Keywords

1 Introduction

The associativity models the independence of the aggregation on the grouping of input values and it allows to investigate binary aggregation operators only (as far as their n-ary extensions are then determined uniquely). It is needless to emphasize the key role of associative operations (t-norms, t-conorms, uninorms, nullnorms, etc.) not only in fuzzy set theory, but also in many areas of application, especially in decision-making under uncertainty [5], image processing [1, 6], fuzzy neural networks [7] and so on. The most important classes of associative, commutative, increasing operations in the framework of fuzzy sets is that of uninorms ([4, 5, 18]), which includes t-norms [10, 17] and t-conorms [10] as two special classes. A large number of methods to construct uninorms (including t-norms and t-conorms) are introduced: Klement et al. [10], Schweizer and Sklar [17], Jenei [8], Ling [13], Maes and De Baets [11], Mas et al. [12], Mesiarová-Zemánková [14,15,16] and so on.

Fig. 1.
figure 1

The structure of \(\oplus \), where the thick line is the boundary between \(\{(i,j)\mid i*'j=0\}\) and \(\{(i,j)\mid i*'j>0)\}\). Inside the blocks it is shown in which sub-interval the operation \(\oplus \) takes its values.

Full size image
Fig. 2.
figure 2

The structure of \(\oplus \), where the thick line is the boundary between \(\{(i,j)\mid i*'j=n\}\) and \(\{(i,j)\mid i*'j<n)\}\). Inside the blocks it is shown in which sub-interval the operation \(\oplus \) takes its values.

Full size image

Kalina et al. [2, 9] introduced a construction method called paving. The main idea is as follows: the unit interval is split into countably many disjoint sub-intervals \((I_i)_{i\in J_n}\) with \(J_n\) an index-set and with the help of an appropriate operation \(*'\) on \(J_n\) and a family of increasing transformations \(\varphi _i:I_i\rightarrow [0,1]\), a new operation \(\oplus \) is defined by

$$\begin{aligned} x\oplus y=\varphi ^{-1}_{i *' j}(\varphi _i(x)*\varphi _j(y)), \quad x\in I_i,~y\in I_j. \end{aligned}$$
(1)

Unfortunately, Kalina et al. only consider discrete representable associative operations as operation \(*'\), which is rather restrictive. For instance, not every discrete t-norm can be generated by some additive generator, and this applies to t-conorms and uninorms. Moreover, the operation \(*'\) in [2] is not always internal on \(J_n\). In this paper, we will consider a general discrete associative operation as operation \(*'\) on \(J_n\), to construct some new associative, commutative and increasing operations. The graphical schema of paving is depicted in Fig. 1 (which depicts the construction of a conjunctive operation \(\oplus \)) and Fig. 2 (which depicts the construction of a disjunctive operation \(\oplus \)).

The paper is organized as follows. In Sect. 2, we present some preliminary notions and results that are necessary for the rest of the paper. Starting from Eq. (1), when \(*\) is a t-norm and \(*'\) is a discrete t-norm, t-superconorm, t-conorm or uninorm, we construct some new associative, commutative and increasing operations in Sect. 3. At the same time, all the dual constructions when \(*\) is a t-conorm are also listed in Sect. 3.

2 Preliminaries

In this section we recall some basic notions and facts that are necessary for the understanding of what follows.

Definition 1

[10]. A decreasing function \(N: [0, 1] \rightarrow [0,1]\) is called a fuzzy negation if \(N(0)=1\) and \(N(1)=0\). Moreover, a fuzzy negation N is called strong if it is involutive, i.e., if \(N(N(x))=x\) for all \(x\in [0,1]\).

Definition 2

[18]. A binary operation \(U:[0, 1]^2\rightarrow [0, 1]\) is called a uninorm if it is associative, commutative, increasing and has a neutral element \(e\in [0,1]\), i.e., \(U(x,e)=x\) for all \(x\in [0,1]\).

A uninorm with neutral element \(e=1\) is a t-norm [10, 17] and a uninorm with neutral element \(e=0\) is a t-conorm [10]. We say that a uninorm U is proper if \(e\in \;]0,1[\). If \(U(1,0)=0\), then U is called conjunctive. If \(U(1,0)=1\), then U is called disjunctive. Conjunctive and disjunctive uninorms are dual to each other. For an arbitrary disjunctive uninorm U and a strong negation N, its N-dual conjunctive uninorm is given by

$$\begin{aligned} U^d_N(x,y) = N (U(N(x), N(y))). \end{aligned}$$
(2)

For an overview of basic properties of uninorms, we refer to [3].

Remark 1

Note that, for a strong negation N, the N-dual operation to a t-norm T defined by \(S(x, y) = N(T(N(x), N(y)))\) is a t-conorm. For more information, see, e.g., [10].

Definition 3

[8]. (i) A binary operation \(\widetilde{T}:[0, 1]^2\rightarrow [0, 1]\) is called a triangular subnorm (t-subnorm, for short), if it is associative, commutative, increasing and fulfills the condition \(\widetilde{T}(x, y) \le \min (x, y)\) for all \((x, y)\in [0, 1]^2\).

(ii) A binary operation \(\widetilde{S}: [0, 1]^2\rightarrow [0, 1]\) is called a triangular superconorm (t-superconorm, for short), if it is associative, commutative, increasing and fulfills the condition \(\widetilde{S}(x, y) \ge \max (x, y)\) for all \((x, y)\in [0, 1]^2\).

Definition 4

Let \(*: [0,1]^2\rightarrow [0,1]\) be a commutative operation. Fix a value \(a\in [0,1]\). We say that \(x\in [0,1]\), \(x\ne a\), is an a-divisor if there exists \(y\in [0,1]\), \(y\ne a\), such that

$$\begin{aligned} x*y=a. \end{aligned}$$
(3)

3 Construction of New Operations

The main idea of our construction method is described in Eq. (1) with the help of a discrete associative operation \(*'\). For the rest of this paper, we adopt the following notations.

Let \(\mathbb {N}\) be the set of all positive integers. We consider an index-set

$$\begin{aligned} J_n =\{0,1,2,\ldots , n\} \end{aligned}$$

for some \(n \in {\mathbb {N}}\).

We will split the interval [0, 1] into \(n+1\) sub-intervals by choosing the end-points of the system of sub-intervals

$$ 0=a_{-1}<a_0<a_{1}<a_{2}<\ldots< a_{n-1}<a_n=1. $$

Because of this partition, we will use half-open intervals, i.e., either left-open or right-open. We will use indexing of the chosen sub-intervals in accordance with the right end-point. For the case of left-open sub-intervals, \(I_i =\;]a_{i-1}, a_i]\); for the case of right-open sub-intervals, \(I_i =[a_{i-1}, a_i[\).

For a fixed system of right-open sub-intervals \((I_i)^n_{i=0}\), \(\varphi _i:I_i\rightarrow [0,1[\) are increasing bijections. For a fixed system of left-open sub-intervals \((I_i)^n_{i=0}\), \(\chi _i:I_i\rightarrow \,]0,1]\) are increasing bijections.

Remark 2

[2]. In order not to get out of the range of the transformations \(\chi _i\) when using left-open sub-intervals, the starting operation \(*\) (the basic paving stone) must be without zero-divisors. Similarly, when using right-open sub-intervals, \(*\) must be without one-divisors.

Here, we consider to construct new associative, commutative and increasing operations from a given one \(*\), and two certain cases of associative, commutative and increasing operations will be taken into account: the case that \(*\) is a t-norm and the case that \(*\) is a t-conorm.

3.1 The Case that \(*\) Is a T-Norm

In this subsection, we construct some new associative, commutative and increasing operations on the unit interval from a t-norm on the unit interval and a discrete t-norm/t-superconorm/t-conorm/uninorm.

Firstly, we construct a new operation \(\oplus \) from a t-norm \(*\) and a discrete t-norm \(*'\) in Eq. (1). Because of the partition of unit interval, we distinguish two cases: when right-open sub-intervals of [0, 1[ and left-open sub-intervals of ]0, 1].

Proposition 1

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete t-norm on \(J_n=\{0,\ldots ,n\}\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j>0\}\). Then the operation \(\oplus _1\) defined by

$$\begin{aligned} x\oplus _1 y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)),&{} \mathrm{if}~x\in I_i,~y\in I_j~\mathrm{and}~i*'j>0, \\ \min (x,y),&{} \mathrm{if}~\max (x,y)=1,\\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(4)

is a t-norm.

In fact, \(\oplus _1\) is not always increasing without the condition that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J,~i*'j>0\}\).

Example 1

Assume that \(J_7=\{0,1,2,\ldots ,7\}\), \((I_i = [i/8, (i+1)/8[)^7_{ i = 0}\) is a partition of [0, 1[. Let \(*\) be the t-norm \(T_M(x,y)=\min (x,y)\) on [0, 1], \(*'\) be the discrete t-norm \(T_M(i,j)=\min (i,j)\) on \(J_7\), \(\varphi _i(x)=\frac{x-a_{i-1}}{a_i-a_{i-1}}\). Define \(x\oplus y\) as follows:

$$\begin{aligned} x\oplus y =\left\{ \begin{array}{ll} \varphi ^{-1}_{\min (i,j)}(\min (\frac{x-a_{i-1}}{a_i-a_{i-1}},\frac{y-a_{j-1}}{a_j-a_{j-1}} )) ,&{} \mathrm{if}~x\in I_i,~y\in I_j,~\mathrm{and}~\min (i,j)>0, \\ \min (x,y) ,&{} \mathrm{if}~\max (x,y)=1,\\ 0 ,&{} \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Consider that \(x=\frac{3}{16}\), \(y=\frac{3}{16}\) and \(z=\frac{1}{4}\), then we have that

$$\begin{aligned} x\oplus y = \varphi ^{-1}_{1}\big (\frac{1}{2} \big )=\frac{3}{16}> \frac{1}{8}=\varphi ^{-1}_{1}(0)=x\oplus z. \end{aligned}$$
(5)

That is, \(\oplus \) is not increasing.

By (4), we can see that for any t-norm \(*\), its values on the upper right boundary of the unit square \([0,1]^2\) have no impact on the properties of \(\oplus _1\). Moreover, It is obvious that associativity, commutativity and monotonicity of \(\oplus _1\) are determined by the corresponding properties of \(*\), respectively. Thus, we can easily obtain that Proposition 1 holds for t-subnorm instead of t-norm.

Example 2

Assume that \(J_n=\{0,1,2,\ldots ,n\}\), \((I_i)^n_{i=0}\) is a partition of [0, 1[ consisting of right-open sub-intervals. Let \(*\) be the t-subnorm \(\widetilde{T}=\max (\min (x,\frac{1}{2})+\min (y,\frac{1}{2})-\frac{3}{4},0)\) on [0, 1], \(*\) be the discrete t-norm \(T_L(i,j)=\max (0,i+j-n)\) on \(J_n\), \(\varphi _i(x)=\frac{x-a_{i-1}}{a_i-a_{i-1}}\). Define \(x\oplus y\) as follows:

$$\begin{aligned} x\oplus y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i+j-n}(\widetilde{T}(\frac{x-a_{i-1}}{a_i-a_{i-1}},\frac{y-a_{j-1}}{a_j-a_{j-1}} )) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~\mathrm{and}~i+j>n, \\ \min (x,y) ,&{} \mathrm{if}~\max (x,y)=1,\\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(6)

is a t-norm.

As stated earlier, \(*\) must be a t-norm without zero-divisors when left-open sub-intervals are taken into account. Similar to Proposition 1, the following proposition can be obtained:

Proposition 2

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm without zero-divisors, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete t-norm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j>0\}\). Then the operation \(\oplus _2\) defined by

$$\begin{aligned} x\oplus _2 y =\left\{ \begin{array}{ll} \min (x,y),&{} \mathrm{if}~\max (x,y)=1,\\ \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i\setminus \{1\},~y\in I_j\setminus \{1\}~\mathrm{and}~i*'j>0, \\ 0 ,&{} \mathrm otherwise, \end{array} \right. \end{aligned}$$
(7)

is a t-norm.

Next, we discuss the construction when \(*\) is a t-norm and \(*'\) is a discrete t-superconorm. Analogously, two cases of right-open sub-intervals of [0, 1[ and left-open sub-intervals of ]0, 1] are taken into account. We start with the case of the right-open sub-intervals.

Proposition 3

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete t-superconorm on \(J_n\) such that \(*'\) is strictly increasing and \(i*'j>\max (i,j)\) on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\). Then the operation \(\oplus _3\) defined by

$$\begin{aligned} x\oplus _3 y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)),&{}\mathrm{if}~x\in I_i,~y\in I_j ~\mathrm{and}~i*'j<n, \\ 1,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(8)

is a t-superconorm.

Without the condition that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\), \(\oplus _3\) is not always increasing. We have the following counterexample.

Example 3

Assume that \(J_7=\{0,1,2,\ldots ,7\}\), \((I_i = [i/8, (i+1)/8[)^7_{ i = 0}\) is a partition of [0, 1[. Let \(*\) be the t-norm \(T_M(x,y)=\min (x,y)\) on [0, 1], \(*'\) be the discrete t-superconorm \(\widetilde{S}=\min (n,\max (i,j)+4)\) on \(J_7\), \(\varphi _i(x)=\frac{x-a_{i-1}}{a_i-a_{i-1}}\). Define \(x\oplus y\) as follows:

$$\begin{aligned} x\oplus y =\left\{ \begin{array}{ll} \varphi ^{-1}_{\widetilde{S}(i,j)}(\min (\frac{x-a_{i-1}}{a_i-a_{i-1}},\frac{y-a_{j-1}}{a_j-a_{j-1}} )) ,&{} \mathrm{if}~x\in I_i,~y\in I_j ~\mathrm{and}~i*'j<n, \\ 1 ,&{} \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Consider that \(x=\frac{1}{16}\), \(y=\frac{1}{8}\) and \(z=\frac{3}{16}\), then we have that

$$\begin{aligned} x\oplus z = \varphi ^{-1}_{5}\big (\frac{1}{2} \big )=\frac{11}{16}> \frac{5}{8}=\varphi ^{-1}_{5}(0)=y\oplus z. \end{aligned}$$
(9)

Obviously, \(\oplus \) is not increasing.

In Eq. (8), let \(x\oplus _3 y=\max (x,y)\) on the domain \(\{(x,y)\mid x,y\in [0,1], \min (x,y)=0\}\). We can easily prove that the operation \(\oplus _3\) is a t-conorm by simple calculations.

Similarly, when left-open sub-intervals are taken into account, \(*\) must be a t-norm without zero-divisors. Then, the following proposition can be obtained:

Proposition 4

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm without zero-divisors, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete t-superconorm on \(J_n\) such that \(*'\) is strictly increasing and \(i*'j>\max (i,j)\) on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\). Then the operation \(\oplus _4\) defined by

$$\begin{aligned} x\oplus _4 y =\left\{ \begin{array}{ll} \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~ \mathrm{and}~i*'j<n, \\ \max (x,y) ,&{} \mathrm{if}~\min (x,y)=0,\\ 1 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(10)

is a t-conorm.

In what follows, we construct a new operation from a t-norm \(*\) and a discrete uninorm \(*'\).

Proposition 5

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete uninorm on \(J_n\) with neutral element h such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~\max (i,j)\le h,i*'j>0\}\) and \(\{(i,j)\mid i,j\in J_n,~\min (i,j)\ge h,i*'j<n\}\). Then the operation \(\oplus _5\) defined by

$$\begin{aligned} x\oplus _5 y =\left\{ \begin{array}{ll} a_i ,&{} \mathrm{if}~\min (x,y)<a_h~\mathrm{and}~a_h\le a_i\le \max (x,y)<a_{i+1},\\ \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)) ,&{} \mathrm{if}~x\in I_i, y\in I_j, \max (i,j)\le h~\mathrm{and}~ i*'j>0, \\ &{}~\mathrm{or}~h<\min (i,j)~\mathrm{and}~i*'j<n, \\ 1 ,&{} \mathrm{if}~x\in I_i, y\in I_j,~h<\min (i,j)~\mathrm{and}~i*'j=n,\\ &{}~\mathrm{or}~\max (x,y)=1, \\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(11)

is associative, commutative and increasing.

In fact, the similar proposition does not hold when \((I_i)^n_{i=0}\) is a partition of ]0, 1] consisting of left-open sub-intervals. A counterexample is as follows:

Example 4

Assume that \(J_4=\{0,1,2,3,4\}\), \((I_i = ]i/5, (i+1)/5])^4_{ i = 0}\) is a partition of ]0, 1]. Let \(*\) be the t-norm \(T_M(x,y)=\min (x,y)\) on [0, 1], \(*'\) be the discrete uninorm U with neutral element 2:

$$\begin{aligned} U(i,j) = \left\{ \begin{array}{ll} T_{L}(i,j) ,&{} \mathrm{if}~0\le i,j\le 2,\\ S_{L}(i,j) ,&{} \mathrm{if}~2\le i,j\le 4,\\ \min (i,j) ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

where \(T_L(i,j)=\max (0,i+j-2)\), \(S_L(i,j)=\min (4,i+j-2)\).

Besides, \(\varphi _i(x)=\frac{x-a_{i-1}}{a_i-a_{i-1}}\). Define \(x\oplus y\) as follows:

$$\begin{aligned} x\oplus y =\left\{ \begin{array}{ll} a_{i+1} ,&{} \mathrm{if}~\frac{3}{5}<\max (x,y)~\mathrm{and}~ a_i<\min (x,y)\le a_{i+1}\le \frac{3}{5},\\ \varphi ^{-1}_{U(i,j)}(\varphi _i(x)*\varphi _j(y)) ,&{} \mathrm{if}~x\in I_i, y\in I_j, \max (i,j)\le 2~\mathrm{and}~ i*'j>0, \\ &{}~\mathrm{or}~2<\min (i,j)~\mathrm{and}~i*'j<4, \\ 0 ,&{} \mathrm{if}~x\in I_i, y\in I_j,~\max (i,j)\le 2~\mathrm{and}~i*'j=0,\\ &{}~\mathrm{or}~\min (x,y)=0, \\ 1 ,&{} \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Consider that \(x=\frac{1}{2}\), \(y=\frac{1}{2}\) and \(z=\frac{4}{5}\), then we have that

$$\begin{aligned} (x\oplus y) \oplus z = a_{U(2,2)}=a_2=\frac{3}{5}\ne \frac{1}{2}=x\oplus a_2=x\oplus (y\oplus z). \end{aligned}$$
(12)

Obviously, \(\oplus \) is not associative.

When \(*\) is a t-norm and \(*'\) is a discrete t-conorm, we can construct some proper uninorms.

Proposition 6

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete t-conorm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\). Then the operation \(\oplus _6\) defined by

$$\begin{aligned} x\oplus _6 y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)) ,&{} \mathrm{if}~x\in I_i\setminus \{a_0\},~y\in I_j\setminus \{a_0\}~ \mathrm{and}~i*'j<n, \\ &{} ~\mathrm{or}~\min (x,y)\in I_0,~\max (x,y)\in I_n,\\ y ,&{} \mathrm{if}~x=a_0,\\ x ,&{} \mathrm{if}~y=a_0,\\ 1 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(13)

is a proper disjunctive uninorm with neutral element \(a_0\) if and only if \(*\) has no zero-divisors.

In what follows, we give an example to illustrate that \(*'\) must be strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\).

Example 5

Assume that \(J_4=\{0,1,2,3,4\}\), \((I_i = [i/5, (i+1)/5[)^4_{ i = 0}\) is a partition of [0, 1[. Let \(*\) be the t-norm \(T_M(x,y)=\min (x,y)\) on [0, 1], \(*'\) be the discrete t-conorm \(S_M=\max (i,j)\) on \(J_4\), \(\varphi _i(x)=\frac{x-a_{i-1}}{a_i-a_{i-1}}\). Define \(x\oplus y\) as follows:

$$\begin{aligned} x\oplus y =\left\{ \begin{array}{ll} \varphi ^{-1}_{\max (i,j)}(\min (\frac{x-a_{i-1}}{a_i-a_{i-1}},\frac{y-a_{j-1}}{a_j-a_{j-1}} )) ,&{} \mathrm{if}~x\in I_i\setminus \{\frac{1}{5}\},~y\in I_j\setminus \{\frac{1}{5}\}~ \mathrm{and}~\max (i,j)<4, \\ &{} ~\mathrm{or}~\min (x,y)\in [0,\frac{1}{5}[,~\max (x,y)\in [\frac{4}{5},1[,\\ y ,&{} \mathrm{if}~x=\frac{1}{5},\\ x ,&{} \mathrm{if}~y=\frac{1}{5},\\ 1 ,&{} \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Consider that \(x=\frac{3}{10}\), \(y=\frac{2}{5}\) and \(z=\frac{1}{2}\), then we have that

$$\begin{aligned} x\oplus z = \varphi ^{-1}_{2}\big (\frac{1}{2} \big )=\frac{1}{2}> \frac{2}{5}=\varphi ^{-1}_{2}(0)=y\oplus z. \end{aligned}$$
(14)

Obviously, \(\oplus \) is not increasing.

Similar to Proposition 6, when the left-open sub-intervals are taken into account, we have the following result:

Proposition 7

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-norm without zero-divisors, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete t-conorm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\). Then the operation \(\oplus _7\) defined by

$$\begin{aligned} x\oplus _7 y =\left\{ \begin{array}{ll} \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~ \mathrm{and}~i*'j<n, \\ &{} ~\mathrm{or}~\min (x,y)\in I_0,~\max (x,y)\in I_n,\\ 0 ,&{} \mathrm{if}~\min (x,y)=0,\\ 1 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$
(15)

is a proper conjunctive uninorm with neutral element \(a_0\).

3.2 The Case that \(*\) Is a T-Conorm

Taking into account the duality between t-norms and t-conorms, the results in the case that \(*\) is a t-conorm are easily obtained.

Proposition 8

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm without one-divisors, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete t-conorm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\). Then the operation \(\oplus ^1\) defined by

$$\begin{aligned} x\oplus ^1 y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)) ,&{} \mathrm{if}~x\in I_i\setminus \{0\},~y\in I_j\setminus \{0\}~\mathrm{and}~i*'j<n, \\ \max (x,y) ,&{} \mathrm{if}~\min (x,y)=0,\\ 1 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

is a t-conorm.

Similar to the case that \(*\) is a t-norm, Proposition 8 holds for t-superconorm instead of t-conorm.

Proposition 9

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete t-conorm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j<n\}\). Then the operation \(\oplus ^2\) defined by

$$\begin{aligned} x\oplus ^2 y =\left\{ \begin{array}{ll} \max (x,y),&{} \mathrm{if}~\min (x,y)=0,\\ \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~\mathrm{and}~i*'j<n, \\ 1 ,&{} \mathrm otherwise, \end{array} \right. \end{aligned}$$

is a t-conorm.

Proposition 10

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm without one-divisors, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete t-subnorm on \(J_n\) such that \(*'\) is strict increasing and \(i*'j<\min (i,j)\) on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j>0\}\). Then operation \(\oplus ^3\) defined by

$$\begin{aligned} x\oplus ^3 y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~ \mathrm{and}~i*'j>0, \\ \min (x,y) ,&{} \mathrm{if}~\max (x,y)=1,\\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

is a t-norm.

Proposition 11

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete t-subnorm on \(J_n\) such that \(*'\) is strictly increasing and \(i*'j<\min (i,j)\) on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j>0\}\). Then the operation \(\oplus ^4\) defined by

$$\begin{aligned} x\oplus ^4 y =\left\{ \begin{array}{ll} \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~ \mathrm{and}~i*'j>0, \\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

is a t-subnorm.

Proposition 12

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm without one-divisors, \((I_i)^n_{i=0}\) be a partition of [0, 1[ consisting of right-open sub-intervals. Assume that \(*'\) is a discrete t-norm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j>0\}\). Then the operation \(\oplus ^5\) defined by

$$\begin{aligned} x\oplus ^5 y =\left\{ \begin{array}{ll} \varphi ^{-1}_{i*'j}(\varphi _i(x)*\varphi _j(y)) ,&{} \mathrm{if}~x\in I_i,~y\in I_j~ \mathrm{and}~i*'j>0, \\ &{} ~\mathrm{or}~\min (x,y)\in I_0,~\max (x,y)\in I_n,\\ 1 ,&{} \mathrm{if}~\max (x,y)=1,\\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

is a proper disjunctive uninorm with neutral element \(a_{n-1}\).

Proposition 13

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete uninorm on \(J_n\) with neutral element h such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~\max (i,j)\le h,i*'j>0\}\) and \(\{(i,j)\mid i,j\in J_n,~\min (i,j)\ge h,i*'j<n\}\). Then the operation \(\oplus ^6\) defined by

$$\begin{aligned} x\oplus ^6 y =\left\{ \begin{array}{ll} a_{i+1} ,&{} \mathrm{if}~\max (x,y)>a_{h-1}~\mathrm{and}~ a_i<\min (x,y)\le a_{i+1}\le a_{h-1},\\ \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i, y\in I_j, \max (i,j)\le h-1~\mathrm{and}~ i*'j>0, \\ &{}~\mathrm{or}~h-1<\min (i,j)~\mathrm{and}~i*'j<n, \\ 0 ,&{} \mathrm{if}~x\in I_i, y\in I_j, \max (i,j)\le h-1~\mathrm{and}~i*'j=0,\\ &{}~\mathrm{or}~\min (x,y)=0, \\ 1 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

is associative, commutative and increasing.

Proposition 14

Let \(*:[0,1]^2\rightarrow [0,1]\) be a t-conorm, \((I_i)^n_{i=0}\) be a partition of ]0, 1] consisting of left-open sub-intervals. Assume that \(*'\) is a discrete t-norm on \(J_n\) such that \(*'\) is strictly increasing on the domain \(\{(i,j)\mid i,j\in J_n,~i*'j>0\}\). Then the operation \(\oplus ^7\) defined by

$$\begin{aligned} x\oplus ^7 y =\left\{ \begin{array}{ll} \chi ^{-1}_{i*'j}(\chi _i(x)*\chi _j(y)) ,&{} \mathrm{if}~x\in I_i\setminus \{a_{n-1}\},~y\in I_j\setminus \{a_{n-1}\}~ \mathrm{and}~i*'j>0, \\ &{} \mathrm{or}~\min (x,y)\in I_0,~\max (x,y)\in I_n,\\ y ,&{} \mathrm{if}~x=a_{n-1},\\ x ,&{} \mathrm{if}~y=a_{n-1},\\ 0 ,&{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

is a proper conjunctive uninorm with neutral element \(a_{n-1}\) if and only if \(*\) has no one-divisors.

Results

Inspired by the construction method of paving, we construct some new associative, commutative and increasing operations on the unit interval from a t-norm on the unit interval and a discrete t-norm/t-superconorm/t-conorm/uninorm. Similarly, we present the dual constructions from a t-conorm and a discrete t-norm/t-subnorm/t-conorm/uninorm.