A new attitude coupled with fuzzy thinking to fuzzy rings and fields

Abstract

In this paper, we shall embark on the study of the algebraic object known as a fuzzy ring which serves as one of the fundamental building blocks for the subject which is called fuzzy abstract algebra. The acceptable definition of fuzzy ring and field are presented with binary operations and on the basis of the specified parameter, called ambiguity rank, which fulfils the basic requirements. The properties of these fuzzy rings and their fundamental qualities are discussed. We should like to stress that these fuzzy algebraic systems and their axioms, must come from the experience of looking at many examples. Namely, they should be rich in meaningful results. Hence, the several illustrative examples are given. The future prospect of this paper is a new attitude to fuzzy basic mathematics, which will be referred to in the end.

Keywords

Fuzzy par fuzzy group fuzzy field ambiguity rank transmission average (TA)

1 Introduction

The fuzzy algebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoretical physics, computer science, control engineering, information science, coding theory, group theory, real analysis, measure theory etc. In 1971, Rosenfeld [24] first introduced the concept of fuzzy subgroups, which was the first fuzzification of any algebraic structure. Thereafter the notion of different fuzzy algebraic structures such as fuzzy ideals in rings and semi-rings etc. have seriously studied by many mathematicians.

In 1982 W. J. Liu [17] introduced the concept of fuzzy ring and fuzzy ideal. In 1985 Ren [23] studied the notions of fuzzy ideal and fuzzy quotient ring. Fuzzy rings and fuzzy ideal in the sense of Liu and Ren were actually a rational extension of Rosenfield’s fuzzy group by starting with an ordinary ring and then define a fuzzy sub-ring based on the ordinary operations of the given ring. Based on the notion of fuzzy space which play the role of universal set in ordinary algebra and using fuzzy binary operation K. A. Dib [11] obtained a new formulation for fuzzy rings and fuzzy ideals. Since then many mathematicians such as Aktaÿs and Cÿaćgman [5], Malik and Mordeson [18, 19], Yuan and Lee [28] have studied about them and more recently in S. Abdullah et al. [1– 4 , 14].

The fuzzy algebraic systems are usually sets on whose elements we can operate algebraically by this we mean that we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set and, in addition, we assume that these fuzzy algebraic operations are subject to certain rules, which are explicitly spelled out in what we call the axioms or postulates defining the system. In this abstract setting we then attempt to prove theorems about these very general structures, always hoping that when these results are applied to a particular, concrete realization of the abstract system there will flow out facts and insights into the example at hand which would have been obscured from us by the mass of inessential information available to us in the particular, special case.

We should like to stress that these fuzzy algebraic systems and the axioms which define them must have a certain naturally about them. They must come from the experience of looking at many examples. Namely, they should be rich in meaningful results. One does not just sit down, list a few axioms, and then proceed to study the system so described. This, admittedly, is done by some, but most mathematicians would dismiss these attempts as poor mathematics.

According to the above mentioned cases, a new attitude coupled with fuzzy thinking to fuzzy rings and fields will be presented with binary operations and on the basis of the specified parameter, called ambiguity rank. With the new proposed attitude, several theorems and examples will be presented to demonstrate that these fuzzy algebraic systems and the axioms which define them have a certain naturally about them. Namely, they will be rich in meaningful results.

In later papers, with new attitudes to the fuzzy ranking and the fuzzy distance, we will have a new encounter to fuzzy equations, fuzzy equations systems, fuzzy differential equations and so on.

The paper is organized as follows. In Section 2, we present the basic definitions and concepts related to the subject. In Section 3, a new definition of fuzzy ring and field with binary operations and on the basis of the specified parameter are provided and its basic properties are investigated in Section 4. Several illustrative examples due to confirm the above mention, about having a certain naturally for fuzzy algebraic systems are given in Section 5. Finally, conclusions and future research are drawn in Section 6.

2 Preliminaries and notations

In this section, some notations and background about the concept are brought.

Definition 2.1. [13] (Fuzzy number) A fuzzy set A in $ℝ$ is called a fuzzy number if it satisfies the following conditions

A is normal,

A _αis a closed interval for every α ∈ (0, 1],

the support of A is bounded.

According to definition of fuzzy number mentioned above, several revised definitions are presented [8, 15].

Pseudo-geometric fuzzy numbers

The pseudo-geometric fuzzy numbers are defined in two cases:

Definition 2.2. [16] (Pseudo-triangular fuzzy number) A fuzzy number $\tilde{A}$ is called a pseudo-triangular fuzzy number if its membership function function $μ_{\tilde{A}} (x)$ is given by $\begin{matrix} μ_{\tilde{A}} (x) = {\begin{matrix} l_{\tilde{A}} (x), \underline{a} \leq x \leq a, \\ r_{\tilde{A}} (x), a \leq x \leq \bar{a}, \\ 0, otherwise . \end{matrix} \end{matrix}$ (1) Where $l_{\tilde{A}} (x)$ and $r_{\tilde{A}} (x)$ are non-decreasing and non-increasing functions, respectively. The pseudo-triangular fuzzy number $\tilde{A}$ is denoted by the quintuplet $\tilde{A} = (\underline{a}, a, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)),$ and the triangular fuzzy number by the quintuplet $(\underline{a}, a, \bar{a}, -, -) .$

Definition 2.3. [16] (Pseudo-trapezoidal fuzzy number) A fuzzy number $\tilde{A}$ is called a pseudo-trapezoidal fuzzy number if its membership function $μ_{\tilde{A}} (x)$ is given by $\begin{matrix} μ_{\tilde{A}} (x) = {\begin{matrix} l_{\tilde{A}} (x), \underline{a} \leq x \leq a_{1}, \\ 1, a_{1} \leq x \leq a_{2}, \\ r_{\tilde{A}} (x), a_{2} \leq x \leq \bar{a}, \\ 0, otherwise . \end{matrix} \end{matrix}$ (2) Where $l_{\tilde{A}} (x)$ and $r_{\tilde{A}} (x)$ are nondecreasing and non increasing functions, respectively. The pseudo-trapezoidal fuzzy number $\tilde{A}$ is denoted by the Senary $\tilde{A} = (\underline{a}, a_{1}, a_{2}, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)),$ and the trapezoidal fuzzy number by the Senary $(\underline{a}, a_{1}, a_{2}, \bar{a}, -, -) .$

Definition 2.4. [12] (Equal fuzzy sets) Two fuzzy sets $\tilde{A}$ and $\tilde{B}$ are said to be equal(denoted $\tilde{A} = \tilde{B}$ ) if and only if $\begin{matrix} \forall x \in X, μ_{\tilde{A}} (x) = μ_{\tilde{B}} (x) . \end{matrix}$ (3)

Definition 2.5. (Euclidean distance) [10] Let X = (x ₁, x ₂, x ₃, . . . , x _n) and Y = (y ₁, y ₂, y ₃, . . . , y _n) are two points in Euclidean n-space, then the distance from X to Y, or from Y to Y is given by: $d (X, Y) = {(Σ_{i = 1}^{n} (x_{i} - y_{i})^{2})}^{\frac{1}{2}} .$

Definition 2.6. Let $\tilde{A}$ be a normal, convex and continuous (NCC) fuzzy set on the universal set U. Then, we define from [16],

$a . c (\tilde{A}) = \frac{1}{2} {\min (core (\tilde{A})) + \max (core (\tilde{A}))},$

$\underline{support} (\tilde{A}) = {x \in U ∣ x \leq a . c (\tilde{A})},$

$\bar{support} (\tilde{A}) = {x \in U ∣ x \geq a . c (\tilde{A})},$

${\underline{p}}_{\tilde{A}} = {x \in support (\tilde{A}) ∣ \inf (support (\tilde{A})) \leq x \leq \min (core (\tilde{A}))},$

${\bar{p}}_{\tilde{A}} = {x \in support (\tilde{A}) ∣ \max (core (\tilde{A})) \leq x \leq \sup (support (\tilde{A}))},$

$\begin{matrix} μ_{\tilde{A}} (x) = {\begin{matrix} {\underline{μ}}_{\tilde{A}} (x), x \in {\underline{p}}_{\tilde{A}}, \\ 1, x \in core (\tilde{A}), \\ {\bar{μ}}_{\tilde{A}} (x), x \in {\bar{p}}_{\tilde{A}}, \\ 0, otherwise . \end{matrix} \end{matrix}$

Now, we shall embark on the study of the algebraic object known as a fuzzy group which serves as one of the fundamental building blocks for the subject which is called fuzzy abstract algebra. Then, we shall have a look at some of the others such as fuzzy ring and field. Aside from the fact that it has become traditional to consider fuzzy groups at the outset, there are natural, cogent reasons for this choice. To begin with, fuzzy groups, being one-operational systems, lend themselves to the simplest formal description.

3 Description of the new definitions for fuzzy ring and field

Before offering a new definitions for the fuzzy group and field, we define a few definitions related to the topic with the examples of them.

Definition 3.1. (Type-1 fuzzy par) Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets. Then,

$\tilde{A} \sim_{1} \tilde{B}$ if and only if

1. $\forall x \in \underline{support} (\tilde{A}), \forall y \in \underline{support} (\tilde{B}); if d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B})) then {\underline{μ}}_{\tilde{A}} (x) = {\underline{μ}}_{\tilde{B}} (x),$

2. $\forall x \in \bar{support} (\tilde{A}), \forall y \in \bar{support} (\tilde{B}); if d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B})) then {\bar{μ}}_{\tilde{A}} (x) = {\bar{μ}}_{\tilde{B}} (x) .$

Example 3.1. Consider the seven fuzzy numbers

$A_{1} = (1, 2, 4, 1 - (x - 2)^{2})^{\frac{1}{2}}, (1 - \frac{1}{4} (x - 2)^{2})^{\frac{1}{2}}), A_{2} = (1, 2, 4, -, -), A_{3} = (6, 7, 9, 1 - (x - 7)^{2})^{\frac{1}{2}}, (1 - \frac{1}{4} (x - 7)^{2})^{\frac{1}{2}}), A_{4} = (3, 4, 5, -, -), A_{5} = (1, 4, 6, \frac{x - 1}{3}, (1 - \frac{1}{4} (x - 4)^{2})^{\frac{1}{2}}), A_{6} = (\frac{1}{4}, 1, \frac{3}{2}, \frac{4 x - 1}{3}, (1 - \frac{1}{4} (4 x - 4)^{2})^{\frac{1}{2}}), A_{7} = (\frac{3}{4}, 1, \frac{5}{4}, -, -) .$

The fuzzy numbers are visualized in the above figure. By using the definition (3.1), we have

A ₁ ∼ ₁ A ₃, :A ₁≁₁ A ₄, :A ₂ ∼ ₁ A ₄, :A ₂≁₁ A ₆, A ₅≁₁ A ₆, :A ₅≁₁ A ₇, :A ₄≁ ₁ A ₇, :A ₇≁ ₁ A ₁ .

Definition 3.2. (Type-2 fuzzy par) Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets. Then,

$\tilde{A} \sim_{2} \tilde{B}$ if and only if

1. $\forall x \in \underline{support} (\tilde{A}), \forall y \in \underline{support} (\tilde{B});$ $if \frac{x}{a . c (\tilde{A})} = \frac{y}{a . c (\tilde{B})} then {\underline{μ}}_{\tilde{A}} (x) = {\underline{μ}}_{\tilde{B}} (x),$ 2. $\forall x \in \bar{support} (\tilde{A}), \forall y \in \bar{support} (\tilde{B});$ $if \frac{x}{a . c (\tilde{A})} = \frac{y}{a . c (\tilde{B})} then {\bar{μ}}_{\tilde{A}} (x) = {\bar{μ}}_{\tilde{B}} (x) .$

Example 3.2. Consider the fuzzy numbers of example (3.1). Then, by using the definition (3.2), we have

A ₁≁ ₂ A ₃, :A ₁≁ ₂ A ₄, :A ₂≁ ₂ A ₄, :A ₂≁ ₂ A ₆, A ₅ ∼ ₂ A ₆, :A ₅≁ ₂ A ₇, :A ₄ ∼ ₂ A ₇, :A ₇≁ ₂ A ₁ .

Definition 3.3. (Fuzzy par) Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets. Then, $\tilde{A} \sim \tilde{B} if and only if \tilde{A} \sim_{1} \tilde{B} or \tilde{A} \sim_{2} \tilde{B} .$

Definition 3.4. (Fuzzy approximation) Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets.Then, $\tilde{A} ≅ \tilde{B} if and only if a . c (\tilde{A}) = a . c (\tilde{B}) .$

Example 3.3. Consider the fuzzy numbers of example (3.1). Then, by using the definition (3.4), we have

A ₁ ≅ A ₂, A ₁≇ ₃, A ₄ ≅ A ₅,

A ₂≇ ₅, A ₆ ≅ A ₇, A ₄≇ ₆ .

Definition 3.5. (Type-1 fuzzy group)A type-1 fuzzy group is an ordered pair $(\tilde{G}, *)$ where $\tilde{G}$ is a set of the NCC fuzzy sets and ∗ is a binary operation on $\tilde{G}$ satisfying the following properties

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, (\tilde{A} * \tilde{B}) * \tilde{C} ≅ \tilde{A} * (\tilde{B} * \tilde{C}),$

$\forall \tilde{A}, \tilde{B} \in \tilde{G}, if \tilde{A} \sim_{1} \tilde{B} then \tilde{A} * \tilde{B} \sim_{1} \tilde{A} or \tilde{A} * \tilde{B} \sim_{1} \tilde{B},$

$\forall \tilde{A} \in \tilde{G} \exists e_{\tilde{A}} \in \tilde{G}; e_{\tilde{A}} \sim_{1} \tilde{A} and \tilde{A} * e_{\tilde{A}} = e_{\tilde{A}} * \tilde{A} = \tilde{A},$

$\forall \tilde{A} \in \tilde{G} \exists \tilde{A^{- 1}} \in \tilde{G}; \tilde{A^{- 1}} \sim_{1} \tilde{A} and \tilde{A} * \tilde{A^{- 1}} = \tilde{A^{- 1}} * \tilde{A} = e_{\tilde{A}} .$

Definition 3.6. (Type-2 fuzzy group)A type-2 fuzzy group is an ordered pair $(\tilde{G}, *)$ where $\tilde{G}$ is a set of the NCC fuzzy sets and ∗ is a binary operation on $\tilde{G}$ satisfying the following properties

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, (\tilde{A} * \tilde{B}) * \tilde{C} ≅ \tilde{A} * (\tilde{B} * \tilde{C}),$

$\forall \tilde{A}, \tilde{B} \in \tilde{G}, if \tilde{A} \sim_{2} \tilde{B} then \tilde{A} * \tilde{B} \sim_{2} \tilde{A} or \tilde{A} * \tilde{B} \sim_{2} \tilde{B},$

$\forall \tilde{A} \in \tilde{G} \exists e_{\tilde{A}} \in \tilde{G}; e_{\tilde{A}} \sim_{2} \tilde{A} and \tilde{A} * e_{\tilde{A}} = e_{\tilde{A}} * \tilde{A} = \tilde{A},$

$\forall \tilde{A} \in \tilde{G} \exists \tilde{A^{- 1}} \in \tilde{G}; \tilde{A^{- 1}} \sim_{2} \tilde{A} and \tilde{A} * \tilde{A^{- 1}} = \tilde{A^{- 1}} * \tilde{A} = e_{\tilde{A}} .$

Definition 3.7. (Fuzzy group)A fuzzy group is an ordered pair $(\tilde{G}, *)$ where $\tilde{G}$ with ∗ binary operation is either type-1 fuzzy group or type-2 fuzzy group.

Definition 3.8. A fuzzy group $(\tilde{G}, *)$ is said to be Abelian if $\tilde{A} * \tilde{B} = \tilde{B} * \tilde{A}$ for all $\tilde{A} and \tilde{B} in \tilde{G}$ . A fuzzy group is said to be non-Abelian if it is not Abelian.

Definition 3.9. (Fuzzy ring)A fuzzy ring is a triplet $(\tilde{G}, *, °) w h e r e \tilde{G}$ is a set of the NCC fuzzy sets, * and ° are binary operations on $\tilde{G}$ satisfying the following properties

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, (\tilde{A} * \tilde{B}) * \tilde{C} ≅ \tilde{A} * (\tilde{B} * \tilde{C}),$

$\forall \tilde{A}, \tilde{B} \in \tilde{G}, if \tilde{A} \sim_{1} \tilde{B} then \tilde{A} * \tilde{B} \sim_{1} \tilde{A} or \tilde{A} * \tilde{B} \sim_{1} \tilde{B},$

$\forall \tilde{A} \in \tilde{G} \exists e_{\tilde{A}} \in \tilde{G}; e_{\tilde{A}} \sim_{1} \tilde{A} and \tilde{A} * e_{\tilde{A}} = e_{\tilde{A}} * \tilde{A} = \tilde{A},$

$\forall \tilde{A} \in \tilde{G} \exists \tilde{A^{- 1}} \in \tilde{G}; \tilde{A^{- 1}} \sim_{1} \tilde{A} and \tilde{A} * \tilde{A^{- 1}} = \tilde{A^{- 1}} * \tilde{A} = e_{\tilde{A}},$

$\forall \tilde{A}, \tilde{B} \in \tilde{G}, \tilde{A} * \tilde{B} = \tilde{B} * \tilde{A},$

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, (\tilde{A} \circ \tilde{B}) \circ \tilde{C} ≅ \tilde{A} \circ (\tilde{B} \circ \tilde{C}),$

$\forall \tilde{A}, \tilde{B} \in \tilde{G}, if \tilde{A} \sim_{2} \tilde{B} then \tilde{A} \circ \tilde{B} \sim_{2} \tilde{A} or \tilde{A} \circ \tilde{B} \sim_{2} \tilde{B},$

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, \tilde{A} \circ (\tilde{B} * \tilde{C}) ≅ (\tilde{A} \circ \tilde{B}) * (\tilde{A} \circ \tilde{C})$ and $(\tilde{B} * \tilde{C}) \circ \tilde{A} ≅ (\tilde{B} \circ \tilde{A}) * (\tilde{C} \circ \tilde{A}) .$

Definition 3.10. (Fuzzy commutative ring)

A fuzzy ring $(\tilde{G}, *, \circ)$ is said to be commutative if $\tilde{A} \circ \tilde{B} = \tilde{B} \circ \tilde{A}$ for all $\tilde{A}, \tilde{B} in \tilde{G} .$

Definition 3.11. (Fuzzy division ring) A fuzzy ring $(\tilde{G}, *, \circ)$ is said to be division if its nonzero(non the neutral elements related ∗ binary operation)elements form a type-2 fuzzy group under ∘ binary operation.

Finally, we make the definition of the ultra-important object known as a fuzzy field.

Definition 3.12. (Fuzzy field) A fuzzy field is a fuzzy commutative division ring.

It is clear that the each problem involved with at least a fuzzy number is fuzzy problem and, if that the all fuzzy numbers involved be crisp, it is crisp problem. In this regard, the parameter as ambiguity rank for both arbitrary fuzzy number will assigned. Then, using of them, we define a ambiguity rank for the fuzzy problems, in order to compare with the crisp problems.

Definition 3.13. (Fuzzy approximation with type-1 ambiguity rank) Let $\tilde{G}$ be a type-1 fuzzy group. Then, type-1 ambiguity rank for $\tilde{A}, \tilde{B}$ is defined with ${ar}_{1 AB} = (l_{11 \tilde{A}, \tilde{B}}, r_{11 \tilde{A}, \tilde{B}}, l_{12 \tilde{A}, \tilde{B}}, r_{12 \tilde{A}, \tilde{B}})_{1}$ as follows:

$l_{11 \tilde{A}, \tilde{B}} = d (\sup {d ({\underline{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]}, \sup {d ({\underline{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]}),$

$r_{11 \tilde{A}, \tilde{B}} = d (\sup {d ({\bar{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]}, \sup {d ({\bar{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]}),$

$l_{12 \tilde{A}, \tilde{B}} = \sup {d ({\underline{μ}}_{\tilde{A}} (x), {\underline{μ}}_{\tilde{B}} (y)) ∣ \forall x \in \underline{support} (\tilde{A}), \forall y \in \underline{support} (\tilde{B}); d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B}))},$

$r_{12 \tilde{A}, \tilde{B}} = \sup {d ({\bar{μ}}_{\tilde{A}} (x), {\bar{μ}}_{\tilde{B}} (y)) ∣ \forall x \in \bar{support} (\tilde{A}), \forall y \in \bar{support} (\tilde{B}); d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B}))} .$

We denote the fuzzy approximation with type-1 ambiguity rank by: $\tilde{A} ≅_{{ar}_{1 AB}} \tilde{B} . ∥$ (4) Example 3.4. Consider the fuzzy numbers of example (3.1). Then, by using the definition (3.13), we have

A ₁ ≅ A ₂, :ar _1A₁A₂ = (0, 0, 0.4142135622, 0.4142135620) ,

A ₁ ∼ ₁ A ₃, :ar _1A₁A₃ = (0, 0, 0, 0) ,

A ₄ ≅ A ₅, :ar _1A₄A₅ = (2, 1, 2, 1.236067975) .

Definition 3.14. (Fuzzy approximation with type-2 ambiguity rank) Let $\tilde{G}$ be a type-2 fuzzy group. Then, type-2 ambiguity rank for $\tilde{A}, \tilde{B}$ is defined with ${ar}_{2 AB} = (l_{21 \tilde{A}, \tilde{B}}, r_{21 \tilde{A}, \tilde{B}}, l_{22 \tilde{A}, \tilde{B}}, r_{22 \tilde{A}, \tilde{B}})_{2}$ as follows:

$l_{21 \tilde{A}, \tilde{B}} = d (\inf {\frac{{\underline{μ}}_{\tilde{A}}^{- 1} (α)}{a . c (\tilde{A})} ∣ α \in (0, 1]}, \inf {\frac{{\underline{μ}}_{\tilde{B}}^{- 1} (α)}{a . c (\tilde{B})} ∣$ α ∈ (0, 1]}) ,

$r_{21 \tilde{A}, \tilde{B}} = d (\inf {\frac{{\bar{μ}}_{\tilde{A}}^{- 1} (α)}{a . c (\tilde{A})} ∣ α \in (0, 1]}, \inf {\frac{{\bar{μ}}_{\tilde{B}}^{- 1} (α)}{a . c (\tilde{B})} ∣$ α ∈ (0, 1]}) ,

$l_{22 \tilde{A}, \tilde{B}} = \sup {d ({\underline{μ}}_{\tilde{A}} (x), {\underline{μ}}_{\tilde{B}} (y)) ∣ \forall x \in \underline{support} (\tilde{A}),$ $\forall y \in \underline{support} (\tilde{B}); \frac{x}{a . c (\tilde{A})} = \frac{y}{a . c (\tilde{B})}},$

$r_{22 \tilde{A}, \tilde{B}} = \sup {d ({\bar{μ}}_{\tilde{A}} (x), {\bar{μ}}_{\tilde{B}} (y)) ∣ \forall x \in \bar{support} (\tilde{A}),$ $\forall y \in \bar{support} (\tilde{B}); \frac{x}{a . c (\tilde{A})} = \frac{y}{a . c (\tilde{B})}} .$

We denote the fuzzy approximation with type-2 ambiguity rank by: $\tilde{A} ≅_{{ar}_{2 AB}} \tilde{B} .$ (5)

Example 3.5. Consider the fuzzy numbers of example (3.1). Then, by using the definition (3.14), we have

A ₁ ≅ A ₂, :ar _2A₁A₂ = (0, 0, 0.4142135622, 0.4142135620) ,

A ₄ ≅ A ₅, :ar _2A₄A₄ = (0.5, 0.25, 0.6666666667, 0.75) ,

A ₄ ∼ ₂ A ₇, :ar _2A₄A₅ = (0, 0, 0, 0) .

Definition 3.15. (Ambiguity rank of type-1 fuzzy group) Let $\tilde{G}$ be a type-1 fuzzy group. Then, ambiguity rank for $\tilde{G}$ is defined with $\tilde{G} {ar}_{1} = (l_{11 \tilde{G}}, r_{11 \tilde{G}}, l_{12 \tilde{G}}, r_{12 \tilde{G}})_{1}$ as follows:

$l_{11 \tilde{G}} = \sup {l_{11 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}},$

$r_{11 \tilde{G}} = \sup {r_{11 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}},$

$l_{12 \tilde{G}} = \sup {l_{12 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}},$

$r_{12 \tilde{G}} = \sup {r_{12 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}} .$

Definition 3.16. (Ambiguity rank of type-2 fuzzy group) Let $\tilde{G}$ be a type-2 fuzzy group. Then, ambiguity rank for $\tilde{G}$ is defined with $\tilde{G} {ar}_{2} = (l_{21 \tilde{G}}, r_{21 \tilde{G}}, l_{22 \tilde{G}}, r_{22 \tilde{G}})_{2}$ as follows:

$l_{21 \tilde{G}} = \sup {l_{21 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}},$

$r_{21 \tilde{G}} = \sup {r_{21 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}},$

$l_{22 \tilde{G}} = \sup {l_{22 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}},$

$r_{22 \tilde{G}} = \sup {r_{22 \tilde{A}, \tilde{B}} ∣ \tilde{A}, \tilde{B} \in \tilde{G}} .$

Definition 3.17. (Ambiguity rank of fuzzy group) Let $\tilde{G}$ be a fuzzy group. Then, ambiguity rank for $\tilde{G}$ is defined either with the $\tilde{G} {ar}_{1}$ or $\tilde{G} {ar}_{2} .$

Remark 1. Another natural characteristic of a fuzzy group $\tilde{G}$ is the number of elements it contains. We call this the order of $\tilde{G}$ and denote it by o( $\tilde{G}$ ). This number is, of course, most interesting when it is finite. In that case we say that $\tilde{G}$ is a finite fuzzy group. If $\tilde{G}$ be a finite fuzzy group. Then, ambiguity rank of $\tilde{G}$ will be defined with the change sup to max in the definitions $\tilde{G} {ar}_{1}$ and $\tilde{G} {ar}_{2} .$

Definition 3.18. (Ambiguity rank of fuzzy field) Let $(\tilde{G}, *, \circ)$ be a fuzzy field. Then, ambiguity rank for $\tilde{G}$ is defined $(\tilde{G} {ar}_{1}, \tilde{G} {ar}_{2}) .$

We have now been exposed to the theories of fuzzy groups, rings and fields for several pages and as yet not a single, solitary fact has been proved about their. It is high time to remedy this situation. Although the first few results we demonstrate are, admittedly, not very exciting (in fact, they are rather dull) they will be extremely useful.

4 Theorems and properties

In this section, we will show that, the algebraic systems are special cases of the fuzzy algebraic systems. Before presenting theorems, we give some definitions and Lemma relevant.

Lemma 4.1 Let $\tilde{G}$ be a type-1 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{1} .$ Then,

$l_{11 \tilde{G}}, r_{11 \tilde{G}}, l_{12 \tilde{G}}, r_{12 \tilde{G}} \geq 0 .$

Proof. It is obvious.

Lemma 4.2. Let $\tilde{G}$ be a type-2 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{2} .$ Then,

$l_{21 \tilde{G}}, r_{21 \tilde{G}}, l_{22 \tilde{G}}, r_{22 \tilde{G}} \geq 0 .$

Proof. It is obvious.

Lemma 4.3. Let $\tilde{G}$ be a type-1 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{1} .$ Then,

$(l_{12 \tilde{G}}, r_{12 \tilde{G}}) = (0, 0) \Rightarrow (l_{11 \tilde{G}}, r_{11 \tilde{G}}) = (0, 0) .$

Proof. Suppose for the sake of contradiction that, $(l_{11 \tilde{G}},$ $r_{11 \tilde{G}}) \neq (0, 0) .$

Thus, the following three modes can be investigated:

$l_{11 \tilde{G}} \neq 0 and r_{11 \tilde{G}} = 0,$

$l_{11 \tilde{G}} = 0 and r_{11 \tilde{G}} \neq 0,$

$l_{11 \tilde{G}} \neq 0 and r_{11 \tilde{G}} \neq 0 .$

case 1. $if l_{11 \tilde{G}} \neq 0 then \exists \tilde{A}, \tilde{B} \in \tilde{G};$

$\sup {d ({\underline{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]} \neq$ $\sup {d ({\underline{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]} .$ Without loss of generality, we may assume that

$\sup {d ({\underline{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]} > \sup {d ({\underline{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]} .$

we define

$x = \sup {d ({\underline{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]},$

$y = \sup {d ({\underline{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]} .$ Thus, we conclude from definition(3.15),

$\exists S \subseteq \underline{support} (\tilde{A}); ∣ S ∣ = x - y .$

So, $\forall t \in S \exists t_{y}; d (t, a . c (\tilde{A})) = d (t_{y}, a . c (\tilde{B})), {\underline{μ}}_{\tilde{A}} (t)$ $> 0, {\underline{μ}}_{\tilde{B}} (t_{y}) = 0 .$

Therefore, $0 < {\underline{μ}}_{\tilde{A}} (t) \leq l_{12 \tilde{G}},$ namely $l_{12 \tilde{G}} \neq 0 .$

This is a contradiction. Similar to above, the case 2 and case 3 will lead to

$r_{12 \tilde{G}} \neq 0 and (l_{12 \tilde{G}}, r_{12 \tilde{G}}) \neq (0, 0),$ respectively. These are inconsistency.

Lemma 4.4. Let $\tilde{G}$ be a type-2 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{2} .$ Then,

$(l_{22 \tilde{G}}, r_{22 \tilde{G}}) = (0, 0) \Rightarrow (l_{21 \tilde{G}}, r_{21 \tilde{G}}) = (0, 0) .$

Proof. Similar to the proof of lemma (4.3).

Lemma 4.5. Let $(\tilde{G}, *)$ be a type-1 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{1} = (0, 0, 0, 0)_{1} .$ Then,

i) $\forall \tilde{A}, \tilde{B} \in \tilde{G}; \tilde{A} \sim_{1} \tilde{B},$

ii) $\forall \tilde{A}, \tilde{B} \in \tilde{G}; \tilde{A} * \tilde{B} \sim_{1} \tilde{A} or \tilde{A} * \tilde{B} \sim_{1} \tilde{B} .$

Proof.

Proof (i) Suppose $\tilde{A}, \tilde{B} \in \tilde{G} .$

Then, we have from $(l_{11 \tilde{G}}, r_{11 \tilde{G}}) = (0, 0) :$ $\begin{matrix} \sup {d ({\underline{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]} = \\ \sup {d ({\underline{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]}, \\ \sup {d ({\bar{μ}}_{\tilde{A}}^{- 1} (α), a . c (\tilde{A})) ∣ α \in (0, 1]} = \\ \sup {d ({\bar{μ}}_{\tilde{B}}^{- 1} (α), a . c (\tilde{B})) ∣ α \in (0, 1]} . \end{matrix}$

So, $\begin{matrix} ∣ {x \in \underline{support} (\tilde{A}) ∣ {\underline{μ}}_{\tilde{A}} (x) > 0} ∣ = \\ ∣ {x \in \underline{support} (\tilde{B}) ∣ {\underline{μ}}_{\tilde{B}} (x) > 0} ∣, \\ ∣ {x \in \bar{support} (\tilde{A}) ∣ {\bar{μ}}_{\tilde{A}} (x) > 0} ∣ = \\ ∣ {x \in \bar{support} (\tilde{B}) ∣ {\bar{μ}}_{\tilde{B}} (x) > 0} ∣ . \end{matrix}$ (Note: ∣ · ∣ = sup (·) - inf (·)).

Therefore, have Prerequisite for $\tilde{A} \sim_{1} \tilde{B} .$

Also, we have from $(l_{12 \tilde{G}}, r_{12 \tilde{G}}) = (0, 0) :$

$\forall x \in \underline{support} (\tilde{A}), \forall y \in \underline{support} (\tilde{B});$

$d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B})) \Rightarrow d ({\underline{μ}}_{\tilde{A}} (x), {\underline{μ}}_{\tilde{B}} (y)) = 0 .$

Namely, $\begin{matrix} {\underline{μ}}_{\tilde{A}} (x) = {\underline{μ}}_{\tilde{B}} (y) . \end{matrix}$ (6)

And,

$\forall x \in \bar{support} (\tilde{A}), \forall y \in \bar{support} (\tilde{B});$

$d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B})) \Rightarrow d ({\bar{μ}}_{\tilde{A}} (x), {\bar{μ}}_{\tilde{B}} (y)) = 0 .$

Namely, $\begin{matrix} {\bar{μ}}_{\tilde{A}} (x) = {\bar{μ}}_{\tilde{B}} (y) . \end{matrix}$ (7)

Finally, of (6), (7) and the definition (3.1) have $\tilde{A} \sim_{1} \tilde{B} .$

Proof (ii) According to the second condition of definition (3.5) and case (i), it is obvious.

Lemma 4.6. Let $(\tilde{G}, *)$ be a type-2 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{2} = (0, 0, 0, 0)_{2} .$ Then,

i) $\forall \tilde{A}, \tilde{B} \in \tilde{G}; \tilde{A} \sim_{2} \tilde{B},$

ii) $\forall \tilde{A}, \tilde{B} \in \tilde{G}; \tilde{A} * \tilde{B} \sim_{2} \tilde{A} or \tilde{A} * \tilde{B} \sim_{2} \tilde{B} .$

Proof. Similar to the proof of lemma (4.5).

Lemma 4.7. Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets. Then, $\tilde{A} ≅ \tilde{B}$ and $\tilde{A} \sim_{1} \tilde{B}$ if and only if $\tilde{A} = \tilde{B} .$

Proof. According to the definitions (3.1) and (2.4), it is obvious.

Lemma 4.8. Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets. Then,

$\tilde{A} ≅ \tilde{B} and \tilde{A} \sim_{2} \tilde{B} if and only if \tilde{A} = \tilde{B} .$

Proof. According to the definitions (3.2) and (2.4), it is obvious.

Lemma 4.9. Let $\tilde{A}, \tilde{B}$ are two NCC fuzzy sets. Then,

\tilde{A} \sim_{1} \tilde{B} and \tilde{A} \sim_{2} \tilde{B} if and only if \tilde{A} = \tilde{B} .

Proof. Suppose $\tilde{A} = \tilde{B}$ then, $\tilde{A} \sim_{1} \tilde{B}$ and $\tilde{A} \sim_{2} \tilde{B}$ are obvious.

Conversely, suppose for the sake of contradiction that, $\tilde{A} \neq \tilde{B} .$

Thus, from definition (2.4) have:

$\exists x; μ_{\tilde{A}} (x) \neq μ_{\tilde{B}} (x) .$

If $d (x, a . c (\tilde{A})) = d (x, a . c (\tilde{B}))$ and $\frac{x}{a . c (\tilde{A})} = \frac{x}{a . c (\tilde{B})}$ then, $a . c (\tilde{A})) = a . c (\tilde{B})) .$

Thus, we have from definition (3.4):

$\tilde{A} ≅ \tilde{B},$ then, according to the primary assumptions and the lemma (4.7), (4.8):

$\tilde{A} = \tilde{B} .$ This is a contradiction.

If $d (x, a . c (\tilde{A})) \neq d (x, a . c (\tilde{B}))$ and $\frac{x}{a . c (\tilde{A})} \neq \frac{x}{a . c (\tilde{B})}$ then,

$a . c (\tilde{A})) \neq a . c (\tilde{B})), x \neq a . c (\tilde{A})) and x \neq a . c (\tilde{B})) .$

Without loss of generality, we may assume that, $x \in \underline{support} (\tilde{A}) .$ Thus,

$\exists y_{1}, y_{2} \in \underline{support} (\tilde{B})$ such that;

$d (x, a . c (\tilde{A})) = d (y_{1}, a . c (\tilde{B})) and \frac{x}{a . c (\tilde{A})} = \frac{y_{2}}{a . c (\tilde{B})} .$

Then, according to the primary assumptions,

${\underline{μ}}_{\tilde{A}} (x) = {\underline{μ}}_{\tilde{B}} (y_{1}) and {\underline{μ}}_{\tilde{A}} (x) = {\underline{μ}}_{\tilde{B}} (y_{2}) .$

Thus, ${\underline{μ}}_{\tilde{B}} (y_{1}) = {\underline{μ}}_{\tilde{B}} (y_{2}) .$

According to the definitions (2.2) or (2.3) have:::y ₁ = y ₂ .

Suppose y = y ₁ = y ₂ that, we have:

$d (x, a . c (\tilde{A})) = d (y, a . c (\tilde{B})) and \frac{x}{a . c (\tilde{A})} = \frac{y}{a . c (\tilde{B})} .$

Thus, $x = a . c (\tilde{A}), y = a . c (\tilde{B})$ . This is a contradiction.

Lemma 4.10. Let $\tilde{G}, *$ be a type-1 fuzzy group with the ambiguity rank $\tilde{G} a r_{1} = (0, 0, 0, 0)_{1}$ . Then, $\forall \tilde{A}, \tilde{B} \in \tilde{G}; e_{\tilde{A}} = e_{\tilde{B}}$

Proof. According to the lemma (4.5) and definition (3.5), we have $\begin{matrix} e_{\tilde{A}} \sim_{1} e_{\tilde{B}}, \end{matrix}$ (8) and $\begin{matrix} {\begin{matrix} \tilde{A} * e_{\tilde{A}} = e_{\tilde{A}} * \tilde{A} = \tilde{A}, \\ \tilde{B} * e_{\tilde{B}} = e_{\tilde{B}} * \tilde{B} = \tilde{B} . \end{matrix} \end{matrix}$ (9)

Then, $\begin{matrix} {\begin{matrix} a . c (\tilde{A}) * a . c (e_{\tilde{A}}) = a . c (e_{\tilde{A}}) * a . c (\tilde{A}) = a . c (\tilde{A}), \\ a . c (\tilde{B}) * a . c (e_{\tilde{B}}) = a . c (e_{\tilde{B}}) * a . c (\tilde{B}) = a . c (\tilde{B}) . \end{matrix} \end{matrix}$ (10)

The system (10) is a system at the crisp space, thus $\begin{matrix} a . c (e_{\tilde{A}}) = a . c (e_{\tilde{B}}) . \end{matrix}$ (11)

Therefore, from the case (11) and definition (3.4), we have $e_{\tilde{A}} ≅ e_{\tilde{B}} .$

Finally, according to the above case and lemma (4.7), $e_{\tilde{A}} = e_{\tilde{B}} .$

Lemma 4.11. Let $(\tilde{G}, *)$ be a type-2 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{2} = (0, 0, 0, 0)_{2} .$ Then, $\forall \tilde{A}, \tilde{B} \in \tilde{G}; e_{\tilde{A}} = e_{\tilde{B}} .$

Proof. Similar to the proof of lemma (4.10).

Now, according to the above lemma can be claimed that, the algebraic systems are special cases of the fuzzy algebraic systems. It is expressed in the following theorems.

Theorem 4.1. Algebraic group is a special case of the type-1 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{1} = (0, 0, 0, 0)_{1} .$

Proof. Let $(\tilde{G}, *)$ be a type-1 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{1} = (0, 0, 0, 0)_{1} .$

Then, we must show that,

i) $\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, (\tilde{A} * \tilde{B}) * \tilde{C} = \tilde{A} * (\tilde{B} * \tilde{C}),$ (12) and

ii) $\exists e \in \tilde{G}; \forall \tilde{A} \in \tilde{G}, \tilde{A} * e = e * \tilde{A} = \tilde{A} .$ (13)

According to the first and second conditions fuzzy group, lemma (4.5) and lemma (4.7), we have the case (12).

Also, we have the case (13) by lemma (4.10) with definition $e = e_{\tilde{A}}$ for each the selecting arbitrary $\tilde{A} \in \tilde{G}$ . It should be mentioned, the inverse condition $\tilde{G}$ is obvious by (13).

Theorem 4.2. Algebraic group is a special case of the type-2 fuzzy group with the ambiguity rank $\tilde{G} {ar}_{2} = (0, 0, 0, 0)_{2} .$

Proof. Similar to the proof of theorem (4.1).

Theorem 4.3. Algebraic field is a special case of the fuzzy field with the ambiguity rank $(\tilde{G} {ar}_{1}, \tilde{G} {ar}_{2}) = ((0, 0, 0, 0)_{1}, (0, 0, 0, 0)_{2}) .$

Proof. A fuzzy field is a triplet $(\tilde{G}, *, \circ)$ where $\tilde{G}$ is a set of the NCC fuzzy sets, $(\tilde{G}, *)$ and $(\tilde{G} - {\tilde{A} ∣ a . c (\tilde{A}) \neq 0}, \circ)$ are the Abelian type-1 fuzzy group and the Abelian type-2 fuzzy group respectively, and with the condition

According to the theories (4.5) and (4.6), is sufficient, we show that,

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in \tilde{G}, \tilde{A} \circ (\tilde{B} * \tilde{C}) = (\tilde{A} \circ \tilde{B}) * (\tilde{A} \circ \tilde{C})$ and $(\tilde{B} * \tilde{C}) \circ \tilde{A} = (\tilde{B} \circ \tilde{A}) * (\tilde{C} \circ \tilde{A}) .$ According to the lemma (4.7) and (4.8), we have

$\tilde{A} \circ (\tilde{B} * \tilde{C}) \sim_{1} (\tilde{A} \circ \tilde{B}) * (\tilde{A} \circ \tilde{C})$ and $(\tilde{B} * \tilde{C}) \circ \tilde{A} \sim_{1} (\tilde{B} \circ \tilde{A}) * (\tilde{C} \circ \tilde{A}),$

$\tilde{A} \circ (\tilde{B} * \tilde{C}) \sim_{2} (\tilde{A} \circ \tilde{B}) * (\tilde{A} \circ \tilde{C})$ and $(\tilde{B} * \tilde{C}) \circ \tilde{A} \sim_{2} (\tilde{B} \circ \tilde{A}) * (\tilde{C} \circ \tilde{A}) .$

Thus, according to the lemma (4.7) and (4.8), have

Corollary. Algebraic ring is a special case of the fuzzy ring with the ambiguity rank $(\tilde{G} {ar}_{1}, \tilde{G} {ar}_{2}) = ((0, 0, 0, 0)_{1}, (0, 0, 0, 0)_{2}) .$

5 Some examples of fuzzy group and field

In order to introduce examples, we first give a few definitions and theorems related to the subject.

As regards fuzzy arithmetic operations using of the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of the α- cuts), we have some problem in subtraction operator, division operator and obtaining the membership functions of operators [6 , 27].

Although with the revised definitions in [26] on subtraction and division, usage of an interval arithmetic for fuzzy operators have been permitted, because it always exists, but its not efficient, it means that result’s support is major agent (dependence effect) and also complex calculations of interval arithmetic in determining the membership function of operators based on the extension principle, are not yet resolved.

Therefore, we eliminate such deficiency with the fuzzy arithmetic operations based on TA for addition, subtraction, multiplication and division as follows:

Definition 5.1. (The fuzzy arithmetic operations based on TA for pseudo-triangular fuzzy numbers).

Consider two pseudo-triangular fuzzy number

$\tilde{A} = (\underline{a}, a, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)),$

$\tilde{B} = (\underline{b}, b, \bar{b}, l_{\tilde{B}} (x), r_{\tilde{B}} (x)) .$

With the following α-cut forms:

$\begin{matrix} \tilde{A} = ⋃_{α} A_{α}, A_{α} = [{\underline{A}}_{α}, {\bar{A}}_{α}], 0 \leq α \leq 1, \\ \tilde{B} = ⋃_{α} B_{α}, B_{α} = [{\underline{B}}_{α}, {\bar{B}}_{α}], 0 \leq α \leq 1 . \end{matrix}$

In the following, we define fuzzy arithmetic operations based on TA for addition, subtraction, multiplication and division:

$\begin{matrix} \tilde{A + B} = ⋃_{α} (\tilde{A + B})_{α}, \\ (\tilde{A + B})_{α} = [\frac{a + b}{2} + (\frac{{\underline{A}}_{α} + {\underline{B}}_{α}}{2}), \\ \frac{a + b}{2} + (\frac{{\bar{A}}_{α} + {\bar{B}}_{α}}{2})] . \end{matrix}$ (14)

$\tilde{- A} = ⋃_{α} (\tilde{- A})_{α}, (\tilde{- A})_{α} = [- 2 a + {\underline{A}}_{α}, - 2 a + {\bar{A}}_{α}] .$ (15)

$\begin{matrix} \tilde{A - B} = ⋃_{α} (\tilde{A - B})_{α}, \\ (\tilde{A - B})_{α} = [\frac{a - 3 b}{2} + (\frac{{\underline{A}}_{α} + {\underline{B}}_{α}}{2}), \\ \frac{a - 3 b}{2} + (\frac{{\bar{A}}_{α} + {\bar{B}}_{α}}{2})] . \end{matrix}$ (16)

$\begin{matrix} \tilde{A . B} = ⋃_{α} (\tilde{A . B})_{α}, \\ (\tilde{A . B})_{α} = \\ {\begin{matrix} [(\frac{b}{2}) {\underline{A}}_{α} + (\frac{a}{2}) {\underline{B}}_{α}, (\frac{b}{2}) {\bar{A}}_{α} + (\frac{a}{2}) {\bar{B}}_{α}], a \geq 0, b \geq 0, \\ [(\frac{b}{2}) {\bar{A}}_{α} + (\frac{a}{2}) {\underline{B}}_{α}, (\frac{b}{2}) {\underline{A}}_{α} + (\frac{a}{2}) {\bar{B}}_{α}], a \geq 0, b \leq 0, \\ [(\frac{b}{2}) {\bar{A}}_{α} + (\frac{a}{2}) {\bar{B}}_{α}, (\frac{b}{2}) {\underline{A}}_{α} + (\frac{a}{2}) {\underline{B}}_{α}], a \leq 0, b \leq 0, \\ [(\frac{b}{2}) {\underline{A}}_{α} + (\frac{a}{2}) {\bar{B}}_{α}, (\frac{b}{2}) {\bar{A}}_{α} + (\frac{a}{2}) {\underline{B}}_{α}], a \leq 0, b \geq 0 . \end{matrix} \end{matrix}$ (17) $\begin{matrix} \tilde{A^{- 1}} = ⋃_{α} {(\tilde{A^{- 1}})}_{α}, {(\tilde{A^{- 1}})}_{α} = [\frac{1}{a^{2}} {\underline{A}}_{α}, \frac{1}{a^{2}} {\bar{A}}_{α}] . \end{matrix}$ (18) $(\tilde{A} B^{- 1})_{α} = \{\begin{array}{l} [(\frac{1}{2 b}) {\underline{A}}_{α} + (\frac{a}{2 b^{2}}) {\underline{B}}_{α}, (\frac{1}{2 b}) {\bar{A}}_{α} + (\frac{a}{2 b^{2}}) {\bar{B}}_{α}], \\ a \geq 0, b > 0, \\ [(\frac{1}{2 b}) {\bar{A}}_{α} + (\frac{a}{2 b^{2}}) {\underline{B}}_{α}, (\frac{1}{2 b}) {\underline{A}}_{α} + (\frac{a}{2 b^{2}}) {\bar{B}}_{α}], \\ a \geq 0, b < 0, \\ [(\frac{1}{2 b}) {\bar{A}}_{α} + (\frac{a}{2 b^{2}}) {\bar{B}}_{α}, (\frac{1}{2 b}) {\underline{A}}_{α} + (\frac{a}{2 b^{2}}) {\underline{B}}_{α}], \\ a \leq 0, b < 0, \\ [(\frac{1}{2 b}) {\underline{A}}_{α} + (\frac{a}{2 b^{2}}) {\bar{B}}_{α}, (\frac{1}{2 b}) {\bar{A}}_{α} + (\frac{a}{2 b^{2}}) {\underline{B}}_{α}], \\ a \leq 0, b > 0. \end{array}$ (19)

Remark 2. Fuzzy arithmetic operation division on pseudo-triangular fuzzy number $\tilde{0} = (\underline{0}, 0, \bar{0}, l_{\tilde{0}} (x),$ $r_{\tilde{0}} (x))$ is not able to define.

Definition 5.2. (The fuzzy arithmetic operations based on TA for pseudo-trapezoidal fuzzy numbers).

Consider two pseudo-trapezoidal fuzzy number

$\tilde{A} = (\underline{a}, a_{1}, a_{2}, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)),$

$\tilde{B} = (\underline{b}, b_{1}, b_{2}, \bar{b}, l_{\tilde{B}} (x), r_{\tilde{B}} (x)) .$

with the following α-cut forms:

$\begin{array}{l} \tilde{A} = ⋃_{α} A_{α}, A_{α} = [{\underline{A}}_{α}, {\bar{A}}_{α}], \\ 0 \leq α \leq 1, A_{1} = [a_{1}, a_{2}], \\ \tilde{B} = ⋃_{α} B_{α}, B_{α} = [{\underline{B}}_{α}, {\bar{B}}_{α}], \\ 0 \leq α \leq 1, B_{1} = [b_{1}, b_{2}] . \end{array}$

Let

$φ = \frac{a_{1} + a_{2}}{2}, ψ = \frac{b_{1} + b_{2}}{2} .$

In the following, we define fuzzy arithmetic operations based on TA for addition, subtraction, multiplication and division:

$\begin{matrix} \tilde{A + B} = ⋃_{α} (\tilde{A + B})_{α}, \\ (\tilde{A + B})_{α} = [\frac{φ + ψ}{2} + (\frac{{\underline{A}}_{α} + {\underline{B}}_{α}}{2}), \\ \frac{φ + ψ}{2} + (\frac{{\bar{A}}_{α} + {\bar{B}}_{α}}{2})] . \end{matrix} ∥$ (20) $\begin{matrix} \tilde{- A} = ⋃_{α} (\tilde{- A})_{α}, (\tilde{- A})_{α} = [- 2 φ + {\underline{A}}_{α}, - 2 φ + {\bar{A}}_{α}] . \end{matrix}$ (21) $\begin{matrix} \tilde{A - B} = ⋃_{α} (\tilde{A - B})_{α}, \\ {(\tilde{A - B})}_{α} = [\frac{φ - 3 ψ}{2} + (\frac{{\underline{A}}_{α} + {\underline{B}}_{α}}{2}), \end{matrix}$ (22) $\frac{φ - 3 ψ}{2} + (\frac{{\bar{A}}_{α} + {\bar{B}}_{α}}{2})] .$

$\begin{matrix} \tilde{A . B} = ⋃_{α} (\tilde{A . B})_{α}, \\ (\tilde{A . B})_{α} = \\ {\begin{matrix} [(\frac{ψ}{2}) {\underline{A}}_{α} + (\frac{φ}{2}) {\underline{B}}_{α}, (\frac{ψ}{2}) {\bar{A}}_{α} + (\frac{φ}{2}) {\bar{B}}_{α}], φ \leq 0, ψ \leq 0, \\ [(\frac{ψ}{2}) {\bar{A}}_{α} + (\frac{φ}{2}) {\underline{B}}_{α}, (\frac{ψ}{2}) {\underline{A}}_{α} + (\frac{φ}{2}) {\bar{B}}_{α}], φ \leq 0, ψ \leq 0, \\ [(\frac{ψ}{2}) {\bar{A}}_{α} + (\frac{φ}{2}) {\bar{B}}_{α}, (\frac{ψ}{2}) {\underline{A}}_{α} + (\frac{φ}{2}) {\underline{B}}_{α}], φ \leq 0, ψ \leq 0, \\ [(\frac{ψ}{2}) {\underline{A}}_{α} + (\frac{φ}{2}) {\bar{B}}_{α}, (\frac{ψ}{2}) {\bar{A}}_{α} + (\frac{φ}{2}) {\underline{B}}_{α}], φ \leq 0, ψ \leq 0 . \end{matrix} \end{matrix}$ (23) $\begin{matrix} \tilde{A^{- 1}} = ⋃_{α} (\tilde{A^{- 1}})_{α}, (\tilde{A^{- 1}})_{α} = [(\frac{1}{φ^{2}}) {\underline{A}}_{α}, (\frac{1}{φ^{2}}) {\bar{A}}_{α}] . \end{matrix}$ (24)

$\begin{matrix} \tilde{A . B^{- 1}} = ⋃_{α} (\tilde{A . B^{- 1}})_{α}, \\ (\tilde{A . B^{- 1}})_{α} = \\ {\begin{matrix} [(\frac{1}{2 ψ}) {\underline{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\underline{B}}_{α}, (\frac{1}{2 ψ}) {\bar{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\bar{B}}_{α}], \\ φ \geq 0, ψ > 0, \\ [(\frac{1}{2 ψ}) {\bar{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\underline{B}}_{α}, (\frac{1}{2 ψ}) {\underline{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\bar{B}}_{α}], \\ φ \geq 0, ψ < 0, \\ [(\frac{1}{2 ψ}) {\bar{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\bar{B}}_{α}, (\frac{1}{2 ψ}) {\underline{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\underline{B}}_{α}], \\ φ \leq 0, ψ < 0, \\ [(\frac{1}{2 ψ}) {\underline{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\bar{B}}_{α}, (\frac{1}{2 ψ}) {\bar{A}}_{α} + (\frac{φ}{2 ψ^{2}}) {\underline{B}}_{α}], \\ φ \leq 0, ψ > 0 . \end{matrix} \end{matrix}$ (25)

Remark 3. The fuzzy arithmetic operation division on fuzzy number $\tilde{0}$ ( $a . c (\tilde{0}) = 0$ ), is not able to define.

As regards fuzzy arithmetic operations using the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of the α - cuts), there existed certain problems in the subtraction operator, division operator and in obtaining the membership functions of operators and above all, the major agent (dependence effect). Therefore we eliminated such deficiency with the fuzzy arithmetic operations based on TA. The following theorems illustrate these claims.

Theorem 5.1. (Lack dependence effect) The result’s support of fuzzy arithmetic operations based on TA (in the domain of the transmission average of support) are smaller than fuzzy arithmetic operations based on the EP (in the domain of the membership function) and the interval arithmetic (in the domain of the α-cuts).

Proof. Let

$\tilde{A} = (\underline{a}, a, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)), \tilde{B} = (\underline{b}, b, \bar{b}, l_{\tilde{B}} (x), r_{\tilde{B}} (x)) .$

According to the fuzzy arithmetic operations based on the extension principle (α-cut) and the

TA, we have:

Based on the extension principle (α-cut):

$\tilde{A} + \tilde{B} = (\underline{a} + \underline{b}, a + b, \bar{a} + \bar{b}, l_{\tilde{A} + \tilde{B}} (x), r_{\tilde{A} + \tilde{B}} (x)),$

$\tilde{A} * \tilde{B} = (\min {\underline{a} \underline{b}, \underline{a} \bar{b}, \bar{a} \underline{b}, \bar{a} \bar{b}}, ab, \max {\underline{a} \underline{b}, \underline{a} \bar{b}, \bar{a} \underline{b}, \bar{a} \bar{b}}, l_{\tilde{A} * \tilde{B}} (x), r_{\tilde{A} * \tilde{B}} (x)) .$

Based on the TA:

$\tilde{A + B} = (\frac{(\underline{a} + b) + (a + \underline{b})}{2}, a + b, \frac{(\bar{a} + b) + (a + \bar{b})}{2}, l_{\tilde{A + B}} (x), r_{\tilde{A + B}} (x)), \tilde{A * B} = (\frac{\underline{a} b + a \underline{b}}{2}, a + b, \frac{\bar{a} b + a \bar{b}}{2}, l_{\tilde{A + B}} (x), r_{\tilde{A + B}} (x)) .$

It is well-known that,

$\frac{(\underline{a} + b) + (a + \underline{b})}{2} \geq \underline{a} + \underline{b}, \frac{(\bar{a} + b) + (a + \bar{b})}{2} \leq \bar{a} + \bar{b}, \frac{(\bar{a} + b) + (a + \bar{b})}{2} \leq \min {\underline{a} \underline{b}, \underline{a} \bar{b}, \bar{a} \underline{b}, \bar{a} \bar{b}} = \underline{a} \underline{b}, \frac{\bar{a} b + a \bar{b}}{2} \leq \max {\underline{a} \underline{b}, \underline{a} \bar{b}, \bar{a} \underline{b}, \bar{a} \bar{b}} = \bar{a} \bar{b} .$

Thus,

$support (\tilde{A + B}) \subseteq support (\tilde{A} + \tilde{B}), support (\tilde{A * B}) \subseteq support (\tilde{A} * \tilde{B}) .$

Remark 4. In the above, let a, :b ≥ 0. Similar to above, we have the same reasons for (a, :b ≤ 0) , (a ≥ 0, b ≤ 0) and (a ≤ 0, :b ≥ 0) .

Remark 5. Although subtraction division addition and multiplication are all defined, similar results were obtained for subtraction and division.

Remark 6. Similar to above, the theorem can be proved for pseudo-trapezoidal fuzzy numbers.

Theorem 5.2. (Fuzzy neutral elements) Let $F_{C} (ℝ)$ is a set of pseudo-geometric fuzzy numbers defined on set of real numbers, then $\forall \tilde{A} \exists! {\tilde{0}}_{\tilde{A}} st; \tilde{A + 0_{\tilde{A}}} = \tilde{0_{\tilde{A}} + A} = \tilde{A} & \tilde{A - A} = {\tilde{0}}_{\tilde{A}},$

$\forall \tilde{A} (a . c (\tilde{A}) \neq 0) \exists! {\tilde{1}}_{\tilde{A}} st; \tilde{A . 1_{\tilde{A}}} = \tilde{1_{\tilde{A}} . A} = \tilde{A} & \tilde{A . A^{- 1}} = {\tilde{1}}_{\tilde{A}} .$

Proof. We have the above cases, according to (14), (17), (20), (23) and with ${\tilde{0}}_{\tilde{A}}$ and ${\tilde{1}}_{\tilde{A}}$ as following:

for $\tilde{A} = (\underline{a}, a, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)) :$ $\begin{matrix} {\tilde{0}}_{\tilde{A}} = (\underline{a} - a, 0, \bar{a} - a, l_{\tilde{A}} (x + a), r_{\tilde{A}} (x + a)), \end{matrix}$ (26) $\begin{matrix} {\tilde{1}}_{\tilde{A}} = {\begin{matrix} (\frac{\underline{a}}{a}, 1, \frac{\bar{a}}{a}, l_{\tilde{A}} (xa), r_{\tilde{A}} (xa)), a > 0, \\ (\frac{\bar{a}}{a}, 1, \frac{\underline{a}}{a}, r_{\tilde{A}} (xa), l_{\tilde{A}} (xa)), a < 0 . \end{matrix} \end{matrix}$ (27) and for $\tilde{A} = (\underline{a}, a_{1}, a_{2}, \bar{a}, l_{\tilde{A}} (x), r_{\tilde{A}} (x)) :$ $\begin{matrix} {\tilde{0}}_{\tilde{A}} = (\underline{a} - φ, \frac{a_{1} - a_{2}}{2}, \frac{a_{2} - a_{1}}{2}), \bar{a} - φ, \\ l_{\tilde{A}} (x + φ), r_{\tilde{A}} (x + φ), \end{matrix}$ (28) $\begin{matrix} {\tilde{1}}_{\tilde{A}} = {\begin{matrix} (\frac{\underline{a}}{φ}, \frac{a_{1}}{φ}, \frac{a_{2}}{φ}, \frac{\bar{a}}{φ}), l_{\tilde{A}} (x φ), r_{\tilde{A}} (x φ), φ > 0, \\ (\frac{\bar{a}}{φ}, \frac{a_{2}}{φ}, \frac{a_{1}}{φ}, \frac{\underline{a}}{φ}), r_{\tilde{A}} (x φ), l_{\tilde{A}} (x φ), φ < 0 . \end{matrix} \end{matrix}$ (29)

Lemma 5.1. Let $F_{c} (ℝ)$ be a set of the pseudo-geometric fuzzy numbers, then

$\forall \tilde{A}, \tilde{B}, \tilde{C} \in F_{c} (ℝ);$

$\tilde{A} \sim_{1} \tilde{B} \Rightarrow \tilde{A + B} \sim_{1} \tilde{A} or \tilde{A + B} \sim_{1} \tilde{B},$

$\tilde{A} \sim_{2} \tilde{B} \Rightarrow \tilde{A . B} \sim_{2} \tilde{A} or \tilde{A . B} \sim_{2} \tilde{B},$

$\tilde{A + B} = \tilde{B + A},$

$\tilde{A . B} = \tilde{B . A},$

$\tilde{A + (B + C)} ≅ \tilde{(A + B) + C},$

$\tilde{A . (B . C)} ≅ \tilde{(A . B) . C},$

$\tilde{A . (B + C)} ≅ \tilde{(A . B) + (A . C)},$

$\tilde{(B + C) . A} ≅ \tilde{(B . A) + (C . A)} .$

Proof. We have the above cases, according to the definitions of (5.1), (5.2) and (3.4).

Finally, according to the above points, a set of the pseudo-geometric fuzzy numbers can be introduced as fuzzy group and field. It is expressed in the following examples.

Example 5.1. Let $F_{c} (ℝ)$ be a set of the pseudo-geometric fuzzy numbers, then $(F_{c} (ℝ), +)$ is a type-1 fuzzy group of type Abelian.

Example 5.2. Let $F_{c} (ℝ)$ be a set of the pseudo-geometric fuzzy numbers and

$F_{c}^{'} (ℝ) = {\tilde{A} \in F_{c} (ℝ) a . c (\tilde{A}) \neq 0},$

then $(F_{c}^{'} (ℝ), .)$ is a type-2 fuzzy group of type Abelian.

Example 5.3. Let $F_{c} (ℝ)$ be a set of the pseudo-geometric fuzzy numbers, then $(F_{c} (ℝ), +, .)$ is a fuzzy field.

6 Conclusion and future research

In this paper, we considered the algebraic objects known as fuzzy ring and field which serve as one of the fundamental building blocks for the subject which is called fuzzy abstract algebra. These are usually sets on whose elements we can operate algebraically. In addition, we assume that these fuzzy algebraic operations are subject to certain rules, which are explicitly spelled out in what we call the axioms or postulates defining the system. In this abstract setting we then attempt to prove theorems about these very general structures, always hoping that when these results are applied to a particular, concrete realization of the fuzzy abstract system there will flow out facts and insights into the example at hand which would have been obscured from us by the mass of inessential information available to us in the particular, special case. We should like to stress that these fuzzy algebraic systems and the axioms which define them must have a certain naturally about them. They must come from the experience of looking at many examples. Namely, they should be rich in meaningful results. One does not just sit down, list a few axioms, and then proceed to study the system so described.

According to the above mentioned cases, a new attitude coupled with fuzzy thinking to fuzzy rings and fields are presented with binary operations and on the basis of the specified parameter, called ambiguity rank. Finally, several illustrative examples and theorems are given due to confirm the above case for having a certain natural behavior for fuzzy algebraic systems.

In later papers, we will have a new look at some other of the fuzzy issues as fuzzy ranking and fuzzy distance. Finally, we will have a new encounter to fuzzy equations, fuzzy equations systems and fuzzy differential equations.

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