2: x_1\le x_2\}\).

Repeating all of the above calculations for the function \(k^2\), we get analogous functional equation:

$$\begin{aligned} f^2_{[x_1,x_2]}&(u_1+v_1,u_2+v_2)= f^2_{[x_1,x_2]}(u_1,u_2)+f^2_{[x_1,x_2]}(v_1,v_2), \end{aligned}$$
(3)

where \(f^2_{[x_1,x_2]}:L^{\overline{\infty }}\rightarrow [-\infty ,\infty ]\) is an unknown function defined by

$$\begin{aligned} f^2_{[x_1,x_2]}(u,v):= h_3\circ k^2_{[x_1,x_2]}([h_1^{-1}(u),h_1^{-1}(v)]), \qquad \qquad (u,v)\in L^{\overline{\infty }}. \end{aligned}$$

Observe that Eqs. (2) and (3) are exactly the same functional Eq. (F), i.e.,

$$\begin{aligned} f(u_1+v_1,u_2+v_2)= f(u_1,u_2)+f(v_1,v_2), \end{aligned}$$

where \(f:L^{\overline{\infty }}\rightarrow [-\infty ,\infty ]\) is an unknown function.

As a summary of this case we see that conjunctive representable uninorms \(U_1\), \(U_3\) leads us to Eq. (F) with the assumption (\(A-\)) on the domain and codomain of a function f, while the case of disjunctive representable uninorms \(U_1\), \(U_3\) leads us to Eq. (F) with the assumption (\(A+\)) on the domain and codomain of function f.

Now let us return to Eq. (DUU-2’). As before, let \(U_1=U_2\) and \(U_3=U_4\) will be representable uninorms generated by \(h_1\) and \(h_3\), respectively. By Lemma 2.11 we know that it is enough to consider again only two cases: when \(U_1\) is conjunctive and \(U_3\) disjunctive, or vice versa - when the \(U_1\) is an disjunctive, and \( U_3 \) conjunctive. We still assume (\(A-\)) on the codomains of generators of conjunctive uninorms and (\(A+\)) on the codomains of generators of disjunctive uninorms. For fixed \([z_1,z_2]\in L^I\) let us define

$$\begin{aligned}&g^{[z_1,z_2]}_1(u,v):= h_3\circ l^{[z_1,z_2]}_1([h_1^{-1}(u),h_1^{-1}(v)]), \qquad \qquad (u,v)\in L^{\overline{\infty }}, \\&g^{[z_1,z_2]}_2(u,v):= h_3\circ l^{[z_1,z_2]}_2([h_1^{-1}(u),h_1^{-1}(v)]), \qquad \qquad (u,v)\in L^{\overline{\infty }}. \end{aligned}$$

Repeating, for functions \(l_1\), \(l_2\), all the calculations which we carried out earlier for functions \(k_1\) and \(k_2\), we obtain that also functions \(g^{[z_1,z_2]}_1\) and \(g^{[z_1,z_2]}_2\) satisfy the functional Eq. (F). This time the case of conjunctive uninorm \(U_1\) and disjunctive uninorm \(U_3\) leads to the Eq. (F) with the assumption (\(A-\)) on the domain of f and (\(A+\)) on the codomain of f, while the case of disjunctive uninorm \(U_1\) and conjunctive uninorm \(U_3\) lead to the Eq. (F) with the assumption (\(A+\)) on the domain of f and (\(A-\)) on the codomain of f.

4 Some New Results Pertaining to Functional Equations

In [3] we solved the additive Cauchy functional equation:

$$\begin{aligned} f(x + y) = f(x)+f(y), \qquad x,y\in [-\infty ,\infty ], \end{aligned}$$

for an unknown function \(f:[-\infty ,\infty ]\rightarrow [-\infty ,\infty ]\). It should be noted that the main problem in this context was with the adequate definition of the additions \(\infty + (-\infty )\) and \((-\infty ) + \infty \). Recently, in [14] we presented solutions of the Eq. (F) for all \((u_1,u_2),(v_1,v_2)\in L^{\overline{\infty }}\), with the assumption (A-), i.e., \((-\infty )+\infty =\infty +(-\infty )=-\infty \) in both sets of domain (formally in both projections) and codomain.

In this article we present new theorem which shows all solutions of Eq. (F) with the assumption (\(A+\)) on the domain (formally in both projections) and codomain of function f.

Theorem 4.1

Let \(L^{\overline{\infty }}=\{(u_1,u_2)\in [-\infty ,\infty ]^2 \; | \; u_1\le u_2\}\). For a function \(f:L^{\overline{\infty }}\rightarrow [-\infty ,\infty ]\) the following statements are equivalent:

  1. (i)

    f satisfies functional Eq. (F) for \((u_1,u_2),(v_1,v_2)\in L^{\overline{\infty }}\), with the assumption \((A+)\), i.e., \((-\infty )+\infty =\infty +(-\infty )=\infty \), in both sets of domain (formally in both projections) and codomain of f.

  2. (ii)

    Either \(f=-\infty \), or \(f=0\), or \(f=\infty \) or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u=\infty , \\ 0, &{} u<\infty , \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} v=\infty , \\ 0, &{} v<\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} \infty , &{} u=\infty , \\ 0, &{} u<\infty , \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} \infty , &{} v=\infty , \\ 0, &{} v<\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u<\infty , \\ \infty , &{} u=\infty , \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} v<\infty , \\ \infty , &{} v=\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} v\in \mathbb {R}, \\ \infty , &{} v\in \{-\infty ,\infty \}, \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u\in \mathbb {R}, \\ \infty , &{} u\in \{-\infty ,\infty \}, \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u,v\in \mathbb {R}, \\ \infty , &{} u=-\infty \text { or } v=\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u<\infty \text { and } v=\infty , \\ 0, &{} v<\infty , \\ \infty , &{} u=\infty , \end{array}\right. } \end{aligned}$$

    or there exists a unique additive function \(c:\mathbb {R} \rightarrow \mathbb {R}\) such that

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u\in \{-\infty ,\infty \}, \\ c(u), &{} u\in \mathbb {R}, \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} v\in \{-\infty ,\infty \}, \\ c(v), &{} v\in \mathbb {R}, \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} \infty , &{} u\in \{-\infty ,\infty \}, \\ c(u), &{} u\in \mathbb {R}, \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} \infty , &{} v\in \{-\infty ,\infty \}, \\ c(v), &{} v\in \mathbb {R}, \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u=-\infty , \\ c(u), &{} u\in \mathbb {R}, \\ \infty , &{} u=\infty , \end{array}\right. } \quad \text { or } \quad f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} v=-\infty , \\ c(v), &{} v\in \mathbb {R}, \\ \infty , &{} v=\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} (u<\infty \text { and } v=\infty ) \text { or } v=-\infty , \\ c(v), &{} v\in \mathbb {R}, \\ \infty , &{} u=\infty , \end{array}\right. } \end{aligned}$$

    or there exist unique additive functions \(c_1,c_2:\mathbb {R} \rightarrow \mathbb {R}\) such that

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u=-\infty \text { or } v=\infty , \\ c_1(u)+c_2(v), &{} u,v\in \mathbb {R}, \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} \infty , &{} u=-\infty \text { or } v=\infty , \\ c_1(u)+c_2(v), &{} u,v\in \mathbb {R}, \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u=-\infty \text { and } v<\infty , \\ c_1(u)+c_2(v), &{} u,v\in \mathbb {R}, \\ \infty , &{} v=\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} (u<\infty \text { and } v=\infty ) \text { or } u=-\infty , \\ c_1(u)+c_2(v), &{} u,v\in \mathbb {R}, \\ \infty , &{} u=\infty , \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u\in \mathbb {R}\text { and } v=\infty , \\ c_1(u)+c_2(v), &{} u,v\in \mathbb {R}, \\ \infty , &{} u\in \{-\infty ,\infty \}, \end{array}\right. } \end{aligned}$$

    or

    $$\begin{aligned} f(u,v)={\left\{ \begin{array}{ll} -\infty , &{} u=-\infty \text { and } v\in \mathbb {R}, \\ c_1(u)+c_2(v), &{} u,v\in \mathbb {R}, \\ \infty , &{} v\in \{-\infty ,\infty \}, \end{array}\right. } \end{aligned}$$

    for all \((u,v)\in L^{\overline{\infty }}\).

5 Conclusions

In this article we presented method for reducing Eqs. (D-UU1) and (D-UU2) to Eq. (F) for implication operations and decomposable uninorms (generated from two conjunctive or disjunctive representable uninorms) in interval-valued fuzzy sets theory. We showed that with this assumption it is enough to solve Eq. (F) for some combinations of the assumptions (\(A-\)) and/or (\(A+\)) in both sets of domain (formally in both projections) and codomain of a function f.

Theorem 4.1 solves the considered functional equation with the assumption (\(A+\)) in both the domain (in fact in both projections) and the codomain of function f. We would like to underline that cases combining (\(A-\)) in the domain and (\(A+\)) in the codomain (and vice versa) were also analyzed by us and we will present them soon.

Now, using Theorem 4.1, we are able to solve Eq. (2) and (3), i.e., we can obtain the description of the two projections of the vertical section \(\mathcal {I}([x_1,x_2],\cdot )\), for fixed \([x_1,x_2]\in L^{\overline{\infty }}\), of the solutions of our main distributivity Eq. (D-UU1) for decomposable uninorms generated from disjunctive representable uninorms. In our future work we will consider these problems in details.