Keywords

1 Introduction

The functional equations involving aggregation operators [4, 5, 9, 19–22] play an important role in theories of fuzzy sets and fuzzy logic. As Ref. [12] pointed out, a new direction of investigations is concerned with distributivity equation and inequalities for uninorms and nullnorms [3, 7, 8, 11–13, 22–25, 30]. Uninorms, introduced by Yager and Rybalov [30], and studied by Fodor et al. [15], are special aggregation operators that have proven to be useful in many fields like fuzzy logic, expert systems, neural networks, utility theory and fuzzy system modeling [14, 16, 18, 26–29]. Uninorm is interesting because its structure is a special combination of a t-norm and a t-conorm having a neutral element lying somewhere in the unit interval.

This paper is mainly devoted to solving the distributivity equations for 2-uninorms over semi-uninorms. Our investigations are motivated by the couple of distributive logical connectives and their generalizations, which are used in fuzzy set theory. Recently, there appeared two kind of their generalizations. One is a 2-uninorm, which generalizes both a nullnorm and a uninorm. Such generalization, further extending to the n-uninorm, was introduced by P. Akella in [2]. A 2-uninorm belongs to the class of increasing, associative and commutative binary operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements. The other is a semi-uninorm—a generalization of a uninorm by omitting commutativity and associativity, which was introduced by Drewniak et al. to study distributivity between a uninorm and a nullnorm [9].

This paper is organized as follows. In Sect. 2, we recall the structures of uninorms, semi-uninorms and 2-uninorms. Over here, we also recall the characterization of three subclasses of 2-uninorms and the functional equation of distributivity. In Sect. 3, the main section, we investigate distributivity for a 2-uninorm over a semi-uninorm and give full characterization for the above-mentioned three subclasses. Section 4 is conclusion and further work.

2 Preliminaries

In this section, we recall basic definitions and facts to be used later in the paper.

Definition 1

([30]) Let \(s\in [0,1]\). A binary operator \(U:[0,1]^{2}\rightarrow [0,1]\) is called a uninorm if it is commutative, associative, non-decreasing in each variable, and there exists an element \(s\in [0,1]\) called neutral element such that \(U(s,x)=x\) for all \(x\in [0,1]\).

It is clear that U becomes a t-norm when \(s=1\), while U becomes a t-conorm when \(s=0\) (see [17]). For any uninorm U, we have \(U(0,1)\in \{0,1\}\), and U is called conjunctive when \(U(0,1)=0\) and disjunctive when \(U(0,1)=1\). By \(\mathbf {U}_{s}\) we denote the family of all uninorms with the neutral element \(s\in [0,1]\). Now we recall the general structure of a uninorm (for more details see [6, 10, 15]) by using the notation \(D_{s}=[0,s)\times (s,1]\cup (s,1]\times [0,s)\) for \(s\in [0,1]\).

Theorem 1

([15]) Let \(s\in [0,1]\). Then, \(U\in \mathbf {U}_{s}\) if and only if

$$\begin{aligned} U={\left\{ \begin{array}{ll} T_{U} &{} \text { if } (x,y)\in [0,s]^2,\\ S_{U} &{} \text { if } (x,y)\in [s,1]^2,\\ C &{} \text { if } (x,y)\in D_{s}, \end{array}\right. } \end{aligned}$$
(1)

where \(T_{U}\) and \(S_{U}\) are respectively isomorphic with a t-norm and a t-conorm, the increasing operation \(C: D_{s}\rightarrow [0,1]\) fulfills \(\min (x,y) \leqslant C(x,y)\leqslant \max (x,y)\) for \((x,y)\in D_{s}\).

Theorem 2

([15]) Let \(U:[0,1]^{2}\rightarrow [0,1]\) be a uninorm with the neutral element \(s\in (0,1)\).

  1. (i)

    If \(U(0,1)=0\) and the function U(x, 1) is continuous except for the point \(x=s\), then \(C=\min \) in Eq. (1) and the class of such uninorms is denoted by \(\mathbf {U}_{s}^{\min }\).

  2. (ii)

    If \(U(0,1)=1\) and the function U(x, 0) is continuous except for the point \(x=s\), then \(C=\max \) in Eq. (1) and the class of such uninorms is denoted by \(\mathbf {U}_{s}^{\max }\).

Definition 2

([6]) An element \(a\in [0,1]\) is called an idempotent element of \(F:[0,1]^{2}\rightarrow [0,1]\) if \(F(a,a)=a\). The operation F is called idempotent if all elements from [0, 1] are idempotent.

Theorem 3

([6]) Let \(s\in [0,1]\). Then, the operations

$$\begin{aligned} U^{\min }_s={\left\{ \begin{array}{ll} \max &{} \text { if } (x,y)\in [s,1]^2,\\ \min &{} \text { elsewhere }, \end{array}\right. } \end{aligned}$$
(2)

and

$$\begin{aligned} U^{\max }_s={\left\{ \begin{array}{ll} \min &{} \text { if } (x,y)\in [0,s]^2,\\ \max &{} \text { elsewhere }, \end{array}\right. } \end{aligned}$$
(3)

are unique idempotent uninorms in \(\mathbf {U}_{s}^{\min }\) and \(\mathbf {U}_{s}^{\max }\), respectively.

Definition 3

([9]) A binary operator \(V:[0,1]^{2}\rightarrow [0,1]\) is called a semi-uninorm if it is non-decreasing in each variable, and there exists an element \(s\in [0,1]\) called neutral element such that \(V(s,x)=x\) for all \(x\in [0,1]\).

It is clear that a commutative and associative semi-uninorm is a uninorm. By \(\mathbf {V}_{s}\) we denote the family of all semi-uninorms with the neutral element \(s\in [0,1]\). A semi-uninorm V is called a semi-t-norm if \(e=1\), while V is called a semi-t-conorm if \(e=0\). For any \(V\in \mathbf {V}_{s}\), we have \(V(0,0)=0\) and \(V(1,1)=1\). Therefore, V is a binary aggregation operator with the neutral element \(s\in [0,1]\). Moreover, a semi-uninorm V is said to be conjunctive if \(V(0,1)=V(1,0)=0\) and disjunctive if \(V(0,1)=V(1,0)=1\). In fact, for any conjunctive semi-uninorm V, it holds that \(V(0,x)=V(x,0)=0\) for all \(x\in [0,1]\), while for any disjunctive semi-uninorm V, it follows that \(V(1,x)=V(x,1)=1\) for all \(x\in [0,1]\).

Now we recall the structure of a semi-uninorm, which has also been given by Drewniak et al. [9] in another form.

Theorem 4

([9]) Let \(s\in [0,1]\). Then, \(V\in \mathbf {V}_{s}\) if and only if

$$\begin{aligned} V={\left\{ \begin{array}{ll} T_{V} &{} \text { if } (x,y)\in [0,s]^2,\\ S_{V} &{} \text { if } (x,y)\in [s,1]^2,\\ C &{} \text { if } (x,y)\in D_{s}, \end{array}\right. } \end{aligned}$$
(4)

where \(T_{V}\) and \(S_{V}\) are respectively isomorphic with a semi-t-norm and a semi-t-conorm, the increasing operation \(C:D_{s}\rightarrow [0,1]\) fulfills \(\min (x,y) \leqslant C(x,y)\leqslant \max (x,y)\) for \((x,y)\in D_{s}\). Moreover, \(T_{V}\) and \(S_{V}\) is called the underlying semi-t-norm and semi-t-conorm of V, respectively.

By \(\mathbf {V}_{s}^{\max }\ (\mathbf {V}_{s}^{\min })\) we denote the family of all semi-uninorms with the same neutral element \(s\in (0,1)\) fulfilling the additional condition: \(V(0,x)=V(x,0)=0 \) for all \(x\in (s,1]\) (\(V(1,x)=V(x,1)=1\) for all \(x\in [0,s)\)) [9].

From Theorems 4 and 7 in [9], we can obtain structures of elements in \(\mathbf {V}_{s}^{\max }\) and \(\mathbf {V}_{s}^{\min }\).

Theorem 5

 ([9]) Let \(V\in \mathbf {V}_{s}\) with the neutral element \(s\in (0,1)\). Then,

  1. (i)

    \(V\in \mathbf {V}_{s}^{\min }\) if and only if

    $$\begin{aligned} V={\left\{ \begin{array}{ll} T_{V} &{} \text { if } (x,y)\in [0,s]^2,\\ S_{V} &{} \text { if } (x,y)\in [s,1]^2,\\ \min &{} \text { if } (x,y)\in D_{s}, \end{array}\right. } \end{aligned}$$
    (5)
  2. (ii)

    \(V\in \mathbf {V}_{s}^{\max }\) if and only if

    $$\begin{aligned} V={\left\{ \begin{array}{ll} T_{V} &{} \text { if } (x,y)\in [0,s]^2,\\ S_{V} &{} \text { if } (x,y)\in [s,1]^2,\\ \max &{} \text { if } (x,y)\in D_{s}, \end{array}\right. } \end{aligned}$$
    (6)

    where \(T_{V}\) and \(S_{V}\) are isomorphic with a semi-t-norm and a semi-t-conorm, respectively.

Theorem 6

 ([9]) The operators \(U^{\min }_s\) and \(U^{\max }_s\) in Theorem 3 are unique idempotent semi-uninorms in \(\mathbf {V}_{s}^{\min }\) and \(\mathbf {V}_{s}^{\max }\), respectively.

Now we recall the definitions and some results of 2-uninorms.

Definition 4

 ([2]) Let \(0\leqslant e\leqslant k\leqslant f\leqslant 1\). An operator \(G:[0,1]^{2}\rightarrow [0,1]\) is called a 2-uninorm, if it is commutative, associative, non-decreasing with respect to both variables, and fulfilling

$$\begin{aligned} G(e,x)=x\hbox { for all }x\in [0,k] \hbox { and } G(f,x)=x\hbox { for all }x\in [k,1]. \end{aligned}$$
(7)

By \(\mathbf {U}_{k(e,f)}\) we denote the class of all 2-uninorms.

Remark 1

 ([12]) Any operator \(G\in \mathbf {U}_{k(e,f)}\) fulfills the condition:

$$\begin{aligned} G(k,x)=k \hbox { for all }x\in [e,f]. \end{aligned}$$
(8)

Lemma 1

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm with \(0\leqslant e\leqslant k\leqslant f\leqslant 1\) and \(k\in (0,1)\). Then, two mappings \(U_{1}\) and \(U_{2}\) defined by, for \(x,y\in [0,1]\),

$$\begin{aligned} U_{1}(x,y)=\frac{G(kx,ky)}{k} \end{aligned}$$
(9)

and

$$\begin{aligned} U_{2}(x,y)=\frac{G(k+(1-k)x,k+(1-k)y)-k}{1-k} \end{aligned}$$
(10)

are uninorms with the neutral elements \(\frac{e}{k}\) and \(\frac{f-k}{1-k}\), respectively.

Lemma 2

 ([12]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm with \(0\leqslant e\leqslant k\leqslant f\leqslant 1\). Then,

  1. (i)

    G(x, 0) is continuous except for the point e if and only if \(U_{1}(x,0)\) is continuous except for the point \(\frac{e}{k}\).

  2. (ii)

    G(x, 1) is continuous except for the point f if and only if \(U_{2}(x,1)\) is continuous except for the point \(\frac{f-k}{1-k}\).

Lemma 3

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm with \(0\leqslant e\leqslant k\leqslant f\leqslant 1\). Then, \(G(0,1)\in \{0,k,1\}\).

Depending on the values of G(0, 1), which plays a zero element role for the operator G, we obtain from the above lemmas that three subclass of operators in \(\mathbf {U}_{k(e,f)}\) are respectively denoted by \(\mathbf {C}^{0}_{k(e,f)}\), \(\mathbf {C}^{k}_{k(e,f)}\), \(\mathbf {C}^{1}_{k(e,f)}\) (or simplifying them \(\mathbf {C}^{0}\), \(\mathbf {C}^{k}\), \(\mathbf {C}^{1}\)).

Theorem 7

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm such that G(x, 1) is discontinuous only at the points e and f. Then, \(G\in \mathbf {C}^0\) and \(G(1,k)=k\) if and only if \(0<e\leqslant k<f\leqslant 1\), and G has the following form

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{c_1}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{c_1}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ T^{c_2}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{c_2}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ \min (x,y) &{} \text { if } (x,y)\in [0,e)\times (e,1]\cup (e,1]\times [0,e)\\ &{} \ \ \ \ \cup [k,f)\times (f,1]\cup (f,1]\times [k,f),\\ k &{} \text { if } (x,y)\in [e,k)\times (k,1]\cup (k,1]\times [e,k), \end{array}\right. } \end{aligned}$$
(11)

where \(T^{c_1}\) and \(T^{c_2}\) are isomorphic with t-norms, \(S^{c_1}\) and \(S^{c_2}\) are isomorphic with t-conorms. We denote the set of all such 2-uninorms by \(\mathbf {C}^0_k\).

Theorem 8

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm such that G(x, 1) is discontinuous only at the point e, and G(x, e) is discontinuous only at the point f. Then, \(G\in \mathbf {C}^0\) and \(G(1,k)=1\) if and only if \(0<e\leqslant k\leqslant f<1\), and G has the following form

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{c}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{c}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ T^{d}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{d}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ \min (x,y) &{} \text { if } (x,y)\in [0,e)\times (e,1]\cup (e,1]\times [0,e),\\ \max (x,y) &{} \text { if } (x,y)\in [e,f)\times (f,1]\cup (f,1]\times [e,f),\\ k &{} \text { if } (x,y)\in [e,k)\times (k,f]\cup (k,f]\times [e,k), \end{array}\right. } \end{aligned}$$
(12)

where \(T^{c}\) and \(T^{d}\) are isomorphic with t-norms, \(S^{c}\) and \(S^{d}\) are isomorphic with t-conorms. We denote the set of all such 2-uninorms by \(\mathbf {C}^0_1\).

Theorem 9

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm such that G(x, 0) is discontinuous only at the points e and f. Then, \(G\in \mathbf {C}^1\) and \(G(0,k)=k\) if and only if \(0\leqslant e<k\leqslant f<1\), and G has the following form

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{d_1}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{d_1}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ T^{d_2}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{d_2}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ \max (x,y) &{} \text { if } (x,y)\in [0,f)\times (f,1]\cup (f,1]\times [0,f)\\ &{} \ \ \ \ \cup [0,e)\times (e,k]\cup (e,k]\times [0,e),\\ k &{} \text { if } (x,y)\in [0,k)\times (k,f]\cup (k,f]\times [0,k), \end{array}\right. } \end{aligned}$$
(13)

where \(T^{d_1}\) and \(T^{d_2}\) are isomorphic with t-norms, \(S^{d_1}\) and \(S^{d_2}\) are isomorphic with t-conorms. We denote the set of all such 2-uninorms by \(\mathbf {C}^1_k\).

Theorem 10

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm such that G(x, 0) is discontinuous only at the point f, and G(x, f) is discontinuous only at the point e. Then, \(G\in \mathbf {C}^1\) and \(G(0,k)=0\) if and only if \(0<e\leqslant k\leqslant f<1\), and G has the following form

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{c}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{c}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ T^{d}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{d}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ \min (x,y) &{} \text { if } (x,y)\in [0,e)\times (e,f]\cup (e,f]\times [0,e),\\ \max (x,y) &{} \text { if } (x,y)\in [0,f)\times (f,1]\cup (f,1]\times [0,f),\\ k &{} \text { if } (x,y)\in [e,k)\times (k,f]\cup (k,f]\times [e,k), \end{array}\right. } \end{aligned}$$
(14)

where \(T^{c}\) and \(T^{d}\) are isomorphic with t-norms, \(S^{c}\) and \(S^{d}\) are isomorphic with t-conorms. We denote the set of all such 2-uninorms by \(\mathbf {C}^1_0\).

Theorem 11

 ([2]) Let \(G\in \mathbf {U}_{k(e,f)}\) be a 2-uninorm such that G(x, 0) is discontinuous only at the point e, and G(x, 1) is discontinuous only at the point f. \(G\in \mathbf {C}^k\) if and only if \(0\leqslant e< k<f\leqslant 1\), and G has the following form

$$\begin{aligned} G(x,y))={\left\{ \begin{array}{ll} T^{d}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{d}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ T^{c}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{c}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ \max (x,y) &{} \text { if } (x,y)\in [0,e)\times (e,k]\cup (e,k]\times [0,e),\\ \min (x,y) &{} \text { if } (x,y)\in [k,f)\times (f,1]\cup (f,1]\times [k,f),\\ k &{} \text { if } (x,y)\in [0,k)\times (k,1]\cup (k,1]\times [0,k), \end{array}\right. } \end{aligned}$$
(15)

where \(T^{c}\) and \(T^{d}\) are isomorphic with t-norms, \(S^{c}\) and \(S^{d}\) are isomorphic with t-conorms.

For convenience, we assume that all of the underlying operators \(T^{c}\), \(T^{d}\), \(T^{c_{1}}\), \(T^{c_{2}}\), \(T^{d_{1}}\), \(T^{d_{2}}\) and \(S^{c}\), \(S^{d}\), \(S^{c_{1}}\), \(S^{c_{2}}\), \(S^{d_{1}}\), \(S^{d_{2}}\) of 2-uninorms in this paper are continuous. Next, we consider the distributivity equation.

Definition 5

([1]) Let \(F,\ G:[0,1]^2\rightarrow [0,1]\). We say that G is distributive over F, if for all \(x,\ y, z\in [0,1]\),

$$\begin{aligned} G(x,F(y,z))=F(G(x,y),G(x,z)). \end{aligned}$$
(16)

Lemma 4

 ([25]) Let \(F:X^2\rightarrow X\) have the right (left) neutral element e in a subset \(\emptyset \not =Y\subset X\) (i.e., \(\forall _{x\in Y}, F(x,e)=x \ (F(e,x)=x))\). If the operation F is distributive over another operation \(G:X^2\rightarrow X\) fulfilling \(G(e,e)=e\), then G is idempotent in Y.

Lemma 5

 ([25]) If an operation \(F:[0,1]^2\rightarrow [0,1]\) with the neutral element \(e\in [0,1]\) is distributive over another operation \(G :[0,1]^2\rightarrow [0,1]\) fulfilling \(G(e,e)=e\), then G is idempotent.

Lemma 6

 ([25]) Every increasing operation \(G:[0,1]^2\rightarrow [0,1]\) is distributive over \(\max \) and \(\min \).

3 Distributivity for a 2-Uninorm over a Semi-uninorm

Lemma 7

Let \(0\leqslant e\leqslant k\leqslant f\leqslant 1\). If a 2-uninorm G is distributive over a semi-uninorm F with the neutral element \(s\in [0,1]\), then \(G(s,s)=s\).

Remark 2

In this paper, we always assume that the semi-uninorm F has neutral element \(s\in (0,1)\) because Ref. [12] has discussed the cases \(s=0\) and \(s=1\). That is, F is a semi-t-norm or a semi-t-conorm.

To completely characterize distributivity for a 2-uninorm G over a semi-uninorm F, according to Theorems from 7 to 11, there are five cases to be consider: (1) \(G\in \mathbf C ^0_k\); (2) \(G\in \mathbf C ^0_1\); (3) \(G\in \mathbf C ^1_k\); (4) \(G\in \mathbf C ^1_0\); (5) \(G\in \mathbf C ^k\). First, let us consider Case (1): \(G\in \mathbf C ^0_k\).

3.1 \(G\in \mathbf {C}^0_k\)

Lemma 8

Let \(G\in \mathbf {C}^0_k\) be a 2-uninorm with \(0<e\leqslant k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then F is idempotent.

So far, we know from Lemma 8 that the structure of F is completely determined. Therefore, the rest of this investigation requires that we characterize the operator G. Furthermore, we can only consider the case \(e<k\) because the case \(e=k\) is fully similar and much easier. Note that the assumption \(0<e<k<f\leqslant 1\) and the order relationship between s and e, k, f, then there are four cases: (1) \(s\in (0,e]\); (2) \(s\in (e,k]\); (3) \(s\in (k,f]\); (4) \(s\in (f,1)\). The following lemma shows that two cases (2) and (4) are impossible.

Lemma 9

Let \(G\in \mathbf {C}^0_k\) be a 2-uninorm with \(0<e<k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then \(s\in (0,e]\) or \(s\in (k,f]\).

Theorem 12

Let \(G\in \mathbf {C}^0_k\) be a 2-uninorm with \(0<e<k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is distributive over F if and only if \(F=U^{\min }_s\) and one of the following two cases holds.

  1. (i)

    \(s\leqslant e\) and the structure of G is

    $$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{c_1}_{1}(x,y) &{} \text { if } (x,y)\in [0,s]^2,\\ T^{c_1}_{2}(x,y) &{} \text { if } (x,y)\in [s,e]^2,\\ S^{c_1}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ U^{c_2}(x,y) &{} \text { if } (x,y)\in [k,1]^2,\\ k &{} \text { if } (x,y)\in [e,k)\times (k,1]\cup (k,1]\times [e,k),\\ \min (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
    (17)

    where \(T^{c_1}_{1}\) and \(T^{c_1}_{2}\) are isomorphic with t-norms, \(S^{c_1}\) is isomorphic with a t-conorm.

  2. (ii)

    \(k<s\leqslant f\) and the structure of G is

    $$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} U^{c_1}(x,y) &{} \text { if } (x,y)\in [0,k]^2,\\ T^{c_2}_{1}(x,y) &{} \text { if } (x,y)\in [k,s]^2,\\ T^{c_2}_{2}(x,y) &{} \text { if } (x,y)\in [s,f]^2,\\ S^{c_2}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ k &{} \text { if } (x,y)\in [e,k)\times (k,1]\cup (k,1]\times [e,k),\\ \min (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
    (18)

    where \(T^{c_2}_{1}\) and \(T^{c_2}_{2}\) are isomorphic with t-norms, \(S^{c_2}\) is isomorphic with a t-conorm.

Theorem 13

Let \(G\in \mathbf {C}^0_k\) be a 2-uninorm with \(0<e\leqslant k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is not distributive over F.

3.2 \(G\in \mathbf {C}^1_k\)

Lemma 10

Let \(G\in \mathbf {C}^1_k\) be a 2-uninorm with \(0\leqslant e< k\leqslant f< 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then F is idempotent.

Next, we only consider the case \(e<k\) because the case \(e=k\) is similar and much easier.

Theorem 14

Let \(G\in \mathbf {C}^1_k\) be a 2-uninorm with \(0\leqslant e< k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is not distributive over F.

Lemma 11

Let \(G\in \mathbf {C}^1_k\) be a 2-uninorm with \(0\leqslant e< k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then \(s\in [e,k)\) or \(s\in [f,1)\).

Theorem 15

Let \(G\in \mathbf {C}^1_k\) be a 2-uninorm with \(0\leqslant e< k<f< 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is distributive over F if and only if \(F=U^{\max }_s\) and one of the following two cases holds.

  1. (i)

    \(e\leqslant s<k\) and the structure of G is

    $$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{d_1}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{d_1}_{1}(x,y) &{} \text { if } (x,y)\in [e,s]^2,\\ S^{d_1}_{2}(x,y) &{} \text { if } (x,y)\in [s,k]^2,\\ U^{d_2}(x,y) &{} \text { if } (x,y)\in [k,1]^2,\\ k &{} \text { if } (x,y)\in [0,k)\times (k,f]\cup (k,f]\times [0,k),\\ \max (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
    (19)

    where \(T^{d_1}\) is isomorphic with a t-norm, \(S^{d_1}_{1}\) and \(S^{d_1}_{2}\) are isomorphic with t-conorms.

  2. (ii)

    \(s\geqslant f\) and the structure of G is

    $$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} U^{d_1}(x,y) &{} \text { if } (x,y)\in [0,k]^2,\\ T^{d_2}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{d_2}_{1}(x,y) &{} \text { if } (x,y)\in [f,s]^2,\\ S^{d_2}_{2}(x,y) &{} \text { if } (x,y)\in [s,1]^2,\\ k &{} \text { if } (x,y)\in [0,k)\times (k,f]\cup (k,f]\times [0,k),\\ \max (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
    (20)

    where \(T^{d_2}\) is isomorphic with a t-norm, \(S^{d_2}_{1}\) and \(S^{d_2}_{2}\) are isomorphic with t-conorms.

3.3 \(G\in \mathbf {C}^0_1\)

Lemma 12

Let \(G\in \mathbf {C}^0_1\) be a 2-uninorm with \(0< e\leqslant k\leqslant f< 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then F is idempotent.

Lemma 13

Let \(G\in \mathbf {C}^0_1\) be a 2-uninorm with \(0<e<k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then \(s\in (0,e]\).

Theorem 16

Let \(G\in \mathbf {C}^0_1\) be a 2-uninorm with \(0<e<k<f<1\), and let \(F\in V_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is distributive over F if and only if \(F=U^{\min }\) and the structure of G is

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{c}_{1}(x,y) &{} \text { if } (x,y)\in [0,s]^2,\\ T^{c}_{2}(x,y) &{} \text { if } (x,y)\in [s,e]^2,\\ S^{c}(x,y) &{} \text { if } (x,y)\in [e,k]^2,\\ U^{d}(x,y) &{} \text { if } (x,y)\in [k,1]^2,\\ k &{} \text { if } (x,y)\in [e,k)\times (k,f]\cup (k,f]\times [e,k),\\ \max (x,y) &{} \text { if } (x,y)\in [e,k)\times (f,1]\cup (f,1]\times [e,k),\\ \min (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
(21)

where \(s\leqslant e\) and \(T^{c}_{1}\) and \(T^{c}_{2}\) are isomorphic with t-norms, \(S^{c}\) is isomorphic with a t-conorm.

Theorem 17

Let \(G\in \mathbf {C}^0_1\) be a 2-uninorm with \(0<e<k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is not distributive over F.

3.4 \(G\in \mathbf {C}^1_0\)

Lemma 14

Let \(G\in \mathbf {C}^1_0\) be a 2-uninorm with \(0< e\leqslant k\leqslant f< 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then F is idempotent.

Next, we only consider the case \(e<k<f\), the other cases \(e=k\) ae \(k=f\) are similar.

Theorem 18

Let \(G\in \mathbf {C}^1_0\) be a 2-uninorm with \(0<e<k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is not distributive over F.

Lemma 15

Let \(G\in \mathbf {C}^1_0\) be a 2-uninorm with \(0<e<k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then \(s\in [f,1)\).

Theorem 19

Let \(G\in \mathbf {C}^1_0\) be a 2-uninorm with \(0<e<k<f<1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is distributive over F if and only if \(F= U^{\max }_s\) and the structure of G is

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} U^{c}(x,y) &{} \text { if } (x,y)\in [0,k]^2,\\ T^{d}(x,y) &{} \text { if } (x,y)\in [k,f]^2,\\ S^{d}_{1}(x,y) &{} \text { if } (x,y)\in [f,s]^2,\\ S^{d}_{2}(x,y) &{} \text { if } (x,y)\in [s,1]^2,\\ k &{} \text { if } (x,y)\in [e,k)\times (k,f]\cup (k,f]\times [e,k),\\ \min (x,y) &{} \text { if } (x,y)\in [0,e)\times (k,f]\cup (k,f]\times [0,e),\\ \max (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
(22)

where \(s\geqslant f\) and \(T^{d}\) is isomorphic with a t-norm, \(S^{d}_{1}\) and \(S^{d}_{2}\) are isomorphic with t-conorms.

3.5 \(G\in \mathbf {C}^k\)

Lemma 16

Let \(G\in \mathbf {C}^k\) be a 2-uninorm with \(0< e\leqslant k< f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then F is idempotent.

Next, without loss of generality, we only consider the case \(e<k\).

Lemma 17

Let \(G\in \mathbf {C}^k\) be a 2-uninorm with \(0<e<k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then \(s\in (k,f]\).

Theorem 20

Let \(G\in \mathbf {C}^k\) be a 2-uninorm with \(0<e<k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(1,x)=F(x,1)=x\) for \(x\in [0,s)\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is distributive over F if and only if \(F=U^{\min }_s\) and the structure of G is

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} U^{d}(x,y) &{} \text { if } (x,y)\in [0,k]^2,\\ T^{c}_{1}(x,y) &{} \text { if } (x,y)\in [k,s]^2,\\ T^{c}_{2}(x,y) &{} \text { if } (x,y)\in [s,f]^2,\\ S^{c}(x,y) &{} \text { if } (x,y)\in [f,1]^2,\\ k &{} \text { if } (x,y)\in [0,k)\times (k,1]\cup (k,1]\times [0,k),\\ \min (x,y) &{} \text { otherwise, } \end{array}\right. } \end{aligned}$$
(23)

where \(k<s\leqslant f\) and \(T^{c}_{1}\) and \(T^{c}_{2}\) are isomorphic with t-norms, \(S^{c}\) is isomorphic with a t-conorm.

Lemma 18

Let \(G\in \mathbf {C}^k\) be a 2-uninorm with \(0<e<k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. If G is distributive over F, then \(s\in [e,k)\).

Theorem 21

Let \(G\in \mathbf {C}^k\) be a 2-uninorm with \(0<e<k<f\leqslant 1\), and let \(F\in \mathbf {V}_{s}\) be a semi-uninorm with the neutral element \(s\in (0,1)\) and \(F(0,x)=F(x,0)=x\) for \(x\in (s,1]\) such that the underlying semi-t-norm \(T_{F}\) and semi-t-conorm \(S_{F}\) are continuous. Then G is distributive over F if and only if \(F=U^{\max }_s\) and one of the structure of G is

$$\begin{aligned} G(x,y)={\left\{ \begin{array}{ll} T^{d}(x,y) &{} \text { if } (x,y)\in [0,e]^2,\\ S^{d}_{1}(x,y) &{} \text { if } (x,y)\in [e,s]^2,\\ S^{d}_{2}(x,y) &{} \text { if } (x,y)\in [s,k]^2,\\ U^{c}(x,y) &{} \text { if } (x,y)\in [k,1]^2,\\ k &{} \text { if } (x,y)\in [0,k)\times (k,1]\cup (k,1]\times [0,k),\\ \max (x,y) &{} \text { otherwise }, \end{array}\right. } \end{aligned}$$
(24)

where \(e\leqslant s<k\) and \(T^{d}\) is isomorphic with a t-norm, \(S^{d}_{1}\) and \(S^{d}_{2}\) are isomorphic with t-conorms.

4 Conclusions and Further Work

In this paper, we have investigated distributivity for 2-uninorms over semi-uninorms. In fact, 2-uninorms cover both uninorms and nullnorms, which form a class of commutative, associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements. While semi-uninorms are generalizations of uninorms by omitting commutativity and associativity. Moreover, all possible solutions of the distributivity equation for the three defined subclasses of 2-uninorms over semi-uninorms are characterized. In future work, we will concentrate on the converse, that is, distributivity for semi-uninorms over 2-uninorms. Indeed, this problem is very difficult because we are not sure that the second operator, namely, the 2-uninorm, is idempotent.