Orthodox Ordered Fuzzy Number A is defined as an ordered pair of continuous real functions specified on the interval [0, 1]; that is,
with
$$f, g : [0,1] \rightarrow \mathbf{R }.$$
In this chapter, the set of all OFNs is denoted by \(\mathscr {R}\). The continuity of both functions implies that their images are bounded intervals, say UP and DOWN, respectively. The following symbols are used to mark boundaries for \(UP = [l_{A}, 1_{A}^{-}]\) and for \(DOWN = [1_{A}^{+}, p_{A}]\). If we further assume that f and g are monotone (and consequently invertible), and add the constant function on the interval \([1_{A}^{-}, 1_{A}^{+}]\) with its value equal to 1, we might define the membership function
$$\begin{aligned} \mu (x)= \left\{ \begin{array}{lll} \mu _{up}(x) &{} \text {if} &{} x\in [l_{A}, 1_{A}^{-}]=[f(0), f(1)], \\ \mu _{down}(x) \;\;\;\;&{} \text {if} \;\;\;\; &{} x\in [1_{A}^{+}, p_{A}]=[g(1), g(0)], \\ 1 &{} \text {if} &{} x\in [1_{A}^{-}, 1_{A}^{+}]. \end{array} \right. \end{aligned}$$
(7.1)
where
-
1.
\(\mu _{up}(x)=:f^{-1}(x)\) and \(\mu _{down}(x)=:g^{-1}(x)\).
-
2.
f is increasing; g is decreasing.
-
3.
\(f\le g\) (pointwise).
Obtained in this way the membership function \(\mu (x), x\in \mathbf{R }\) represents a mathematical object that refers to a convex fuzzy number in the classical sense [10, 34]. However, we can observe here some limitations. This is because some membership functions already known in the classical theory of fuzzy numbers (cf. [10, 18, 46]) cannot be obtained by taking inverses of continuous functions f and g in the process described above. These are the functions that are piecewise constant; that is, \(\mu _{up}\) and \(\mu _{down}\) are not strictly monotone. The lack of strict monotonicity implies that functions inverse to \(\mu _{up}\) and \(\mu _{down}\) do not exist in the classical sense. To cope with this problem Kosiński offered to accept some limitations assuming that for both functions \(\mu _{up}\) and \(\mu _{down}\) there exist a finite (or at most countable) number of such constancy subintervals, and then the inverse functions are piecewise continuous and monotone with a finite (or at most countable) number of discontinuity points [36]. In this way we can employ a class of functions larger than continuous ones. This is the class of real-valued functions of bounded (finite) variation, BV [41].
2.1 Step-Ordered Fuzzy Numbers
In 2006 Kosiński introduced a generalization of the original definition of OFNs to make the algebra a more efficient tool in dealing with imprecise, fuzzy quantitative terms [36].
Definition 1
By an OFN A (in the generalized form) we mean an ordered pair (f, g) of functions such that \(f,g:[0,1] \rightarrow \mathbf{R }\) are of bounded variation, that is, \(f,g \in BV\).
Let \(\mathscr {R}_{BV}\) denote the set of all generalized OFNs, that is, those that meet Definition 1. Notice that all convex fuzzy numbers are contained in this new space, \(\mathscr {R}\subset \mathscr {R}_{BV}\). Operations for generalized OFNs are defined in a similar way to operations for orthodox OFNs, the norm, however, will change into the norm of the Cartesian product of the space of functions of bounded variations.
An important consequence of this generalization is the possibility of introducing a subspace of an OFN composed of pairs of step functions [37]. First, a natural number K is fixed and [0, 1) is split into \(K - 1\) subintervals \([a_{i}, a_{i+1})\); that is,
$$ \bigcup \limits _{i=1}^{K-1}[a_{i}, a_{i+1})=[0,1), $$
where
$$0 = a_1< a_2< \cdots < a_{K} = 1. $$
Now, define a step function f of resolution K by putting value \(u_i \in \mathbf{R }\) on each subinterval \([a_{i}, a_{i + 1})\). Each such function f is identified with a K-dimensional vector; that is,
$$f \sim {{\varvec{u}}} = (u_1, u_2, \ldots ,u_K)\in \mathbf{R }^K,$$
where the Kth value \(u_K\) corresponds to \(y = 1\); that is, \(f(1) = u_K\). Taking a pair of such functions we have an OFN from \(\mathscr {R}_{BV}\).
Definition 2
By a Step-Ordered Fuzzy Number A of resolution K we mean an ordered pair (f, g) of functions such that \(f, g : [0,1] {\rightarrow } \mathbf{R }\) are step functions of resolution K.
We use \(\mathscr {R}_K\) for denotation of the set of elements satisfying the above definition. The example of an SOFN (also called Step Kosiński’s fuzzy number, SKFN) and its membership relation (represented by a curve) are shown in Figs. 7.1 and 7.2. The set \(\mathscr {R}_K \subset \mathscr {R}_{BV}\) has been extensively elaborated in [22, 35].
We can identify \(\mathscr {R}_K\) with the Cartesian product of \(\mathbf{R }^K \times \mathbf{R }^K\) because each K-step function is represented by its K values. It is obvious that each element of the space \(\mathscr {R}_K\) may be regarded as an approximation of elements from \(\mathscr {R}_{BV}\); by increasing the number K of steps we are getting a better approximation. The norm of \(\mathscr {R}_K\) is assumed to be the Euclidean one of \(\mathbf{R }^{2K}\), thus we have an inner-product structure at our disposal.
Now let \(\mathscr {B}\) be the set of two binary values: 0, 1 and let us introduce the particular subset \(\mathscr {N}\) of \(\mathscr {R}_K\)
$$\begin{aligned} {\mathscr {N}} = \{A =(\underline{u},\underline{v}) \in \mathscr {R}_K : \underline{u}\in {\mathscr {B}}^K\, , \underline{v}\in {\mathscr {B}}^K \}. \end{aligned}$$
(7.2)
It means that each such component of the vector \(\underline{u}\) as well as of \(\underline{v}\) has value 1 or 0. Because each element of \(\mathscr {N}\) is represented by a 2K-dimensional binary vector the cardinality of the set \(\mathscr {N}\) is \(2^{2K}\). The set \({\mathscr {N}}\) consists of all binary SOFN, also called Binary Step Kosiński’s Fuzzy Numbers (BSKFN).
Definition 3
By a BSKFN A of resolution K we mean an ordered pair (f, g) of functions such that \(f, g : [0,1] {\rightarrow } {\mathscr {B}}\) are step functions of resolution K.
2.2 Lattice on \(\mathscr {R}_K\)
Let us consider the set \(\mathscr {R}_K\) of SOFNs with operations
$$A \wedge B =:F \;\;\;and\;\;\;A \vee B =:G$$
defined for each two fuzzy numbers \(A = (f_A,g_A), B = (f_B,g_B)\) by the relations:
$$\begin{aligned} F=(f_F,g_F), \, {\text {if }} \, f_F=\sup \{f_A, f_B\}\, , g_F=\sup \{g_A, g_B\}\, , \end{aligned}$$
(7.3)
$$\begin{aligned} F=(f_F,g_F), \, {\text {if }} \, f_F=\inf \{f_A, f_B\}\, , g_F=\inf \{g_A, g_B\}\, . \end{aligned}$$
(7.4)
Notice that \(\vee \) and \(\wedge \) are actually operations in \(\mathscr {R}_K\); that is, they are defined for all A, \(B \in {\mathscr {R}_K}\) and the result of the operations is in \({\mathscr {R}_K}\). Next, let us observe that operation \(\vee \) is
The same properties characterize the operation \(\wedge \). Moreover, these two operations are connected by the absorption law:
$$\begin{aligned}&A \wedge (A \vee B) = A\wedge (sup\{f_A,f_B\},sup\{g_A,g_B\})=\\&(inf\{f_A,sup\{f_A,f_B\}\},inf\{g_A,sup\{g_A,g_B\}\})=(f_A,g_A) =A \end{aligned}$$
and similarly for
$$ A \vee (A \wedge B) = A. $$
The absorption laws ensure that the set \({\mathscr {R}_K}\) with an order \(\le \) defined as
$$\begin{aligned} A \le B \;\;\; \text {iff} \;\;\; B=A \vee B \end{aligned}$$
(7.5)
is a partial order within which meets and joins are given through the operations \(\vee \) and \(\wedge \). It is easy to show that for every \(A,B\in \mathscr {R}_K\) it holds that \(A \vee B = B\) iff \(B - A\ge 0\). Moreover, joints and meets exist for every two elements of \(\mathscr {R}\). The following theorem is the consequence of the above reasoning.
Theorem 1
The algebra \(({\mathscr {R}_K},\vee ,\wedge )\) is a lattice.
2.3 Complements and Negation on \(\mathscr {N}\)
Now let us consider the subset \(\mathscr {N}\) of \(\mathscr {R}_K\) defined in Sect. 7.2.1. As we have already noted above, every element of \(\mathscr {N}\) can be represented by a binary vector and thereby \(\mathscr {N}\) is isomorphic to the space of Boolean vectors. Below, we use the notation \(A_{(a_1,a_2,\ldots ,a_{2K})}\) for a number A represented by vector \((a_1,a_2,\dots ,a_{2K})\) and we show that \(\mathscr {N}\) is a Boolean algebra.
It is easy to observe that all subsets of \(\mathscr {N}\) have both a join and a meet in \(\mathscr {N}\). In fact, for every pair of numbers from the set \(\{0,1\}\) we can determine max and min and it is always 0 or 1. Therefore \(\mathscr {N}\) creates a complete lattice. In such a lattice we can distinguish the greatest element \(\underline{1} = A_{(1,1,\ldots ,1)}\) and the least element \(\underline{0} = A_{(0,0,\ldots ,0)}\).
Theorem 2
The algebra \(({\mathscr {N}},\vee ,\wedge )\) is a complete lattice.
In a lattice in which the greatest and the least elements exist it is possible to define complements. We say that two elements A and B are complements of each other if and only if
$$A \vee B = \underline{1} \; \;\; \text {and} \;\;\; A \wedge B = \underline{0}. $$
The complement of a number A is marked with \(\lnot A\) and is defined as follows.
Definition 4
Let \(A_{(a_1,a_2,\dots ,a_{2K})} \in {\mathscr {N}}\) be a SOFN. Then the complement of \(A_{(a_1,a_2,\dots ,a_{2K})}\) equals
$$\lnot A_{(a_1,a_2,\dots ,a_{2K})} = A_{(1 - a_1,1 - a_2,\dots ,1 -a_{2K})}.$$
A bounded lattice for which every element has a complement is called a complemented lattice. The structure of Step-Ordered Fuzzy Numbers \(({\mathscr {N}},\vee ,\wedge )\) forms complete and complemented lattices in which complements are unique. In fact it is a Boolean algebra. An example of such an algebra is depicted in Fig. 7.3. A set of universe is created by numbers
$$ {\mathscr {N}}=\{A_{(a_1,a_2,a_3,a_4)} : a_i \in \{0,1\} \;\;\; \text {for} \;\;\; i=1,2,3,4\}. $$
The complements of elements are:
$$\begin{aligned} \lnot A_{(0,0,0,0)}=A_{(1,1,1,1)}, \lnot A_{(0,1,0,0)}=A_{(1,0,1,1)}, \lnot A_{(1,1,0,0)}=A_{(0,0,1,1)} \mathrm{etc.}. \end{aligned}$$
Now we can rewrite the definition of the complement in terms of a new mapping.
Definition 5
For any \(A \in {\mathscr {N}}\) we define its negation as
$$ N(A):= (1 - a_1,1 - a_2,\dots ,1 - a_{2K})\,\;\;\; {\text {for}} \, \;\;\; A = (a_1,a_2,\dots ,a_{2K}) . $$
It is obvious, from Definitions 4 and 5, that the negation of a given number A is its complement. Moreover, the operator N is a strong negation, because it is involutive:
$$ N(N(A)) = A \, {\text { for any }}\, A\in {\mathscr {N}}\, .$$
One can refer here to known facts from the theory of fuzzy implications (cf. [6, 7, 21]) and write the strong negation N in terms of the standard strong negation \(N_I\) on the unit interval \(I = [0, 1]\) defined by \(N_I(x) = 1 - x\, , x\in I\), namely \(N((a_1,a_2,\dots ,a_{2K})) = ((N_I(a_1), N_I(a_2),\dots ,N_I(a_{2K}))\).
2.4 Fuzzy Implication on BSOFN
The implication operator holds center stage in the inference mechanisms of any logic. Thus, the obvious question is whether and how one can define an implication on an OFN. Studies on this issue were initiated in the works by Kacprzak and Kosiński in 2011 [28, 38]. The aim was to propose an implication operation on Ordered Fuzzy Numbers analogous to classical implication that preserves its main properties. In the literature we can find several different definitions of fuzzy implications. Some of them are built from basic fuzzy logic connectives. In Sect. 7.2.2 conjunction and disjunction operations for any two-order fuzzy numbers are defined. However, the main problem is the negation operation. In Sect. 7.2.3 complements for SOFNs from the set \(\mathscr {N}\) are constructed. Thus given disjunction and complement, implication can be defined in the standard way. Such a new operator on the set \(\mathscr {N}\) was introduced by Kacprzak and Kosiński and is called 2K-fuzzy implication [28, 29, 38]. The set of all OFNs is not a complete lattice, therefore the way of defining implication is still an open question.
In the classical Zadeh fuzzy logic the definition of a fuzzy implication on an abstract lattice \({\mathscr {L}} = (L,\le _{L})\) is based on the notation from fuzzy set theory introduced in [21].
Definition 6
Let \({\mathscr {L}} = (L,\le _{L}, 0_L, 1_L)\) be a complete lattice. A mapping \({\mathscr {I}}:L^2\rightarrow L\) is called a fuzzy implication on \(\mathscr {L}\) if it is decreasing with respect to the first variable, increasing with respect to the second variable, and fulfills the border conditions
$$\begin{aligned} {\mathscr {I}}(0_L, 0_L)={\mathscr {I}}(1_L,1_L) = 1_L\, , {\mathscr {I}}(1_L,0_L) =0_L\, . \end{aligned}$$
(7.6)
Now, possessing the lattice structure of \({\mathscr {R}_{\mathscr {K}}}\) and the Boolean structure of our lattice \(\mathscr {N}\), we can repeat most of the definitions known in Zadeh’s fuzzy set theory. The first one is the Kleene-Dienes operation, called 2K-fuzzy implication [28]
$$\begin{aligned} {\mathscr {I}}_{b}(A, B) = N(A)\vee B\, , {\text { for any}} \;\;\; A , B\in {\mathscr {N}}\, . \, \end{aligned}$$
(7.7)
In other words, the result of the binary implication \({\mathscr {I}}_{b}(A, B)\), denoted in [28] by \(A \rightarrow B\), is equal to the result of operation sup for the number B and the complement of A:
$$ A \rightarrow B = sup\{\lnot A, B \}. $$
For illustration, let us assume two numbers \(A_{(0,1,1,0)}\) and \(A_{(0,1,0,1)}\). The implication
$$ A_{(0,1,1,0)} \rightarrow A_{(0,1,0,1)} $$
equals
$$ N(A_{(0,1,1,0)}) \vee A_{(0,1,0,1)} = A_{(1,0,0,1)} \vee A_{(0,1,0,1)}= A_{(1,1,0,1)}. $$
Examples of other implications are given in Table 7.1.
Table 7.1 Examples of implications for SOFN
2K-fuzzy implication satisfies the basic property of logical implication: it returns false if and only if the first term is true, and the second term is false.
Proposition 1
Let us consider the Boolean algebra \(({\mathscr {N}}, \vee , \wedge , \lnot , \underline{1}, \underline{0})\). The values of the 2K-fuzzy implication on the greatest and the least elements of this algebra are given in Table 7.2.
Table 7.2 Table of values of implications for the least element and the greatest elements of \(\mathscr {N}\)
In fact, because \(\lnot \;\underline{0} = \underline{1}\) and \(\lnot \;\underline{1} = \underline{0}\) it holds that:
-
\(\underline{0} \rightarrow \underline{0} = N(\underline{0}) \vee \underline{0} = \underline{1}\vee \underline{0} = \underline{1}\).
-
\(\underline{0} \rightarrow \underline{1} = N(\underline{0}) \vee \underline{1} = \underline{1}\vee \underline{1} = \underline{1}\).
-
\(\underline{1} \rightarrow \underline{0} = N(\underline{1}) \vee \underline{0} = \underline{0}\vee \underline{0} = \underline{0}\).
-
\(\underline{1} \rightarrow \underline{1} = N(\underline{1}) \vee \underline{1} = \underline{0}\vee \underline{1} = \underline{1}\).
Next we may introduce the Zadeh implication by
$$\begin{aligned} {\mathscr {I}}_{Z}(A, B) = (A \wedge B)\vee N(A)\, , {\text { for any }} A , B\in {\mathscr {N}}\, . \end{aligned}$$
(7.8)
In our lattice \(\mathscr {R}_K\) the arithmetic operations are well defined, therefore we may introduce the counterpart of the Łukasiewicz implication by
$$\begin{aligned} {\mathscr {I}}_{L}(A, B) = C\, , {\text {where }} \, C=1 \wedge (1-A+B) \, . \end{aligned}$$
(7.9)
When calculating the right-hand side of (7.9) we have to regard all numbers as elements of \(\mathscr {R}_K\), because by adding step fuzzy number A from \(\mathscr {N}\) to the crisp number 1 we may leave the subset \(\mathscr {N}\subset \mathscr {R}_K\). However, the operation \(\wedge \) takes us back to the lattice \(\mathscr {N}\). It is obvious that in our notation \(1_N = 1\). The explicit calculation is: if \(C = (c_1, c_2,\dots , c_{2K)}), A = (a_1,a_2,\dots ,a_{2K}), B = (b_1,b_2,\dots ,b_{2K})\), then \(c_i = min\{1, 1 - a_i + b_i\}\), where \(1\le i \le 2K\).
It is obvious that all implications \({\mathscr {I}}_{b}, {\mathscr {I}}_{Z}\), and \( {\mathscr {I}}_{L}\) satisfy the border conditions (7.6) as well as the fourth condition of the classical binary implication, namely \({\mathscr {I}}(0_N, 1_N) = 1_N\).
2.5 Applications
Initially, OFNs were designed to deal with optimization problems when data are fuzzy [14, 15, 17, 20]. When Kacprzak and Kosiński observed that a subspace of OFNs, called SOFN, may be equipped with a lattice structure, it turned out that OFNs have a much wider field of application. The ability to define Boolean operations such as conjunction, disjunction, and, more important, diverse types of implications, has become the basis for creating a new logical system. In consequence, it turned out that SOFNs can be used not only for evaluation of linguistic statements such as, “A patient is fat” or “A car is fast,” but also for approximate reasoning on such imprecise notions.
One of the important applications is employing SOFNs in multiagent systems for modeling agents’ beliefs about fuzzy expressions [27]. This can be helpful in evaluating features of multiagent systems concerning agents’ fuzzy beliefs. If some sentence is expressed by an agent in a multiagent system then we could try to evaluate the level of truth for an agent’s belief about another agent’s belief. This is the first step in the application of the fuzzy logic that stands behind the SOFN.
Just before his death, Kosiński with his coworkers Kacprzak and Wȩgrzyn-Wolska showed another application of SOFNs in specification and automatic verification of diversity of opinion [30]. As a consequence, SOFN can also be used in reasoning about communicating software agents or boots that are decision-support systems. For example, we can analyze activity of agents that assists clients with their decisions in e-shops, that is, agents which support users of a system in making decisions and choosing the right product.