An extension of several properties for fuzzy t -norm and vague t -norm

Abstract

Rosenfeld defined a fuzzy subgroup of a given group as a fuzzy subset with two special conditions and Mustafa Demirci proposed the idea of fuzzifying the operations on a group through a fuzzy equality and a fuzzy equivalence relation. This paper mainly focuses on fuzzy subsets and vague sets of monoids with several extended algebraic properties. Firstly, we generalize some algebraic properties of t-norms to fuzzy t-norms, this allows for a broader analysis and classification of fuzzy t-norms, enabling their wider application. Furthermore, we explore specific research on the properties of vague t-norms. Finally, selected conclusions about fuzzy t-norms are extended to bounded lattices.

Keywords

uninorm nullnorm aggregation function fuzzy monoid vague monoid

1 Introduction

1.1 Brief review of fuzzy subgroup

The fuzzification or relaxation of logical connectives can be effectively applied in solving practical problems such as imprecision, lack of accuracy, or the presence of noise. Rosenfeld firstly defined a fuzzy subgroup of group G which is a fuzzy subset of G with two special conditions attached [1]. On this basis, many mathematicians have obtained rich results [12 , 16]. Considering that many scholars are very interested in fuzzified algebraic structures, including groups, rings, actions, they begin to search for fuzzy cases of various algebraic structures based on Rosenfeld’s method [1, 5].

In Rosenfeld’s related work, only a subset of the group G is fuzzy, but the operations on it are still classical. Thus Mustafa Demirci proposed the idea of fuzzifying the operations on the group, which uses tools such as fuzzy equality and fuzzy functions [14]. Later, Biswas introduced the notion of vague groups [3 , 15].

1.2 Brief analysis of common aggregation functions

There is evidence that t-norm and t-conorm as two kinds of fuzzy logic operations play a crucial role in fuzzy sets theory [8, 11]. To further enrich the properties of the aggregation operators, Yager and Rybalov proposed the concpet of uninorms [9, 18]. The identity element of uninorm can take any number in the unit interval, not just zero and one in the case of t-norms and t-conorms. In addition, Calvo et al. introduced the notion of nullnorms in 2001 [19]. Uninorms and nullnorms are generalizations of t-norms and t-conorms, and on the other side, there exists close connection between them [2].

1.3 The motivation of this paper

In order to better handle issues such as imprecision, lack of accuracy or noise, the relaxation or fuzzification of logical connectors plays a crucial role. At present, there exist many researches on fuzzy t-norm (fuzzy t-conorm) and vague t-norm (vague t-conorm). More specifically, Boixader and Recasens [6] proposed fuzzy submonoid and further construct the concepts of T-fuzzy t-subnorms (T-fuzzy t-subconorms). On the other hand, they studied vague t-norms (vague t-conorms), where a vague operation ∘ (x, y, z) on set X represents the degree in which z is x ∘ y. However, there is relatively little research on the properties of fuzzy and vague t-norm (t-conorm) in existing literature. The main reason is that the operations have become abstract, and the original properties cannot be directly taken back. Therefore, from the perspective of algebraic properties, it is meaningful to explore the strict monotonicity, cancellation law, Archimedean and limit property of fuzzy and vague t-norms. Meantime, in light of different algebraic properties, fuzzy t-norm and vague t-norm can be divided into different categories, which may facilitate the direct extraction of different types of fuzzy and vague t-norm in practical problems. The study of algebraic properties related to fuzzy t-norm and vague t-norm can be regarded as a further supplement and improvement in theoretical aspects, which is helpful for scholars’ subsequent research on fuzzy t-norm and vague t-norm.

In this paper, we firstly introduce the properties of fuzzy t-norms, such as strict monotonicity, cancellation law, conditional cancellation law, Archimedean and limit property. In the case of t-norms, these properties are closely related to each other. After fuzzification, some inter-pushing relationships are still maintained. In addition, the specific research on the above properties are explored on vague t-norm. And then, the concept of bounded lattice [20] is introduced and the contents of the previous two subsections are generalized to the bounded lattice.

The remainder of this paper is organized as follows. In Section 2, some basic concepts and properties of fuzzy set theory and algebraic structure are introduced. In Section 3, the fuzzification of properties such as strict monotonicity, cancellation law, conditional cancellation law, Archimedean and limit property are introduced on fuzzy t-norms. In Section 4, vague t-norms and their properties are proposed. Section 5 further generalizes the content of previous two sections to bounded lattice. Finally, we state some conclusions in Section 7.

2 Preliminaries

This section recapitulates some well-known concepts that shall be used in the sequel.

Definition 1. [7] t-norm A t-norm is a binary operation T : [0, 1] ² → [0, 1], for all x, y, z ∈ [0, 1], which satisfies the following conditions:

Commutativity: T (x, y) = T (y, x);

Associativity: T (T (x, y) , z) = T (x, T (y, z));

Monotonicity: T is non-decreasing in each argument;

Boundary condition: T (x, 1) = x .

If we focus on the algebraic structure, the binary operation t-norm with prefix expression T can also be expressed by the infix binary operator *, then the four axioms (T1)-(T4) can be followed as: $\begin{matrix} x * y = y * x; \\ x * y \leq x * z, if y \leq z; \\ (x * y) * z = x * (y * z); \\ 1 * x = x . \end{matrix}$

Remark 1. The associativity (T2) allows us to extend each t-norm T in a unique way to an n-ary operation in the usual way by induction, defining for each n-tuple (x₁, x₂, ⋯ , x_n) ∈ [0, 1] ⁿ $T (x_{1}, x_{2}, \dots, x_{n}) = T (T (x_{1}, x_{2}, \dots, x_{n - 1}), x_{n}) .$ If, in particular, we have x₁ = x₂ = ⋯ = x_n = x, we shall briefly write $x_{T}^{(n)} = T (x, x, \dots, x) .$ Finally we put, by convention, for each x ∈ [0, 1] $x_{T}^{(0)} = 1 and x_{T}^{(1)} = x .$

Example 1. t-norm Here are four common t-norms:

Minimum: T_M (x, y) = min(x, y),

Product: T_P (x, y) = xy,

Łukasiewicz t-norm: T_L (x, y) = max(x + y - 1, 0),

Drastic product:

$T_{D} (x, y) = {\begin{matrix} 0, & if x, y \in [0, 1), \\ \min (x, y), & otherwise . \end{matrix}$

Definition 2. [7] A t-conorm is a binary operation S : [0, 1] ² → [0, 1], for all x, y, z ∈ [0, 1], which satisfies the following conditions:

Commutativity: S (x, y) = S (y, x);

Associativity: S (S (x, y) , z) = S (x, S (y, z));

Monotonicity: S is non-decreasing in each argument;

Boundary condition: S (x, 0) = x .

Example 2. The following show four common t-conorms:

Maximum: S_M (x, y) = max(x, y),

Probabilistic sum: S_P (x, y) = x + y - xy,

Łukasiewicz t-conorm: S_L (x, y) = min(x + y, 1),

Drastic sum:

$S_{D} (x, y) = {\begin{matrix} 1 & if x, y \in (0, 1], \\ \max (x, y) & otherwise . \end{matrix}$

Definition 3. [7] For an arbitrary t-norm T, we consider the following properties for all x, y, z ∈ [0, 1]:

T is said to be strictly monotone if $T (x, y) < T (x, z) whenever x > 0 and y < z .$

T satisfies the cancellation law if $T (x, y) = T (x, z) \Rightarrow x = 0 or y = z .$

T satisfies the conditional cancellation law if $T (x, y) = T (x, z) > 0 \Rightarrow y = z .$

T is called Archimedean if $\begin{matrix} \forall (x, y) \in (0, 1)^{2}, there is an \\ n \in ℕ with x_{T}^{(n)} < y . \end{matrix}$

T has the limit property if $for all x \in (0, 1), lim_{n \to \infty} x_{T}^{(n)} = 0 .$

Example 3. Notice that T_L, T_P are Archimedean t-norm, but T_M is not.

From an algebraic point of view, a t-norm is actually a commutative monoid with identity element 1.

Definition 4. [7] An aggregation function is a mapping A : [0, 1] ⁿ → [0, 1], n is a positive integer greater than 1, such that,

A (x₁, ⋯ , x_n) ≤ A (y₁, ⋯ , y_n) , whenever x_i ≤ y_i, forall i ∈ 1, ⋯ , n .

A (0, ⋯ , 0) =0 and A (1, ⋯ , 1) =1 .

Definition 5. [7] A uninorm is a binary function U : [0, 1] ² → [0, 1], for all x, y, z ∈ [0, 1], which satisfies the following conditions:

Commutativity: U (x, y) = U (y, x);

Associativity: U (U (x, y) , z) = U (x, U (y, z));

Monotonicity: U is non-decreasing in each place;

Identity element: U (x, e) = x, e ∈ [0, 1].

Remark 2. In particular, a uninorm is a t-norm when e = 1 and a t-conorm when e = 0. A uninorm U is called conjunctive if U (1, 0) =0 and a uninorm U is called disjunctive if U (1, 0) =1. A conjunctive (resp. disjunctive) uninorm U is said to be locally internal on the boundary if it satisfies U (1, x) ∈ {1, x} (resp. U (0, x) ∈ {0, x}) for all x ∈ [0, 1].

Example 4. The following are several classes of common unnorms:

$U_{\min}$ , $U_{\max}$ : those given by minimum and maximum in A (e) = {(x, y) : min(x, y) < e < max(x, y)}, respectively.

$U_{ide}$ : U (x, x) = x for all x ∈ [0, 1].

$U_{rep}$ : those that have an additive generator.

$U_{\cos}$ : continuous in the open square (0, 1) ².

Theorem 1. [9] Let U : [0, 1] ² → [0, 1] be a uninorm with neutral element e ∈ (0, 1). Then the sections x ↪ U (x, 1) and x ↪ U (x, 0) are continuous at each point except perhaps for e if and only if U is given by one of the following formulas.

If U (0, 1) =0, then $\begin{matrix} U (x, y) = {\begin{matrix} eT (\frac{x}{e}, \frac{y}{e}), \\ if (x, y) \in [0, e]^{2}; \\ e + (1 - e) S (\frac{x - e}{1 - e}, \frac{y - e}{1 - e}), \\ if (x, y) \in [e, 1]^{2}; \\ \min (x, y), \\ if (x, y) \in A (e) . \end{matrix} \end{matrix}$ The set of uninorms as above will be denoted by $U_{\min}$ .

If U (0, 1) =1, then $U_{max}$ has the same structure, changing minimum by maximum in A (e).

Definition 6. [7] A nullnorm is a binary function F : [0, 1] ² → [0, 1], for all x, y, z ∈ [0, 1], which satisfies the following conditions:

Commutativity: F (x, y) = F (y, x);

Associativity: F (F (x, y) , z) = F (x, F (y, z));

Monotonicity: F is non-decreasing in each place;

Absorbing element: there exists an absorbing element k ∈ [0, 1] , F (k, x) = k and the following statements hold, $F (0, x) = x for all x \leq k,$ $F (1, x) = x for all x \geq k .$

In general, k is always given by F (0, 1).

In addition, some concepts related to fuzzy sets are also necessary.

Definition 7. [7] Let μ : X → [0, 1] be a mapping. We call μ a fuzzy subset on X and μ (x) the membership function of μ.

The set composed of all fuzzy subsets on X is denoted as $F (X)$ . The classical set and the characteristic function are special cases of the fuzzy subset and the membership function respectively.

Definition 8. [14] Let X, Y be two sets. A fuzzy relation f from X to Y is a fuzzy subset of X × Y = {(x, y) ∣ x ∈ X, y ∈ Y}, that is, $f : X \times Y \to [0, 1],$ where f (x, y) represents the degree of the element x has a f relationship with element y. Especially, when X = Y, f is called a fuzzy relationship on X.

Later we will use some knowledge about lattice.

Definition 9. [20] A lattice is a nonempty set L equipped with a partial order ≤ such that each two elements x, y ∈ L have a greatest lower bound, called meet or infimum, denoted by x ∧ y, as well as a smallest upper bound, called join or supremum, denoted by x ∨ y.

If any number of elements have a smallest upper bound and a greatest lower bound in a lattice L, we say that L is complete. For x, y ∈ L, the symbol x < y means that x ≤ y and x ≠ y. If x ≤ y or y < x, then we say that x and y are comparable. Otherwise, we say that x and y are incomparable, which denoted as x ∥ y.

The inclusion of two fuzzy sets μ, ν on S is defined as follows: μ ⊆ ν if and only if for all x ∈ S, μ (x) ≤ ν (x). Clearly the set of all fuzzy sets on S is a complete lattice $L$ under this ordering. We shall denote the supremum and infimum in $L$ by ∪ and ∩.

3 Fuzzy t-norm with some properties

3.1 Fuzzy semigroups and fuzzy groups

We already know that a t-norm is a monoid, so the essence of defining fuzzy t-norms is to define fuzzy monoids. In order to introduce the definition of fuzzy t-norms, first we need to give the definition of fuzzy semigroups and groups, which are proposed by Rosenfeld [1]. At the same time, in order to simplify the statement of the propositional proofs of some fuzzy t-norm in the future, some knowledge to be used is given as propositions together with the definitions of fuzzy semigroups and groups.

Definition 10. [1] Let (S, ∘) be a semigroup and let μ be a fuzzy subset of S. μ will be called a fuzzy semigroup of S if, for all x, y ∈ S, $\min (μ (x), μ (y)) \leq μ (x \circ y) .$

The classic subsemigroup is actually a special case when fuzzy subset μ is the characteristic function.

Proposition 1. [1] Let (S, ∘) be a semigroup and let μ be a fuzzy subset of S. Let μ be into {0, 1}, so that μ is the characteristic function of a subset T ⊂ S if and only if T is a semigroup.

Example 5. Characteristic functions of the empty set ∅ and the complete set S, μ_id (x) =0 and μ_id (x) =1, x ∈ S, can be fuzzy semigroup of any semigroup S, which is easy to verify by definition. We call them trivial fuzzy subsemigroups.

It can be found that the intersection of any number of fuzzy subsemigroups is still a fuzzy subsemigroup. The infimum of a set of fuzzy subsets {μ_i} is denoted as ∩μ_i.

Proposition 2. [1] Let S be a semigroup. The intersection of any number of fuzzy subsemigroups of S is still a fuzzy subsemigroup of S.

Definition 11. [1] Let (S, ∘) be a group with identity element e. A fuzzy subset μ of S is called fuzzy subgroup of S if μ satisfies:

min(μ (x) , μ (y)) ≤ μ (x ∘ y) , ∀ x, y ∈ S .

μ (x^-1) ≥ μ (x) , ∀ x ∈ S .

Proposition 3. [1] Let S be a group and μ a fuzzy subset of S. If the range of μ is {0, 1}, that is, μ is a characteristic function of some subset T of S, then μ is the fuzzy subgroup of S if and only if T is a subgroup of S.

It can be found that the intersection of any number fuzzy subgroups is still a fuzzy subgroup.

Proposition 4. [1] Let S be a group. The intersection of any number of fuzzy subgroups of S is still a fuzzy subgroup of S.

Now we present the definition of fuzzy monoids, which takes the minimum t-norm T_M as T in [6].

Definition 12. [6] Let (G, ∘) be a monoid with identity element e. A fuzzy subset μ of G is called a fuzzy submonoid of G if it satisfies:

min(μ (x) , μ (y)) ≤ μ (x ∘ y) , ∀ x, y ∈ G .

μ (e) =1.

3.2 Generalization of some properties of t-norm

With the previous definition of fuzzy monoids, we can naturally define fuzzy t-norm, and then we generalize some properties originally belonging to t-norm.

Definition 13. Let T be a t-norm. A fuzzy submonoid of ([0, 1] , T) will be called a fuzzy t-subnorm of T.

Proposition 5. Let T be a t-norm. μ is a fuzzy t-subnorm of T, then μ (T (x, y)) =0 ⇒ μ (x) =0 or μ (y) =0 for any x, y ∈ [0, 1].

Proof. By Definition 13, it can be seen that for any x, y ∈ [0, 1], $\min (μ (x), μ (y)) \leq μ (T (x, y)) = 0,$ so μ (x) =0 or μ (y) =0. □

Different t-norms may have different fuzzy t-subnorms.

Example 6. The function μ_id (x) = x for any x ∈ [0, 1] can only be fuzzy t-subnorm of T_M. In fact, any other t-norm T satisfies T (x, y) < min(x, y) ((x, y) ∈ [0, 1] ²), the fuzzy t-subnorm cannot be μ_id, otherwise it contradicts μ (T (x, y)) = T (x, y) ≥ min(μ (x) , μ (y)) = min(x, y).

Example 7. By Definition 13, μ (x) =1, x ∈ [0, 1] is a fuzzy t-subnorm of any t-norm.

To better study fuzzy t-norms, we further extend the relevant properties of t-norms to fuzzy t-norms.

Definition 14. Let μ be a fuzzy t-subnorm of a t-norm T for any x, y, z ∈ [0, 1]. We consider the following properties:

μ is said to be fuzzy strictly monotone if $μ (T (x, y)) > μ (T (x, z)), 0 < x < 1, y < z .$

μ satisfies the fuzzy cancellation law if $μ (T (x, y)) = μ (T (x, z)) \Rightarrow x = 0 or y = z .$

μ satisfies the fuzzy conditional cancellation law if $\begin{matrix} μ (T (x, y)) = μ (T (x, z)) > μ (0) \\ \Rightarrow μ (y) = μ (z) . \end{matrix}$

μ is said to be Archimedean if $\begin{matrix} for each x, y \in (0, 1), \exists n \in ℕ, \\ s . t . μ (x_{T}^{(n)}) < μ (y) . \end{matrix}$

μ satisfies the fuzzy limit property if $for all x \in (0, 1), lim_{n \to \infty} μ (x_{T}^{(n)}) = μ (0) .$

Remark 3.

The fuzzy strictly monotone direction of t-subnorm μ here is opposite to the strictly monotone direction of the t-norm. Because min(μ (x) , μ (y)) ≤ μ (T (x, y)) for any x, y ∈ [0, 1], which contradicts with μ (T (x, y)) < μ (T (x, 1)) , μ (T (x, y)) < μ (T (1, y)).

The case of x = 1 in fuzzy strict monotonicity is also meaningless, since z = 1, μ (T (x, z)) =1.

In addition, we can get the fuzzy cancellation law from the fuzzy strict monotonicity, and the fuzzy conditional cancellation law can obtained from the fuzzy cancellation law, as in the case of the t-norm.

Proposition 6. If μ is fuzzy strictly monotone, then μ satisfies the fuzzy cancellation law.

Proof. Assume x, y, z ∈ [0, 1]. First, if x > 0, y < z, then μ (T (x, y)) > μ (T (x, z)). Similarly, if x > 0, y > z, then μ (T (x, y)) < μ (T (x, z)). So, when μ (T (x, y)) = μ (T (x, z)), we have x = 0 or y = z.

□ Proposition 7. If μ satisfies the fuzzy cancellation law, then μ satisfies the fuzzy conditional cancellation law.

Proof. Assume x, y, z ∈ [0, 1]. If μ (T (x, y)) = μ (T (x, z)) > μ (0), we have x = 0 or y = z by the fuzzy cancellation law. If x = 0, then μ (T (x, y)) = μ (T (x, z)) = μ (0), which is contradictory. In summary, μ (T (x, y)) = μ (T (x, z)) > μ (0) ⇒ y = z . □ In addition, the strict monotonicity of T has restrictions on the fuzzy t-subnorm μ.

Proposition 8. If the t-norm T is strictly monotone and μ is a fuzzy t-subnorm of the t-norm T, then μ (x) cannot be strictly decreasing on [0, 1].

Proof. The t-norm T is strictly monotone, then for any x, y ∈ (0, 1), $T (x, y) < T (x, 1) = x, T (x, y) < y .$ If μ (x) is strictly decreasing on [0, 1], then for all x, y ∈ (0, 1) , μ (T (x, y)) > max(μ (x) , μ (y)), which contradicts the definition. □ However, “μ (x) is not strictly decreasing on [0, 1]” is not equal to “the fuzzy t-subnorm μ is not fuzzy strictly monotone”. Furthermore, we can see that only when t-norm T is strictly monotone, the fuzzy t-subnorm μ is likely to be fuzzy strictly monotone.

Proposition 9. If the t-norm T is not strictly monotone, then the fuzzy t-subnorm is also not fuzzy strictly monotone.

Proof. If the t-norm T is not strictly monotone, then there exists x > 0, 0 ≤ y < z ≤ 1, s.t. T (x, y) = T (x, z). This means μ (T (x, y)) = μ (T (x, z)). Thus μ is not fuzzy strictly monotone. □

4 Vague t-norm with some properties

In this section, we firstly introduce some related concepts, and then generalize several properties of t-norm.

4.1 Vague monoid

The conjunctive operation ∧ always stands for the minimum operation between two real numbers.

Definition 15. [14] A mapping E_X : X × X → [0, 1] is called a fuzzy equality on X if it satisfies the following conditions:

E_X (x, y) =1 ⇔ x = y, ∀ x, y ∈ X.

E_X (x, y) = E_X (y, x) , ∀ x, y ∈ X.

E_X (x, y) ∧ E_X (y, z) ≤ E_X (x, z) , ∀ x, y, z ∈ X.

For two non-empty crisp sets X and Y, let E_X, E_Y be two fuzzy equalities on X and Y, respectively. Then a fuzzy relation f on X × Y (a subset of X × Y) is called a fuzzy function from X to Y w.r.t. E_X and E_Y, denoted by the usual notation f : X → Y, if and only if the characteristic function μ : X × Y → [0, 1] of f holds the following two conditions:

∀x ∈ X, ∃ y ∈ Y, μ (x, y) >0 .

∀x, y ∈ X, ∀ z, w ∈ Y, μ (x, z) ∧ μ (y, w) ∧ E_X (x, y) ≤ E_Y (z, w) .

A fuzzy function f is called a strong fuzzy function if and only if it additionally satisfies: ∀x ∈ X, ∃ y ∈ Y, μ (x, y) =1.

Definition 16. [6] A strong fuzzy function f : X × X → X w.r.t. a fuzzy equality E_X×X on X × X and a fuzzy equality E_X on X is said to be a vague binary operation on X w.r.t. E_X×X and E_X.

With vague binary operations, we can define vague semigroups and vague monoids.

Definition 17. [14] Let ∘ be a vague binary operation on X w.r.t. a fuzzy equality E_X×X on X × X and a fuzzy equality E_X on X. μ : X × X × X → [0, 1] is the characteristic function of ∘, then

X together with ∘, denoted by (X, ∘), is called a vague semigroup if and only if μ fulfills the condition: $\begin{matrix} μ (b, c, d) \land μ (a, d, m) \land μ (a, b, q) \\ \land μ (q, c, w) \leq E_{X} (m, w), \end{matrix}$ for any a, b, c, d, m, q, w ∈ X .

A vague semigroup (X, ∘) is a vague monoid if and only if there exists an (two-sided) identity element e ∈ X such that $μ (e, a, a) \land μ (a, e, a) = 1,$ for any a ∈ X.

A vague monoid (X, ∘) is a vague group if and only if there exists an (two-sided) inverse element a^-1 ∈ X such that $μ (a^{- 1}, a, e) \land μ (a, a^{- 1}, e) = 1,$ for any a ∈ X.

A vague semigroup (X, ∘) is said to be abelian (commutative) if and only if $μ (a, b, m) \land μ (b, a, w) \leq E_{X} (m, w),$ for any a, b, m, w ∈ X.

We know that in a group G, the left and right cancellation law hold, that is, xy = xz ⇒ y = z, yx = zx ⇒ y = z for any x, y, z ∈ G. In fact, every element in group G has an inverse element, just multiply both sides of the equation by x^-1 to the left or right at the same time. We also have a generalized cancellation law in the vague group.

Proposition 10. [14] Let ∘ be the vague binary operation on the fuzzy equality E_X×X and E_X on the set X, μ : X × X × X → [0, 1] be its membership function. (X, ∘) is a vague group. Then

μ (a, b, u) ∧ μ (a, c, u) ≤ E_X (b, c) , ∀ a, b, c, u ∈ X .

μ (b, a, u) ∧ μ (c, a, u) ≤ E_X (b, c) , ∀ a, b, c, u ∈ X .

Proof.

Since ∘ is a strong fuzzy function, for any a, b, c, u ∈ X, there exists v ∈ X such that μ (a^-1, u, v) =1. From the definition of associativity in vague group, we have that $\begin{matrix} μ (a, b, u) & = μ (a, b, u) \land μ (a^{- 1}, u, v) \\ \land μ (a^{- 1}, a, e) \land μ (e, b, b) \\ \leq E_{X} (v, b) \end{matrix}$ and $\begin{matrix} μ (a, c, u) & = μ (a, c, u) \land μ (a^{- 1}, u, v) \\ \land μ (a^{- 1}, a, e) \land μ (e, c, c) \\ \leq E_{X} (v, c) . \end{matrix}$ Thus, μ (a, b, u) ∧ μ (a, c, u) ≤ E_X (b, v) ∧ E_X (v, c) ≤ E_X (b, c).

According to symmetry, it can be proved in a similar way.

□

4.2 Vague t-norm with some extended properties

If we slightly change the conditions in the definition of fuzzy equality, we get a fuzzy equality respect to t-norm.

Definition 18. Let T be a t-norm and X a set. A mapping E_X : X × X → [0, 1] will be called T-fuzzy equality if and only if the following conditions are satisfied:

E_X (x, x) =1, ∀ x ∈ X

E_X (x, y) = E_X (y, x) , ∀ x, y ∈ X

T (E_X (x, y) , E_X (y, z)) ≤ E_X (x, z) , ∀ x, y, z ∈ X .

If E_X (x, y) =1 ⇒ x = y for any x, y ∈ X, then it is said that T-fuzzy equality E_X separates points.

In order to obtain the vague t-norm, we give the definition of the T-vague binary operation similar to the vague binary operation.

Definition 19. [6] Let T be a t-norm and E a T-fuzzy equality on X. The mapping $\tilde{\circ} : M \times M \times M \to [0, 1]$ is a T-vague binary operation, if for all x, y, z, x′, y′, z′ ∈ M

$T (\tilde{\circ} (x, y, z), E (x, x^{'}), E (y, y^{'}), E (z, z^{'})) \leq$

$\tilde{\circ} (x^{'}, y^{'}, z^{'}) .$

$T (\tilde{\circ} (x, y, z), \tilde{\circ} (x, y, z^{'})) \leq E (z, z^{'}) .$

$\forall x, y \in M, \exists z \in M, \tilde{\circ} (x, y, z) = 1 .$

Definition 20. [6] Let T be a t-norm, E a T-fuzzy equality on monoid M and $\tilde{\circ}$ is the T-vague binary operation on M w.r.t. E. Then $(M, \tilde{\circ})$ is a T-vague monoid if and only if it satisfies

For any x, y, z, d, m, q, w ∈ M, $\begin{matrix} T (\tilde{\circ} (y, z, d), \tilde{\circ} (x, d, m), \tilde{\circ} (x, y, q), \tilde{\circ} (q, z, w)) \\ \leq E (m, w) . \end{matrix}$

For each a ∈ M, there exists an element e ∈ M such that $T (\tilde{\circ} (e, a, a), \tilde{\circ} (a, e, a)) = 1 .$

Definition 21. [6] Let T be a t-norm, E a T-fuzzy equality on [0, 1] and $\tilde{T}$ a T-vague binary operation on [0, 1] w.r.t. E. Then a T-vague monoid $([0, 1], \tilde{T})$ is called a T-vague t-norm if and only if $E (T (x, y), z) = \tilde{T} (x, y, z) .$

From the above definition, we show that the T-vague t-norm is commutative.

Proposition 11. [6] Let T be a t-norm, E a T-fuzzy equality on [0, 1], $\tilde{T}$ a T-vague binary operation on [0, 1] w.r.t. E. $([0, 1], \tilde{T})$ is T-vague t-norm, then it must be commutative, i.e., $T (\tilde{T} (a, b, m), \tilde{T} (b, a, w)) \leq E (m, w) .$

Proof. From the definition of the T-vague equality and the T-vague t-norm, for all a, b, m, w ∈ [0, 1], we can conclude that

$\begin{matrix} T (\tilde{T} (a, b, m), \tilde{T} (b, a, w)) \\ = T (E (T (a, b), m), E (T (b, a), w)) \\ = T (E (T (a, b), m), E (T (a, b), w)) \\ \leq E (m, w) . \end{matrix}$ □ Since the T-vague t-norm is commutative, the first two positions of the operation $\tilde{T}$ have the same status. Therefore, the definitions of strict monotonicity and cancellation law only give the first one situation of the location.

Definition 22. Let T be a t-norm, E a T-fuzzy equality on [0, 1], $\tilde{T}$ a T-vague binary operation on [0, 1] w.r.t. E and $([0, 1], \tilde{T})$ a T-vague t-norm.

$\tilde{T}$ is said to be vague strictly monotone if $x < y, \tilde{T} (x, z, a) = \tilde{T} (y, z, b) \Rightarrow a < b,$ where x, y, z, a, b ∈ [0, 1].

$\tilde{T}$ satisfies the vague cancellation law if $\begin{matrix} \forall x, a, b, c \in [0, 1], \tilde{T} (a, x, c) \\ = \tilde{T} (b, x, c) \Rightarrow a = b . \end{matrix}$

Similarly, we can get the vague cancellation law from the vague strictly monotonicity as in the case of t-norm.

Proposition 12. Let T be a t-norm, E a T-fuzzy equality on [0, 1], $\tilde{T}$ a the T-vague binary operation on [0, 1] w.r.t. E and $([0, 1], \tilde{T})$ a T-vague t-norm. If $\tilde{T}$ is vague strictly monotone, then it must satisfy the vague cancellation law.

Proof. If $\tilde{T}$ is vague strictly monotone, then $x < y and \tilde{T} (x, z, a) = \tilde{T} (y, z, b) \Rightarrow a < b,$ where x, y, z, a, b ∈ [0, 1]. Assume that $\tilde{T} (a, x, c) = \tilde{T} (b, x, c) and a \neq b$ , if a > b, we have c < c, which conflicts with the definition of vague strictly monotone. And if a < b, we also have c < c. Therefore, $\tilde{T} (a, x, c) = \tilde{T} (b, x, c) \Rightarrow a = b, \forall x, a, b, c \in [0, 1] .$ □

5 t-norm on bounded lattice with algebraic properties

Finally, we generalize some conclusions obtained to a special partially ordered set, which called bounded lattice.

5.1 Bounded lattice

In this section, we recall some basic notions and results related to lattices and t-norms on a bounded lattice [20]. A bounded lattice is a lattice (L, ≤) which has the top element 1 and the bottom element 0, that is, there exist two elements 1, 0 ∈ L such that 0 ≤ x ≤ 1 for all x ∈ L.

Definition 23. Let (L, ≤ , 0, 1) be a bounded lattice and a, b ∈ L with a ≤ b. The subinterval [a, b] is defined by $[a, b] = {x \in L : a \leq x \leq b} .$ Other subintervals such as [a, b), (a, b] and (a, b) can be defined similarly. Obviously, ([a, b] , ≤) is a bounded lattice with the top element b and the bottom element a. Let (L, ≤ , 0, 1) be a bounded lattice, [a, b] be a subinterval of L, T₁ and T₂ be two binary operations on [a, b] ². If there holds T₁ (x, y) ≤ T₂ (x, y) for all (x, y) ∈ [a, b] ², then we say that T₁ is less than or equal to T₂ or, equivalently, that T₂ is greater than or equal to T₁, and written as T₁ ≤ T₂.

Definition 24. [20] Let (L, ≤ , 0, 1) be a bounded lattice and [a, b] be a subinterval of L. A binary operation T : [a, b] × [a, b] → [a, b] is said to be a t-norm on [a, b] if for any x, y, z ∈ [a, b], the following conditions are fulfilled:

If y ≤ z, then T (x, y) ≤ T (x, z).

T (x, T (y, z)) = T (T (x, y) , z).

T (x, y) = T (y, x).

T (x, b) = x.

For the sake of brevity and without loss of generality, we set a = 0 and b = 1 in the above definition. In order to be able to generalize the conclusions about fuzzy t-norm and vague t-norm to bounded lattices, we need to give the concept of fuzzy sets on lattices.

Definition 25. [7] Let L be a lattice and X be a non-empty set. We call the mapping A : X → L an L-subset of X, and call A (x)(x ∈ X) the degree of membership of x to A, which is interpreted as the degree to which the object x belongs to the L-subset A.

Definition 26. [2] Let T be a t-norm on bounded lattice G = (L, ≤ , 0, 1), μ an L-subset of [0, 1]. Then μ is a fuzzy t-subnorm of G if and only if

μ (x) ∧ μ (y) ≤ μ (T (x, y)) , ∀ x, y ∈ L .

μ (1) =1.

5.2 Generalization of the previous conclusions

We have extended the strict monotonicity and cancellation laws of t-norm to fuzzy and vague cases, and this work can be similarly generalized to bounded lattices.

Definition 27. For a fuzzy t-subnorm μ of a bounded lattice L with respect to a t-norm T, we consider the following properties:

μ is said to be fuzzy strictly monotone if $\begin{matrix} x \in L ∖ {0, 1}, y < z, then μ (T (x, y)) \\ > μ (T (x, z)) . \end{matrix}$

μ satisfies the fuzzy cancellation law if $μ (T (x, y)) = μ (T (x, z)) \Rightarrow x = 0 or y = z .$

μ satisfies the fuzzy conditional cancellation law if $\begin{matrix} μ (T (x, y)) = μ (T (x, z)) > μ (0) \\ \Rightarrow μ (y) = μ (z) . \end{matrix}$

μ is said to be fuzzy Archimedean if $\forall x, y \in L ∖ {0, 1}, \exists n \in ℕ, s . t . μ (x_{T}^{(n)}) < μ (y) .$

μ has the fuzzy limit property if $\begin{matrix} \forall x \in L ∖ {0, 1}, there exists \\ lim_{n \to \infty} μ (x_{T}^{(n)}) = μ (0) . \end{matrix}$

We can also get the fuzzy cancellation law from the fuzzy strict monotonicity, and from the fuzzy cancellation law deduce the fuzzy conditional cancellation law, as in the case of the t-norms.

Proposition 13. If μ is fuzzy strictly monotone, then μ satisfies the fuzzy cancellation law.

Proof. The proof is similar to Proposition 6. □

Proposition 14. If μ satisfies the fuzzy cancellation law, then μ satisfies the fuzzy conditional cancellation law.

Proof. The proof is similar to Proposition 7. □

Definition 28. Let T be a t-norm on a bounded lattice L and X a set. The mapping E_X : X × X → L is called T-fuzzy equality on X with respect to L if and only if the following conditions hold:

E_X (x, x) =1, ∀ x ∈ X .

E_X (x, y) = E_X (y, x) , ∀ x, y ∈ X .

T (E_X (x, y) , E_X (y, z)) ≤ E_X (x, z) , ∀ x, y, z ∈ X .

If E_X (x, y) =1 ⇒ x = y, then T-fuzzy equality E_X seprates points.

Definition 29. Let T be a t-norm on a bounded lattice L. E is a T-fuzzy equality on M. Then $\tilde{\circ} : M \times M \times M \to L$ is called a T-vague binary operation on M if and only if for all x, y, z, x′, y′, z′ ∈ M

$T (\tilde{\circ} (x, y, z), E (x, x^{'}), E (y, y^{'}), E (z, z^{'})) \leq$

$\tilde{\circ} (x^{'}, y^{'}, z^{'}) .$

$T (\tilde{\circ} (x, y, z), \tilde{\circ} (x, y, z^{'})) \leq E (z, z^{'}) .$

$\forall x, y \in M, \exists z \in M, \tilde{\circ} (x, y, z) = 1 .$

Definition 30. Let T be a t-norm on a bounded lattice L. E is a T-fuzzy equality on a monoid M and

\tilde{\circ}

is a T-vague binary operation on M. Then

(M, \tilde{\circ})

is called T-vague monoid if and only if

For all x, y, z, d, m, q, w ∈ M, $\begin{matrix} T (\tilde{\circ} (y, z, d), \tilde{\circ} (x, d, m), \tilde{\circ} (x, y, q), \tilde{\circ} (q, z, w)) \\ \leq E (m, w) . \end{matrix}$

There exists an element e ∈ M such that for each a ∈ M, $T (\tilde{\circ} (e, a, a), \tilde{\circ} (a, e, a)) = 1 .$

Definition 31. Let T be a t-norm on a bounded lattice L, E a T-fuzzy equality on a monoid L,

\tilde{T}

a T-vague binary operation on L. A T-vague monoid

([0, 1], \tilde{T})

is said to be T-vague t-norm if and only if

E (T (x, y), z) = \tilde{T} (x, y, z) .

Definition 32. Let T be a t-norm on a bounded lattice L. E is a T-fuzzy equality on a monoid L,

\tilde{T}

is a T-vague binary operation on L and

(L, \leq, 0, 1, \tilde{T})

is T-vague t-norm.

$\tilde{T}$ is vague strictly monotone if for any x, y, z, a, b ∈ M, x < y and $\tilde{T} (x, z, a) = \tilde{T} (y, z, b) \Rightarrow a < b .$

$\tilde{T}$ satisfies vague cancellation law if for any x, a, b, c ∈ M, $\tilde{T} (a, x, c) = \tilde{T} (b, x, c) \Rightarrow a = b .$

Proposition 15. Let T be a t-norm on a bounded lattice L, E a T-fuzzy equality on a monoid L,

\tilde{T}

a T-vague binary operation on L and

(L, \leq, 0, 1, \tilde{T})

a T-vague t-norm. If

\tilde{T}

is vague strictly monotone, then

\tilde{T}

satisfies vague cancellation law.

Proof. The proof is similar to Proposition 12. □

6 Conclusion

In this paper, we generalize the properties of t-norms, such as strict monotonicity, cancellation law, conditional law, Archimedean and limit property to fuzzy t-norms so that we can analyze and classify the fuzzy t-norms in a way. Just as strict monotonicity leads to the cancellation law and the fuzzy cancellation law deduce the fuzzy conditional cancellation law in the t-norms, we get the same results after extending the properties. For the same purpose, vague properties of strict monotonicity, cancellation law, conditional cancellation law are proposed for vague t-norms. And then, we generalize the related properties to the fuzzy t-norms on bounded lattice. In addition, we define the concepts of fuzzy monoids using aggregate functions, uninorms, and nullnorms. Then we analyze some of their features such as the structure of core and the constraints of fuzzy monoids. Some important examples are analyzed to facilitate a more intuitive understanding.

The discussion in this article can be directly extended to t-conorms. In the future, further exploration of the properties of fuzzified t-norms and t-conorms will be undertaken.

Declarations

No potential conflict of interest was reported by the authors.

Footnotes

Acknowledgments

The authors are extremely grateful to the editor and anonymous referees for their valuable comments and helpful suggestions which helped to improve the presentation of this paper. This research was supported by the National Natural Science Foundation of China (Grant no. 12101500) and the Chinese Universities Scientific Fund (Grant nos. 2452018054 and 2452022370).

References

Rosenfeld

, Fuzzy groups, Journal of Mathematical Analysis and Applications 35(3) (1971), 512–517.

Ajmal

and Thomas

K.V.

, The lattices of fuzzy subgroups and fuzzy normal subgroups[J], Information Sciences 76(1-2) (1994), 1–11.

Biswas

, Vague groups[J], International Journal of Computational Cognition, 2006.

Chen

S.M.

, Analyzing Fuzzy System Reliability Based on the Vague Set Theory[M], IEEE, 2007.

Boixader

and Recasens

, Fuzzy actions, Fuzzy Sets and Systems 339 (2018), 17–30.

Boixader

, Recasens

, Vague and fuzzy t-norms and t-conorms, Fuzzy Sets and Systems (2021).

Klement

E.P.

, Mesiar

, Pap

, Triangular norms, Springer (2003).

Zimmermann

H.J.

, Sebastian

H.J.

, Fuzzy Set and Its Applications 2 (1996), 1120.

Fodor

J.C.

, Yager

R.R.

and Rybalov

, Structure of Uninorms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5(04) (1997), 411–427.

10.

Khan

, Ahmad

and Biswas

, On vague groups, International Journal of Computational Cognition 5(1) (2007), 27–30.

11.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

12.

Akgul

, Some properties of fuzzy groups, Journal of Mathematical Analysis and Applications 133 (1988), 93–100.

13.

Demirci

, Fuzzy functions and their fundamental properties, Fuzzy Sets and Systems 106(2) (1999), 239–246.

14.

Demirci

, Vague Groups, Journal of Mathematical Analysis and Applications 230(1) (1999), 142–156.

15.

Ramakrishna

, Vague normal groups, International Journal of Computational Cognition 6(2) (2008), 10–13.

16.

Das

P.S.

, Fuzzy groups and level subgroups, Journal of Mathematical Analysis and Applications 84(1) (1981), 264–269.

17.

Rasuli

and Narghi

, t-norms over q-fuzzy subgroups of group, Jordan Journal of Mathematics and Statistics (JJMS) 12 (2019), 1–13.

18.

Yager

R.R.

, Rybalov

A.N.

, Uninorm aggregation operators, Fuzzy Sets and Systems (1996).

19.

Calvo

, De Baets

and Fodor

, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets and Systems 120 (2001), 385–394.

20.

, Zhang

, et al. Ordinal sum of two binary operations being a t-norm on bounded lattice, IEEE Transactions on Fuzzy Systems 30 (2020), 1762–1772.