α-fuzzy ideals of C-algebra

ABSTRACT

In this paper, we study α-fuzzy ideals in C-algebras whose truth degrees are in a complete Heyting algebra. We provide several characterizations for fuzzy ideals to be an α-fuzzy ideal. One of the key contributions of this manuscript is the investigation of the smallest α-fuzzy ideal containing a given fuzzy ideal. By establishing the existence and properties of this smallest α-fuzzy ideal, we shed light on the structure and behavior of α-fuzzy ideals in C-algebras. Furthermore, we prove that the class of α-fuzzy ideals forms a complete lattice. We obtain the closure operator on the class of fuzzy ideals FI(A), where the closed elements correspond to the α-fuzzy ideals. We explore the conditions under which every fuzzy ideal in a given C-algebra becomes an α-fuzzy ideal. Finally, we study the space of prime α-fuzzy ideals in C-algebras and we derive a necessary and sufficient condition for this space to be a T₁ space.

KEYWORDS:

• C-algebra
• fuzzy ideal
• α-fuzzy ideal
• prime α-fuzzy ideal

MATHEMATICS SUBJECT CLASSIFICATION:

• 03G25
• 03G05
• 08A72

1. Introduction

According Guzman and Squier (1990), a C-algebra is an algebra with two binary operations $\wedge$ and $\vee$, and one unary operation ${}^{\prime }$ satisfying most of the properties of operations in Boolean algebra. It is observed that the family of C-algebras forms a variety, that is, it is closed under the three class operators, and it is a variety generated by the three-element algebra $\mathit{C}=\{\mathit{F},\mathit{U},\mathit{T}\}$. The variety of C-algebras, known by the name “the algebra of conditional logic”, is the algebraic model for the three-valued conditional logic. Kalesha Vali et al. (2010) and Vali and Swamy (2011) studied the concepts of ideals, prime ideals and maximal ideals of C-algebras. They proved that the set of all ideals can be made an algebraic lattice. Moreover, they studied the space of prime ideals of a C-algebra with respect to the hull-kernel topology. Rao (2013b), studied the notion of annihilator ideals of C-algebras and proved that the class of all annihilator ideals forms a complete Boolean algebra. Later, this author studied the notion of annulates and α-ideals of C-algebras analogous to that of a distributive lattice Rao (2013a).

In classical set theory, an object or individual is either a member or not a member of a set, and there is no in-between. This means that there is a clear boundary between members and non-members. However, many real-life problems cannot be solved by this theory because they require more complex solutions. For example, some problems may require an object to be both a member and non-member of a set at the same time. Therefore, classical set theory is not sufficient to solve all real-life problems.

In classical set theory, it is well known that individuals are either a member or not a member of a set. So, there exists a clear, and sharp, boundary between members and non-members. Thus, in classical set theory, an object or an individual is not allowed to be a member of a set and nonmember of the set at the same time. But many of our real-life problems cannot be handled by such ordinary set theory. In 1965, the famous mathematician called Zadeh (1965), introduced the elegant theory known as the theory of fuzzy sets. He has defined the notion of fuzzy subsets of a set (the universe of discourses) in such a way that could describe vagueness mathematically by assigning a grade of membership (a number between 0 and 1). This value in fact expressed the degree or the extent to which an individual belongs to the given fuzzy set. Goguen (1967) suggested that the unit interval $[0,1]$ is insufficient to represent the truth values of general fuzzy statements. Swamy and Swamy (1988) initiated that complete lattices satisfying the infinite meet distributivity are the most suitable candidates to have the truth values of general fuzzy statements.

In 1971, Rosenfeld (1971) has applied the concept of fuzziness in group theory. He has defined closurity of fuzzy sets under a group binary operation and inversion operation, which enables him to introduce fuzzy subgroups of a group. This laid a foundation and opens the door for several algebraists to work on fuzzy sub-algebras of different algebraic structures. To mention some of them: invariant fuzzy sub-groups of a group (see Anthony and Sherwood (1979); Das (1981); Dixit et al. (1990)), fuzzy sub-rings and fuzzy ideals of a ring (see Dixit et al. (1991); Liu (1982); Mukherjee and Sen (1987)), fuzzy ideals and fuzzy filters of a lattice (see Bo and Wangming (1990); Swamy and Raju (1998)), fuzzy ideals and fuzzy filters of MS-algebras (see Alaba and Alemayehu (2018); Alaba et al. (2018); Alaba and Alemayehu (2019b); Alaba and Alemayehu (2019a)), etc.

Alaba and Norahun (2019) studied the concept of α-fuzzy ideals and the space of prime α-fuzzy ideals of a distributive lattice with the help of annulates and proved that the class of all α-fuzzy ideals forms a complete distributive lattice. Alaba and Addis (2018) studied the concept of fuzzy ideals of C-algebras. Furthermore, Addis (2020) studied the notion of prime fuzzy ideals of a C-algebra and its space.

The purpose of this paper is to study the notion of α-fuzzy ideals of C-algebras analogous to distributive lattices. The basic properties of α-fuzzy ideals are also studied. A special class of fuzzy ideals called α-fuzzy ideals are also studied. One of the key contributions of this manuscript is the investigation of the smallest α-fuzzy ideal containing a given fuzzy ideal. By establishing the existence and properties of this smallest α-fuzzy ideal, we shed light on the structure and behavior of α-fuzzy ideals in C-algebras. We prove that the class of α-fuzzy ideals of a C-algebra A forms a complete lattice, providing a valuable insight into the lattice-theoretic aspects of these ideals. This result is significant as it demonstrates the existence of supremum and infimum operations for α-fuzzy ideals, enabling us to reason about the algebraic structure of these ideals in a systematic manner. Additionally, we obtained the closure operator on the class of fuzzy ideals FI(A), where the closed elements correspond to the α-fuzzy ideals of A. Moreover, we provide a set of equivalent conditions for a given C-algebra, under which every fuzzy ideal becomes an α-fuzzy ideal. These conditions serve as a valuable criterion for determining when the notion of α-fuzzy ideals coincides with that of fuzzy ideals in C-algebras. This result contributes to our understanding of the relationship between fuzzy ideals and α-fuzzy ideals, providing a deeper insight into the algebraic structure of C-algebras. The image and inverse image of α-fuzzy ideals under the homomorphism mapping are also studied. Finally, we study the topological space of prime α-fuzzy ideals of C-algebras. Some of its properties are also studied. We prove that the collection $\{\mathit{\kappa }(\mathit{\eta }):\mathit{\eta }\in \mathit{FI}(\mathit{A})\}$ is a topology on Ω and $\mathcal{B}=\{\mathit{\kappa }(c_{\mathit{\beta }}):\mathit{c}\in \mathit{A},\mathit{\beta }\in \mathit{L}-\{0\}\}$ is a basis for a topology on Ω. We also show that the space Ω is a T₀ space. Moreover, we established a necessary and sufficient condition that the space Ω is a T₁ space.

2. Preliminaries

Throughout this paper, A stands for a C-algebra.

Definition 2.1.

Guzman and Squier (1990)

An algebra $(\mathit{A},\vee ,\wedge {,}^{\prime })$ of type $(2,2,1)$ is called a C-algebra, if it satisfies the following axioms:

1. $j^{″}=\mathit{j}$

2. $(\mathit{j}\wedge \mathit{k}{)}^{\prime }=j^{\prime }\vee k^{\prime }$.

3. $(\mathit{j}\wedge \mathit{k})\wedge \mathit{l}=\mathit{j}\wedge (\mathit{k}\wedge \mathit{l})$.

4. $\mathit{j}\wedge (\mathit{k}\vee \mathit{l})=(\mathit{j}\wedge \mathit{k})\vee (\mathit{j}\wedge \mathit{l})$.

5. $(\mathit{j}\vee \mathit{k})\wedge \mathit{l}=(\mathit{j}\wedge \mathit{l})\vee (j^{\prime }\wedge \mathit{k}\wedge \mathit{l})$.

6. $\mathit{j}\vee (\mathit{j}\wedge \mathit{k})=\mathit{j}$.

7. $(\mathit{j}\wedge \mathit{k})\vee (\mathit{k}\wedge \mathit{j})=(\mathit{k}\wedge \mathit{j})\vee (\mathit{j}\wedge \mathit{k})$ for all $\mathit{j},\mathit{k},\mathit{l}\in \mathit{A}$.

Example 2.2.

Let $\mathit{A}=\{\mathit{a},\mathit{b},\mathit{c},\mathit{d},\mathit{e},\mathit{f}\}$. Define the following operations $\vee ,\wedge$ and ${}^{{}^{\prime }}$ as follows:

$\begin{array}{l}\begin{array}{lllllll}\vee & \mathit{a}& \mathit{b}& \mathit{c}& \mathit{d}& \mathit{e}& \mathit{f}\\ \mathit{a}& \mathit{a}& \mathit{c}& \mathit{c}& \mathit{f}& \mathit{a}& \mathit{f}\\ \mathit{b}& \mathit{c}& \mathit{b}& \mathit{c}& \mathit{d}& \mathit{b}& \mathit{c}\\ \mathit{c}& \mathit{c}& \mathit{c}& \mathit{c}& \mathit{c}& \mathit{c}& \mathit{c}\\ \mathit{d}& \mathit{f}& \mathit{d}& \mathit{f}& \mathit{d}& \mathit{d}& \mathit{f}\\ \mathit{e}& \mathit{a}& \mathit{b}& \mathit{c}& \mathit{d}& \mathit{e}& \mathit{f}\\ \mathit{f}& \mathit{f}& \mathit{f}& \mathit{f}& \mathit{f}& \mathit{f}& \mathit{f}\end{array}\begin{array}{lllllll}\wedge & \mathit{a}& \mathit{b}& \mathit{c}& \mathit{d}& \mathit{e}& \mathit{f}\\ \mathit{a}& \mathit{a}& \mathit{e}& \mathit{a}& \mathit{e}& \mathit{e}& \mathit{a}\\ \mathit{b}& \mathit{b}& \mathit{b}& \mathit{b}& \mathit{b}& \mathit{b}& \mathit{b}\\ \mathit{c}& \mathit{c}& \mathit{b}& \mathit{c}& \mathit{b}& \mathit{b}& \mathit{c}\\ \mathit{d}& \mathit{e}& \mathit{b}& \mathit{b}& \mathit{d}& \mathit{e}& \mathit{d}\\ \mathit{e}& \mathit{e}& \mathit{e}& \mathit{e}& \mathit{e}& \mathit{e}& \mathit{e}\\ \mathit{f}& \mathit{a}& \mathit{b}& \mathit{c}& \mathit{d}& \mathit{e}& \mathit{f}\end{array}\begin{array}{ll}\mathit{x}& x^{\prime }\\ \mathit{a}& \mathit{d}\\ \mathit{b}& \mathit{c}\\ \mathit{c}& \mathit{b}\\ \mathit{d}& \mathit{a}\\ \mathit{e}& \mathit{f}\\ \mathit{f}& \mathit{e}\end{array}\end{array}$

Then $(\mathit{A},\vee ,\wedge {,}^{{}^{\prime }})$ is a C-algebra.

Lemma 2.3.

Guzman and Squier (1990)

Every $\mathit{C}-$algebra satisfies the following identities.

1. $\mathit{j}\wedge \mathit{j}=\mathit{j}$.

2. $\mathit{x}\wedge x^{\prime }=x^{\prime }\wedge \mathit{x}$.

3. $\mathit{j}\wedge \mathit{k}\wedge \mathit{j}=\mathit{j}\wedge \mathit{k}$.

4. $\mathit{j}\wedge j^{\prime }\wedge \mathit{k}=\mathit{j}\wedge j^{\prime }$.

5. $\mathit{j}\wedge \mathit{k}=(j^{\prime }\vee \mathit{k})\wedge \mathit{j}$.

6. $\mathit{j}\wedge \mathit{k}=\mathit{j}\wedge (\mathit{k}\vee j^{\prime })$.

7. $\mathit{j}\wedge \mathit{k}=\mathit{j}\wedge (j^{\prime }\vee \mathit{k})$.

8. $\mathit{j}\wedge \mathit{k}\wedge j^{\prime }=\mathit{j}\wedge \mathit{k}\wedge k^{\prime }$.

9. $(\mathit{j}\vee \mathit{k})\wedge \mathit{j}=\mathit{j}\vee (\mathit{k}\wedge \mathit{j})$.

10. $\mathit{j}\wedge (j^{\prime }\vee \mathit{j})=(j^{\prime }\vee \mathit{j})\wedge \mathit{j}=(\mathit{j}\vee j^{\prime })\wedge \mathit{j}$.

The dual statements of the above identities are also valid in a C-algebra.

Definition 2.4.

Guzman and Squier (1990)

An element n of a C-algebra A is called a left zero for $\wedge$ if $\mathit{n}\wedge \mathit{j}=\mathit{n}$ for all $\mathit{j}\in \mathit{A}$.

Definition 2.5.

(Kalesha Vali et al., 2010)

A nonempty subset I of a C-algebra A is called an ideal of A, if

1. $\mathit{j},\mathit{k}\in \mathit{I}\Rightarrow \mathit{j}\vee \mathit{k},\mathit{j}\wedge \mathit{l}\in \mathit{I}$

2. $\mathit{j}\in \mathit{I}\Rightarrow \mathit{l}\wedge \mathit{j}\in \mathit{I}$ for each $\mathit{l}\in \mathit{A}$.

It can also be observed that $\mathit{j}\wedge \mathit{l}\in \mathit{I}$ for all $\mathit{j}\in \mathit{I}$ and all $\mathit{l}\in \mathit{A}$. For any subset $\mathit{S}\subseteq \mathit{A}$, the smallest ideal of A containing S is called the ideal of A generated by S and is denoted by $⟨\mathit{S}]$. Note that:

$\begin{array}{ll}& ⟨\mathit{S}]=\{\bigvee (j_{\mathit{i}}\wedge k_{\mathit{i}}):j_{\mathit{i}}\in \mathit{A},k_i\in \mathit{S},\mathit{i}=1,\dots ,\\ & & \mathit{n}\mathrm{for}\mathrm{some}\mathit{n}\in Z^{+}\}.\end{array}$

If $\mathit{S}=\{\mathit{j}\}$, then we write $⟨\mathit{j}]$ for $⟨\mathit{S}]$. In this case, $⟨\mathit{j}]=\{\mathit{k}\wedge \mathit{j}:\mathit{k}\in \mathit{A}\}$. Moreover, it is observed in Guzman and Squier (1990) that the set $I_0=\{\mathit{j}\wedge j^{\prime }:\mathit{j}\in \mathit{A}\}$ is the smallest ideal in A.

Definition 2.6.

Rao (2013a)

For any element, a of A the ideal $(\mathit{a}{)}^{\ast }=\{\mathit{x}\in \mathit{A}:\mathit{x}\wedge \mathit{a}\text{is a left zero}\}$ is called an annulate of a.

Definition 2.7.

Rao (2013a)

An ideal I of A is called an α-ideal if $(\mathit{x}{)}^{\ast \ast }\subseteq \mathit{I}$ for all $\mathit{x}\in \mathit{I}$.

Definition 2.8.

Guzman and Squier (1990)

Let $(\mathit{A},\vee ,\wedge {,}^{{}^{\prime }},I_0)$ and $(A^{{}^{\prime }},\vee ,\wedge {,}^{{}^{\prime }},I_0^{{}^{\prime }})$ be two C-algebras. Then, a mapping $\mathit{f}:\mathit{A}\to A^{{}^{\prime }}$ is called a homomorphism if it satisfies the following conditions:

1. $\mathit{f}(\mathit{j}\vee \mathit{k})=\mathit{f}(\mathit{j})\vee \mathit{f}(\mathit{k})$,

2. $\mathit{f}(\mathit{j}\wedge \mathit{k})=\mathit{f}(\mathit{j})\wedge \mathit{f}(\mathit{k})$,

3. $\mathit{f}(j^{{}^{\prime }})=\mathit{f}(\mathit{j}{)}^{{}^{\prime }}$ for all $\mathit{j},\mathit{k}\in \mathit{A}.$

Where $I_0=\{\mathit{j}\in \mathit{A}:\mathit{j}\text{is a left zero for}\wedge \}$ and $I_0^{{}^{\prime }}=\{\mathit{k}\in A^{{}^{\prime }}:\mathit{k}\text{is a left zero for}\wedge \}$.

I₀ and $I_0^{{}^{\prime }}$ are the smallest ideals of A and $A^{{}^{\prime }}$, respectively, and $\mathit{Kerf}=\{\mathit{j}\in \mathit{A}:\mathit{f}(\mathit{j})\in I_0^{{}^{\prime }}\}$.

By an L-fuzzy subset of A, we mean a mapping $\mathit{\eta }:\to \mathit{L}$, where L in this paper is assumed to be a non-trivial complete lattice satisfying the infinite meet distributive law:

$\begin{array}{l}\mathit{\gamma }\wedge (\underset{\mathit{\beta }\in \mathit{M}}{\bigvee }\mathit{\beta })=\underset{\mathit{\beta }\in \mathit{M}}{\bigvee }(\mathit{\gamma }\wedge \mathit{\beta })\end{array}$

for all $\mathit{\gamma }\in \mathit{L}$ and any $\mathit{M}\subseteq \mathit{L}$. If there is no confusion, from now on we simply say fuzzy subsets instead of L-fuzzy subsets. For each $\mathit{\gamma }\in \mathit{L}$, the $\mathit{\gamma }-$level set of η denoted by η_γ is a subset of A given by:

$\begin{array}{l}{\eta }_{\mathit{\gamma }}=\{\mathit{x}\in \mathit{A}:\mathit{\gamma }\le \mathit{\eta }(\mathit{x})\}.\end{array}$

A fuzzy subset η of A is said to be nonempty if there is $\mathit{x}\in \mathit{A}$ such that $\mathit{\eta }(\mathit{x})\ne 0$.

Definition 2.9.

Alaba and Addis (2018)

A fuzzy subset γ of A is called a fuzzy ideal of A if:

1. $\mathit{\gamma }(\mathit{n})=1$, for all $\mathit{n}\in I_0$,

2. $\mathit{\gamma }(\mathit{j}\vee \mathit{k})\ge \mathit{\gamma }(\mathit{j})\wedge \mathit{\gamma }(\mathit{k})$,

3. $\mathit{\gamma }(\mathit{j}\vee \mathit{k})\ge \mathit{\gamma }(\mathit{k})$ for all $\mathit{j},\mathit{k}\in \mathit{A}.$

We denote the class of all fuzzy ideals of A by FI(A).

Lemma 2.10.

Alaba and Addis (2018)

Let γ be a fuzzy ideal of A. Then, the following holds for all$\mathit{j},\mathit{k}\in \mathit{A}$.

1. $\mathit{\gamma }(\mathit{j}\wedge \mathit{k})\ge \mathit{\gamma }(\mathit{j})$,

2. $\mathit{\gamma }(\mathit{j}\wedge \mathit{k})\ge \mathit{\gamma }(\mathit{k}\wedge \mathit{j})$,

3. $\mathit{\gamma }(\mathit{j}\wedge \mathit{l}\wedge \mathit{k})\ge \mathit{\gamma }(\mathit{j}\wedge \mathit{k})$, for each $\mathit{l}\in \mathit{A}$,

4. $\mathit{\gamma }(\mathit{j})\ge \mathit{\gamma }(\mathit{j}\vee \mathit{k})$ and hence $\mathit{\gamma }(\mathit{j})\wedge \mathit{\gamma }(\mathit{k})\wedge \mathit{\lambda }(\mathit{k}\vee \mathit{j})$.

5. If $\mathit{l}\in (\mathit{j}]$, then $\mathit{\gamma }(\mathit{l})\ge \mathit{\gamma }(\mathit{j})$.

For any fuzzy subset η of A, the fuzzy ideal generated by η is denoted by $⟨\mathit{\eta }⟩$.

Theorem 2.11.

Alaba and Addis (2018)

If λ and ν are fuzzy ideals of a C-algebra, then their supremum of is given by:

$\begin{array}{l}(\mathit{\lambda }\vee \mathit{\nu })(\mathit{x})=\mathit{Sup}\{{\bigwedge }_{i=1}^n[\mathit{\lambda }(b_{\mathit{i}})\vee \mathit{\nu }(b_{\mathit{i}})]:\mathit{x}={\bigvee }_{i=1}^n{b}_{\mathit{i}},b_{\mathit{i}}\in \mathit{A}\}.\end{array}$

Definition 2.12.

Addis (2020)

A non-constant fuzzy ideal λ of A is called prime fuzzy ideal of A if for any two fuzzy ideals $\mathit{\theta },\mathit{\eta }$ of A, $\mathit{\theta }\cap \mathit{\eta }\subseteq \mathit{\lambda }\Rightarrow \mathit{\theta }\subseteq \mathit{\lambda }$ or $\mathit{\eta }\subseteq \mathit{\lambda }$.

Theorem 2.13.

Addis (2020)

A non-constant fuzzy ideal η of A is a prime fuzzy ideal if and only if $\mathit{Im}\mathit{\eta }=\{1,\mathit{\beta }\},\mathit{\beta }\in \mathit{L}-\{1\}$ and ${\eta }_{\ast }=\{\mathit{x}\in \mathit{L}:\mathit{\eta }(\mathit{x})=1\}$ is a prime ideal of A.

Definition 2.14.

Addis (2020)

A non-empty fuzzy subset η of A is said to be a multiplicatively closed fuzzy subset, if

$\begin{array}{l}\mathit{\eta }(\mathit{c}\wedge \mathit{d})\ge \mathit{\eta }(\mathit{c})\wedge \mathit{\eta }(\mathit{d}),\text{ for all }\mathit{c},\mathit{d}\in \mathit{A}.\end{array}$

Lemma 2.15.

Addis (2020)

Let η be a fuzzy ideal of A, $\mathit{a}\in \mathit{A}$ and $\mathit{\beta }\in \mathit{L}-\{1\}$. If $\mathit{\eta }(\mathit{a})\le \mathit{\beta }$, then there exists a prime fuzzy ideal λ of A such that $\mathit{\eta }\subseteq \mathit{\lambda }$ and $\mathit{\lambda }(\mathit{a})\le \mathit{\beta }$.

Theorem 2.16.

Addis (2020)

Let η be a fuzzy ideal of A and λ be a multiplicatively closed fuzzy subset of A and $\mathit{\beta }\in \mathit{L}$. If $\mathit{\eta }\cap \mathit{\lambda }\le \mathit{\beta }$, then there exists a prime fuzzy ideal θ of A such that

$\begin{array}{l}\mathit{\eta }\subseteq \mathit{\theta }\text{ and }\mathit{\theta }\cap \mathit{\lambda }\le \mathit{\beta }.\end{array}$

Definition 2.17.

Rosenfeld (1971)

Let f be a function from X into Y; µ be a fuzzy subset of X; and θ be a fuzzy subset of Y.

1. The image of µ under f, denoted by $\mathit{f}(\mathit{\mu })$, is a fuzzy subset of Y defined by: for each $\mathit{y}\in \mathit{Y}$,

   $\begin{array}{l}\mathit{f}(\mathit{\mu })(\mathit{y})=\{\begin{array}{l}\mathit{sup}\{\mathit{\mu }(\mathit{x}):\mathit{x}\in f^{-1}(\mathit{y})\},\$\mathit{if}\$f^{-1}(\mathit{y})\ne \mathit{\varphi }\\ 0,\text{otherwise},\end{array}.\end{array}$

2. The preimage of θ under f, denoted by $f^{-1}(\mathit{\theta })$, is a fuzzy subset of X defined by: for each $\mathit{x}\in \mathit{X}$,

   $\begin{array}{l}{f}^{-1}(\mathit{\theta })(\mathit{x})=\mathit{\theta }(\mathit{f}(\mathit{x})).\end{array}$

Definition 2.18.

Norahun et al. (2021)

For any fuzzy subset η of A, f is said to be a fuzzy annihilator preserving if $\mathit{f}({\eta }^{\ast })=(\mathit{f}(\mathit{\eta }){)}^{\ast }$.

Theorem 2.19.

Norahun et al. (2021)

If $\mathit{Kerf}=I_0$ and f is onto, then f⁻¹ preserves fuzzy annihilator.

The class of fuzzy ideals of a C-algebra A is denoted by FI(A).

3. α-fuzzy ideals

The purpose of this section is to study the concept of α-fuzzy ideals of C-algebras analogous to that of distributive lattices. Basic properties α-fuzzy ideals are also studied. We also study a special class of fuzzy ideals called α-fuzzy ideals. We have shown that these fuzzy ideals form a complete distributive lattice. We provide a set of equivalent conditions for a fuzzy ideal to be an α-fuzzy ideal. Furthermore, we study the image and inverse image of α-fuzzy ideals under a homomorphism mapping.

Definition 3.1.

A fuzzy ideal η of A is called an α-fuzzy ideal, if:

$\begin{array}{l}\underset{\mathit{e}\in (\mathit{d}{)}^{\ast \ast }}{\bigwedge }\mathit{\eta }(\mathit{e})\ge \mathit{\eta }(\mathit{d})\text{ for all }\mathit{d}\in \mathit{A}.\end{array}$

Theorem 3.2.

A fuzzy subset η of A is an α-fuzzy ideal if and only η_β is an α-fuzzy ideal of A for all $\mathit{\beta }\in \mathit{L}$.

Proof.

Suppose $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Clearly ${\eta }_{\mathit{\beta }}\in \mathit{FI}(\mathit{A})$. To show $(\mathit{d}{)}^{\ast \ast }\subseteq {\eta }_{\mathit{\beta }}$, let $\mathit{e}\in (\mathit{d}{)}^{\ast \ast }$. Then $\mathit{\eta }(\mathit{e})\ge \mathit{\eta }(\mathit{d})\ge \mathit{\beta }$. Thus $\mathit{e}\in {\eta }_{\mathit{\beta }}$ for all $\mathit{e}\in (\mathit{d}{)}^{\ast \ast }$ and hence $(\mathit{d}{)}^{\ast \ast }\subseteq {\eta }_{\mathit{\beta }}$. Therefore, ${\eta }_{\mathit{\beta }}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ for all $\mathit{\beta }\in \mathit{L}$.

Conversely, suppose that ${\eta }_{\mathit{\beta }}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ for all $\mathit{\beta }\in \mathit{L}$. Clearly, $\mathit{\eta }\in \mathit{FI}(\mathit{A})$. To show $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, let $\mathit{e}\in (\mathit{d}{)}^{\ast \ast }$ and $\mathit{\eta }(\mathit{d})=\mathit{\beta }$. Then $\mathit{d}\in {\eta }_{\mathit{\beta }}$. Since $\mathit{e}\in (\mathit{d}{)}^{\ast \ast }$ and ${\eta }_{\mathit{\beta }}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, $(\mathit{d}{)}^{\ast \ast }\subseteq {\eta }_{\mathit{\beta }}$. Which implies that $\mathit{\eta }(\mathit{e})\ge \mathit{\eta }(\mathit{d})$ for all $\mathit{e}\in (\mathit{d}{)}^{\ast \ast }$. Thus $\underset{\mathit{e}\in (\mathit{d}{)}^{\ast \ast }}{\bigwedge }\mathit{\eta }(\mathit{e})\ge \mathit{\eta }(\mathit{d})$ and hence $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Corollary 3.3.

For a non-empty subset I of A, I is an α-ideal if and only if ${\chi }_{\mathit{I}}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Theorem 3.4.

Let $\mathit{\eta }\in \mathit{FI}(\mathit{A})$. Define a fuzzy subset η^α on A by:

$\begin{array}{l}{\eta }^{\mathit{\alpha }}(\mathit{c})=\bigvee \{\mathit{\eta }(\mathit{d}):(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast },\mathit{d}\in \mathit{A}\}.\end{array}$

Then ${\eta }^{\mathit{\alpha }}\in \mathit{FI}(\mathit{A})$.

Proof.

Let $\mathit{z}\in I_0$. Then ${\eta }^{\mathit{\alpha }}(\mathit{z})=1$. For any $\mathit{c},\mathit{d}\in \mathit{A}$,

$\begin{array}{lll}{\eta }^{\mathit{\alpha }}(\mathit{c})\wedge {\eta }^{\mathit{\alpha }}(\mathit{d})& =& \bigvee \{\mathit{\eta }(\mathit{e}):(\mathit{c}{)}^{\ast }=(\mathit{e}{)}^{\ast },\mathit{e}\in \mathit{A}\}\\ & & \wedge \bigvee \{\mathit{\eta }(\mathit{f}):(\mathit{d}{)}^{\ast }=(\mathit{f}{)}^{\ast },\mathit{f}\in \mathit{A}\}\\ & \le & \bigvee \{\mathit{\eta }(\mathit{e}\vee \mathit{f}):(\mathit{c}{)}^{\ast }=(\mathit{e}{)}^{\ast },(\mathit{d}{)}^{\ast }=(\mathit{f}{)}^{\ast }\}\end{array}$

Since $(\mathit{c}{)}^{\ast }=(\mathit{e}{)}^{\ast }$ and $(\mathit{d}{)}^{\ast }=(\mathit{f}{)}^{\ast }$, we have $(\mathit{c}\vee \mathit{d}{)}^{\ast }=(\mathit{e}\vee \mathit{f}{)}^{\ast }$. Thus,

$\begin{array}{lll}{\eta }^{\mathit{\alpha }}(\mathit{c})\wedge {\eta }^{\mathit{\alpha }}(\mathit{d})& \le & \bigvee \{\mathit{\eta }(\mathit{e}\vee \mathit{f}):(\mathit{c}\vee \mathit{d}{)}^{\ast }=(\mathit{e}\vee \mathit{f}{)}^{\ast }\}\\ & \le & \bigvee \{\mathit{\eta }(\mathit{g}):(\mathit{c}\vee \mathit{d}{)}^{\ast }=(\mathit{g}{)}^{\ast },\mathit{g}\in \mathit{A}\}\\ & =& {\eta }^{\mathit{\alpha }}(\mathit{c}\vee \mathit{d})\end{array}$

To show ${\eta }^{\mathit{\alpha }}(\mathit{c}\wedge \mathit{d})\ge {\eta }^{\mathit{\alpha }}(\mathit{c})\vee {\eta }^{\mathit{\alpha }}(\mathit{d})$, it is enough to show that ${\eta }^{\mathit{\alpha }}(\mathit{c})\le {\eta }^{\mathit{\alpha }}(\mathit{c}\wedge \mathit{a})$ for any $\mathit{a}\in \mathit{A}$. Now, ${\eta }^{\mathit{\alpha }}(\mathit{c})=\bigvee \{\mathit{\eta }(\mathit{e}):(\mathit{c}{)}^{\ast }=(\mathit{e}{)}^{\ast },\mathit{e}\in \mathit{A}\}$. Since η is a fuzzy ideal and $(\mathit{c}{)}^{\ast }=(\mathit{e}{)}^{\ast }$, we get $\mathit{\eta }(\mathit{e}\wedge \mathit{d})\ge \mathit{\eta }(\mathit{e})$ and $(\mathit{c}\wedge \mathit{d}{)}^{\ast }=(\mathit{e}\wedge \mathit{d}{)}^{\ast }$. Which implies that ${\eta }^{\mathit{\alpha }}(\mathit{c})\le \bigvee \{\mathit{\eta }(\mathit{e}\wedge \mathit{d}):(\mathit{c}\wedge \mathit{d}{)}^{\ast }=(\mathit{e}\wedge \mathit{d}{)}^{\ast }\}={\eta }^{\mathit{\alpha }}(\mathit{c}\wedge \mathit{d})$. Thus, ${\eta }^{\mathit{\alpha }}(\mathit{c}\wedge \mathit{d})\ge {\eta }^{\mathit{\alpha }}(\mathit{c})\vee {\eta }^{\mathit{\alpha }}(\mathit{d})$ and hence ${\eta }^{\mathit{\alpha }}\in \mathit{FI}(\mathit{A})$.

Theorem 3.5.

If $\mathit{\eta }\in \mathit{FI}(\mathit{A})$, then η^α is the smallest α-fuzzy ideal containing η.

Proof.

Clearly, $\mathit{\eta }\subseteq {\eta }^{\mathit{\alpha }}$. Now, we proceed to show ${\eta }^{\mathit{\alpha }}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Let $\mathit{y}\in \mathit{A}$. Then

$\begin{array}{lll}\underset{\mathit{c}\in (\mathit{y}{)}^{\ast \ast }}{\bigwedge }{\eta }^{\mathit{\alpha }}(\mathit{c})& =& \underset{\mathit{c}\in (\mathit{y}{)}^{\ast \ast }}{\bigwedge }(\bigvee \{\mathit{\eta }(\mathit{d}):(\mathit{c}{)}^{\ast \ast }=(\mathit{d}{)}^{\ast \ast },\mathit{d}\in \mathit{A}\})\\ & =& \underset{\mathit{c}\in (\mathit{y}{)}^{\ast \ast }}{\bigwedge }(\bigvee \{\mathit{\eta }(\mathit{d}):(\mathit{c}{)}^{\ast \ast }=(\mathit{d}{)}^{\ast \ast }\subseteq (\mathit{y}{)}^{\ast \ast }\})\\ & \ge & \underset{\mathit{c}\in (\mathit{y}{)}^{\ast \ast }}{\bigwedge }(\bigvee \{\mathit{\eta }(\mathit{d}):(\mathit{d}{)}^{\ast \ast }=(\mathit{y}{)}^{\ast \ast }\})\\ & =& \underset{\mathit{c}\in (\mathit{y}{)}^{\ast \ast }}{\bigwedge }{\eta }^{\mathit{\alpha }}(\mathit{y})\\ & =& {\eta }^{\mathit{\alpha }}(\mathit{y}).\end{array}$

Thus, ${\eta }^{\mathit{\alpha }}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. To show η^α is the smallest α-fuzzy ideal containing η, let $\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ such that $\mathit{\eta }\subseteq \mathit{\lambda }$ and $\mathit{y}\in \mathit{A}$. Then ${\eta }^{\mathit{\alpha }}(\mathit{y})=\bigvee \{\mathit{\eta }(\mathit{c}):(\mathit{y}{)}^{\ast }=(\mathit{c}{)}^{\ast }\}\le \bigvee \{\mathit{\lambda }(\mathit{c}):(\mathit{y}{)}^{\ast }=(\mathit{c}{)}^{\ast }\}$. Since $\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ and $(\mathit{y}{)}^{\ast \ast }=(\mathit{c}{)}^{\ast \ast }$, $\mathit{\lambda }(\mathit{c})=\mathit{\lambda }(\mathit{y})$. Which implies ${\eta }^{\mathit{\alpha }}\subseteq \mathit{\lambda }$. Hence, η^α is the smallest α-fuzzy ideal containing η.

Corollary 3.6.

A fuzzy ideal η of A is an α-fuzzy ideal if and only if $\mathit{\eta }={\eta }^{\mathit{\alpha }}$.

Example 3.7.

If we define a fuzzy subset γ on the C-algebra A in Example 2.2 as,

$\begin{array}{l}\mathit{\gamma }(\mathit{b})=\mathit{\gamma }(\mathit{e})=1,\mathit{\gamma }(\mathit{d})=0.6,\text{ and }\mathit{\gamma }(\mathit{a})=\mathit{\gamma }(\mathit{c})=\mathit{\gamma }(\mathit{f}))=0.4.\end{array}$

then we can easily show that γ is an α fuzzy ideal of A.

Theorem 3.8.

If $\mathit{\eta }\in \mathit{FI}(\mathit{A})$, then

$\begin{array}{l}[{\eta }^{\mathit{\alpha }}{]}_{\mathit{\beta }}=\bigcup \{\bigcap _{\mathit{\gamma }\in \mathit{M}}[{\eta }_{\mathit{\gamma }}{]}^{\mathit{\alpha }}:\mathit{\beta }\le \bigvee \mathit{M},\mathit{M}\subseteq \mathit{L}\}.\end{array}$

Proof.

Put $\mathit{X}=\bigcup \{\bigcap _{\mathit{\gamma }\in \mathit{M}}[{\eta }_{\mathit{\gamma }}{]}^{\mathit{\alpha }}:\mathit{\beta }\le \bigvee \mathit{M},\mathit{M}\subseteq \mathit{L}\}$ and $\mathit{Y}=[{\eta }^{\mathit{\alpha }}{]}_{\mathit{\beta }}$. To show X = Y, let $\mathit{c}\in \mathit{Y}$. Then $\mathit{\beta }\le \bigvee \{\mathit{\eta }(\mathit{d}):(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast }\}$. Let us put $\mathit{M}=\{\mathit{\eta }(\mathit{d}):(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast }\}$. Then $\mathit{M}\subseteq \mathit{L}$ and $\mathit{\beta }\le \bigvee \mathit{M}$. If $\mathit{\gamma }\in \mathit{M}$, then $\mathit{\gamma }=\mathit{\eta }(\mathit{e})$ for some $\mathit{e}\in \mathit{A}$ such that $(\mathit{e}{)}^{\ast }=(\mathit{c}{)}^{\ast }$. Which implies that $\mathit{b}\in {\eta }_{\mathit{\gamma }}$. Thus $\mathit{c}\in ({\eta }_{\mathit{\gamma }}{)}^{\mathit{\alpha }}$ for all $\mathit{\gamma }\in \mathit{M}$ and . Which implies $\mathit{c}\in \bigcap _{\mathit{\gamma }\in \mathit{M}}({\eta }_{\mathit{\gamma }}{)}^{\mathit{\alpha }}$. Therefore, $\mathit{Y}\subseteq \mathit{X}$. To show the other inclusion, let $\mathit{c}\in \mathit{X}$. Then, there is $\mathit{M}\subseteq \mathit{L}$ such that $\mathit{\beta }\le \bigvee \mathit{M}$ and $\mathit{c}\in \bigcap _{\mathit{\gamma }\in \mathit{M}}({\eta }_{\mathit{\gamma }}{)}^{\mathit{\alpha }}$. Thus, for each $\mathit{\gamma }\in \mathit{M}$ there is $\mathit{d}\in {\eta }_{\mathit{\gamma }}$ such that $(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast }$. Since $\bigvee \{\mathit{\eta }(\mathit{d}):(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast }\}\ge \mathit{\eta }(\mathit{d})$, we get that ${\eta }^{\mathit{\alpha }}(\mathit{c})\ge \mathit{\eta }(\mathit{d})$ for all $\mathit{d}\in {\eta }_{\mathit{\gamma }},\mathit{\gamma }\in \mathit{M}$. Thus, ${\eta }^{\mathit{\alpha }}(\mathit{c})\ge \bigvee \mathit{M}\ge \mathit{\beta }$ and $\mathit{c}\in \mathit{Y}$. Therefore, X = Y.

Lemma 3.9.

Let $\mathit{\eta },\mathit{\lambda }\in \mathit{FI}(\mathit{A})$. Then

$\begin{array}{l}(\mathit{\eta }\cap \mathit{\lambda }{)}^{\mathit{\alpha }}={\eta }^{\mathit{\alpha }}\cap {\lambda }^{\mathit{\alpha }}.\end{array}$

Lemma 3.10.

If $\mathit{\eta }\in \mathit{FI}(\mathit{A})$, then the mapping $\mathit{\eta }⟶{\eta }^{\mathit{\alpha }}$ is a closure operator on FI(A). That is,

1. $\mathit{\eta }\subseteq {\eta }^{\mathit{\alpha }}$,

2. ${\eta }^{\mathit{\alpha }}\subseteq {\eta }^{\mathit{\alpha \alpha }}$,

3. $\mathit{\eta }\subseteq \mathit{\lambda }\Rightarrow {\eta }^{\mathit{\alpha }}\subseteq {\lambda }^{\mathit{\alpha }}$ for all $\mathit{\eta },\mathit{\lambda }\in \mathit{FI}(\mathit{A})$.

Theorem 3.11.

Let $\mathit{\eta }\in \mathit{FI}(\mathit{A})$. Then $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ if and only if for each $\mathit{e},\mathit{f}\in \mathit{A}$, $(\mathit{e}{)}^{\ast }=(\mathit{f}{)}^{\ast }$ imply that $\mathit{\eta }(\mathit{e})=\mathit{\eta }(\mathit{f})$.

Proof.

Suppose that $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Then $\mathit{\eta }(\mathit{e})=\mathit{\eta }(\mathit{f})$ for each $\mathit{e},\mathit{f}\in \mathit{A}$, $(\mathit{e}{)}^{\ast }=(\mathit{f}{)}^{\ast }$. Conversely, suppose that the condition holds. To show ${\eta }^{\mathit{\alpha }}=\mathit{\eta }$, let $\mathit{e},\mathit{f}\in \mathit{A}$ such that $(\mathit{e}{)}^{\ast }=(\mathit{f}{)}^{\ast }$. Then ${\eta }^{\mathit{\alpha }}(\mathit{e})=\bigvee \{\mathit{\eta }(\mathit{g}):(\mathit{e}{)}^{\ast }=(\mathit{f}{)}^{\ast },\mathit{g}\in \mathit{A}\}$. Since $(\mathit{e}{)}^{\ast }=(\mathit{g}{)}^{\ast }$, by the assumption, ${\eta }^{\mathit{\alpha }}(\mathit{e})=\mathit{\eta }(\mathit{f})$ for each $\mathit{e},\mathit{g}\in \mathit{A}$. Thus, by Corollary 3.6, $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Example 3.12.

Let λ be a multiplicatively closed fuzzy subset of A with

$\begin{array}{l}\bigvee \{\mathit{\lambda }(\mathit{c}):\mathit{c}\in \mathit{A}\}=1.\end{array}$

If we define a fuzzy subset λ on A as:

$\begin{array}{l}\mathit{\eta }(\mathit{c})=\bigvee \{\mathit{\lambda }(\mathit{d}):\mathit{d}\wedge \mathit{c}\in I_0,\mathit{d}\in \mathit{A}\},\end{array}$

then $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Proof.

Now, we proceed to show $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Clearly $\mathit{\eta }(\mathit{z})=1$ for any $\mathit{z}\in I_0$. Suppose $\mathit{c},\mathit{d}\in \mathit{A}$. Then

$\begin{array}{lll}\mathit{\eta }(\mathit{c})\wedge \mathit{\eta }(\mathit{d})& =& \bigvee \{\mathit{\lambda }(\mathit{e}):\mathit{e}\wedge \mathit{c}\in I_0,\mathit{f}\in \mathit{A}\}\\ & & \wedge \bigvee \{\mathit{\lambda }(\mathit{f}):\mathit{f}\wedge \mathit{d}\in I_0,\mathit{f}\in \mathit{A}\}\\ & =& \bigvee \{\mathit{\lambda }(\mathit{e}\wedge \mathit{f}):\mathit{e}\wedge \mathit{c},\mathit{f}\wedge \mathit{d}\in I_0,\mathit{e},\mathit{f}\in \mathit{A}\}\end{array}$

Since $\mathit{e}\wedge \mathit{c},\mathit{f}\wedge \mathit{d}\in I_0$, we have $(\mathit{e}\wedge \mathit{f})\wedge (\mathit{c}\vee \mathit{d})\in I_0$. Which implies that

$\begin{array}{lll}\mathit{\eta }(\mathit{c})\wedge \mathit{\eta }(\mathit{d})& \le & \bigvee \{\mathit{\lambda }(\mathit{e}\wedge \mathit{f}):(\mathit{e}\wedge \mathit{f})\wedge (\mathit{c}\vee \mathit{d})\in I_0,\mathit{e},\mathit{f}\in \mathit{A}\}\\ & \le & \bigvee \{\mathit{\lambda }(\mathit{g}):\mathit{g}\wedge (\mathit{c}\vee \mathit{d})\in I_0,\mathit{g}\in \mathit{A}\}\\ & =& \mathit{\eta }(\mathit{c}\vee \mathit{d})\end{array}$

Again, $\mathit{\eta }(\mathit{c})=\bigvee \{\mathit{\lambda }(\mathit{e}):\mathit{e}\wedge \mathit{c}\in I_0,\mathit{e}\in \mathit{A}\}$. If $\mathit{e}\wedge \mathit{c}\in I_0$, then $\mathit{a}\wedge \mathit{c}\wedge \mathit{d}\in I_0$ for any $\mathit{d}\in \mathit{A}$. Thus $\mathit{\eta }(\mathit{c}\wedge \mathit{d})\ge \mathit{\eta }(\mathit{c})\vee \mathit{\eta }(\mathit{d})$ and hence $\mathit{\eta }\in \mathit{FI}(\mathit{A})$.

To show $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, let $\mathit{e},\mathit{g}\in \mathit{A}$ such that $(\mathit{e}{)}^{\ast }=(\mathit{g}{)}^{\ast }$. Then

$\begin{array}{lll}\mathit{\eta }(\mathit{e})& =& \bigvee \{\mathit{\lambda }(\mathit{f}):\mathit{f}\wedge \mathit{e}\in I_0,\mathit{f}\in \mathit{A}\}\\ & =& \bigvee \{\mathit{\lambda }(\mathit{f}):\mathit{f}\wedge \mathit{g}\in I_0\}\\ & =& \mathit{\eta }(\mathit{g})\end{array}$

Thus, $\mathit{\eta }(\mathit{e})=\mathit{\eta }(\mathit{g})$ for each $\mathit{e},\mathit{g}\in \mathit{A}$ such that $(\mathit{e}{)}^{\ast }=(\mathit{g}{)}^{\ast }$. Therefore, $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Theorem 3.13.

If $\mathit{\eta }\in \mathit{FI}(\mathit{A})$, then $({\eta }^{\ast }{)}^{\mathit{\alpha }}={\eta }^{\ast }$, where ${\eta }^{\ast }$ is a fuzzy annihilator of η.

Proof.

Now, we need to show that $({\eta }^{\ast }{)}^{\mathit{\alpha }}\subseteq {\eta }^{\ast }$. Since ${\chi }_{{\mathit{I}}_0}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ and ${\eta }^{\ast }\cap \mathit{\eta }\subseteq {\chi }_{{\mathit{I}}_0}$, by Lemma 3.9, $({\eta }^{\ast }{)}^{\mathit{\alpha }}\cap \mathit{\eta }\subseteq {\chi }_{{\mathit{I}}_0}$. Thus, $({\eta }^{\ast }{)}^{\mathit{\alpha }}\subseteq {\eta }^{\ast }$. Therefore, ${\eta }^{\ast }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Corollary 3.14.

Every fuzzy annihilator ideal is an α-fuzzy ideal.

Lemma 3.15.

If $\mathit{\eta },\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, the supremum of η and λ is given by:

$\begin{array}{l}\mathit{\eta }\underset{―}{\vee }\mathit{\lambda }=(\mathit{\eta }\vee \mathit{\lambda }{)}^{\mathit{\alpha }}.\end{array}$

Theorem 3.16.

The set $\mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.

Proof.

Clearly, $(\mathit{F}{I}^{\mathit{\alpha }}(\mathit{A}),\subseteq )$ is a partially ordered set. For $\mathit{\eta },\mathit{\gamma }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, $\mathit{\eta }\cap \mathit{\gamma },\mathit{\eta }\underset{―}{\vee }\mathit{\gamma }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Thus $(\mathit{F}{I}^{\mathit{\alpha }}(\mathit{A}),\cap ,\underset{―}{\mathit{vee}})$ is a lattice.

Let $\mathit{\nu },\mathit{\delta },\mathit{\gamma }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Then

$\begin{array}{lll}\mathit{\nu }\underset{―}{\vee }(\mathit{\delta }\cap \mathit{\gamma })& =& ((\mathit{\nu }\vee \mathit{\delta })\cap (\mathit{\nu }\vee \mathit{\gamma }){)}^{\mathit{\alpha }}\\ & =& (\mathit{\nu }\vee \mathit{\delta }{)}^{\mathit{\alpha }}\cap (\mathit{\nu }\vee \mathit{\gamma }{)}^{\mathit{\alpha }}\\ & =& (\mathit{\nu }\underset{―}{\vee }\mathit{\delta })\cap (\mathit{\nu }\underset{―}{\vee }\mathit{\gamma }).\end{array}$

Hence $\mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ is a distributive lattice.

Since I₀ and A are α-ideals, ${\chi }_{{\mathit{I}}_0},{\chi }_{\mathit{A}}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Let $\{{\eta }_{\mathit{i}}:\mathit{i}\in \mathrm{\Delta }\}\subseteq \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$. Then $\bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, $\bigcap _{\mathit{i}\in \mathit{I}}{\eta }_{\mathit{i}}\subseteq (\bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}{)}^{\mathit{\alpha }}$ and $\bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}\subseteq {\eta }_{\mathit{i}}$ for all $\mathit{i}\in \mathrm{\Delta }$. Which implies that $(\bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}{)}^{\mathit{\alpha }}\subseteq \bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}$. Thus $(\bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}{)}^{\mathit{\alpha }}=\bigcap _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}$ and hence $(\mathit{F}{I}^{\mathit{\alpha }}(\mathit{A}),\cap ,\underset{―}{\vee })$ is a complete distributive lattice.

In Rao (2013b), M. S. Rao observed that, for any C-algebras A and $A^{{}^{\prime }}$ with smallest ideals I₀ and $I_0^{{}^{\prime }}$, respectively and $\mathit{f}:\mathit{A}⟶A^{{}^{\prime }}$ a homomorphism. f and f⁻¹ are said to be annihilator preserving if $\mathit{f}(C^{\ast })=(\mathit{f}(\mathit{C}){)}^{\ast }$ and $f^{-1}(D^{\ast })=(f^{-1}(\mathit{D}){)}^{\ast }$, where $\mathit{C}\subseteq \mathit{A}$ and $\mathit{D}\subseteq A^{{}^{\prime }}$. For any $\mathit{c}\in \mathit{A}$, $\mathit{f}((\mathit{c}])\subseteq (\mathit{f}(\mathit{c})]$. Moreover, if f is onto, then $\mathit{f}((\mathit{c}))=(\mathit{f}(\mathit{c}))$.

Theorem 3.17.

Let $\mathit{f}:\mathit{A}⟶A^{{}^{\prime }}$ be an annihilator preserving epimorphism. If $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, then $\mathit{f}(\mathit{\eta })\in \mathit{F}{I}^{\mathit{\alpha }}(A^{{}^{\prime }})$.

Proof.

If $\mathit{\eta }\in \mathit{FI}(\mathit{A})$, then $\mathit{f}(\mathit{\eta })\in \mathit{FI}(A^{{}^{\prime }})$. Now, we proceed to show

$\begin{array}{l}\underset{\mathit{d}\in (\mathit{c}{)}^{\ast \ast }}{\bigwedge }\mathit{f}(\mathit{\eta })(\mathit{d})\ge \mathit{f}(\mathit{\eta })(\mathit{c})\text{ for all }\mathit{c}\in A^{{}^{\prime }}.\end{array}$

Let $\mathit{d}\in A^{{}^{\prime }}$ and $\mathit{d}\in (\mathit{c}{)}^{\ast \ast }$. Then, there exist $\mathit{a},\mathit{e}\in \mathit{A}$ such that $\mathit{f}(\mathit{a})=\mathit{d}$ and $\mathit{f}(\mathit{e})=\mathit{c}$.

Consider the following:

$\begin{array}{ll}\mathit{f}(\mathit{\eta })(\mathit{d})& =\bigvee \{\mathit{\eta }(\mathit{a}):\mathit{f}(\mathit{a})=\mathit{d}\}\text{ and }\mathit{f}(\mathit{\eta })(\mathit{c})\\ & =\bigvee \{\mathit{\eta }(\mathit{e}):\mathit{f}(\mathit{e})=\mathit{c}\}.\end{array}$

Put,

$\begin{array}{ll}\mathit{C}& =\{\mathit{\eta }(\mathit{a})=\mathit{\gamma }:\mathit{f}(\mathit{a})=\mathit{d}\}\text{ and}\\ \mathit{D}& =\{\mathit{\eta }(\mathit{e})=\mathit{\beta }:\mathit{f}(\mathit{e})=\mathit{c}\}.\end{array}$

Now, we proceed to show $\bigvee \mathit{D}\le \bigvee \mathit{C}$. If $\mathit{\beta }\in \mathit{D}$, then there is $\mathit{z}\in \mathit{A}$ such that $\mathit{\eta }(\mathit{z})=\mathit{\beta }$ and $\mathit{f}(\mathit{z})=\mathit{c}$. Since $\mathit{d}\in (\mathit{c}{)}^{\ast \ast }$ and $\mathit{f}(\mathit{a})=\mathit{d}$, we have $\mathit{f}(\mathit{a})\in (\mathit{f}(\mathit{z}){)}^{\ast \ast }$. Since f is annihilator preserving epimorphism, it yields $\mathit{f}(\mathit{a})\in \mathit{f}((\mathit{z}{)}^{\ast \ast })$. Which implies that there is $\mathit{r}\in (\mathit{z}{)}^{\ast \ast }$ such that $\mathit{f}(\mathit{r})=\mathit{d}=\mathit{f}(\mathit{a})$. This implies $\mathit{r}\wedge \mathit{a}\in (\mathit{z}{)}^{\ast \ast }$ and $\mathit{f}(\mathit{r}\wedge \mathit{a})=\mathit{d}$. Since $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ and $\mathit{r}\wedge \mathit{a}\in (\mathit{z}{)}^{\ast \ast }$, $\mathit{\eta }(\mathit{r}\wedge \mathit{a})\ge \mathit{\eta }(\mathit{z})$. Which implies $\mathit{\eta }(\mathit{r}\wedge \mathit{a})\ge \mathit{\beta }$ and $\mathit{f}(\mathit{r}\wedge \mathit{a})=\mathit{d}$. Thus, $\bigvee \mathit{D}\le \bigvee \mathit{C}$ and hence $\mathit{f}(\mathit{\eta })\in \mathit{F}{I}^{\mathit{\alpha }}(A^{{}^{\prime }})$.

Theorem 3.18.

Let $\mathit{f}:\mathit{A}⟶A^{{}^{\prime }}$ be an annihilator preserving epimorphism. If $\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(A^{{}^{\prime }})$, then $f^{-1}(\mathit{\lambda })\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Proof.

Suppose $\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(A^{{}^{\prime }})$. Let $\mathit{i},\mathit{j}\in \mathit{A}$ and $(\mathit{i}{)}^{\ast }=(\mathit{j}{)}^{\ast }$. Since f is annihilator preserving, we have $(\mathit{f}(\mathit{i}){)}^{\ast }=(\mathit{f}(\mathit{j}){)}^{\ast }$. Let $f^{-1}(\mathit{\lambda })(\mathit{i})=\mathit{\beta }$. Then $\mathit{\lambda }(\mathit{f}(\mathit{i}))=\mathit{\beta }$. Since $\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(A^{{}^{\prime }})$, $\mathit{\lambda }(\mathit{f}(\mathit{j}))=\mathit{\beta }$. Which implies $f^{-1}(\mathit{\lambda })(\mathit{j})=\mathit{\beta }$. Thus $f^{-1}(\mathit{\lambda })(\mathit{i})=f^{-1}(\mathit{\lambda })(\mathit{j})$ for each $\mathit{i},\mathit{j}\in \mathit{A}$ such that $(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast }$. Therefore, $f^{-1}(\mathit{\lambda })\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Lemma 3.19.

Rao (2013a)

Let A₁ and A₂ be two C-algebras. For any $\mathit{a}\in A_1$ and $\mathit{b}\in A_2$, $(\mathit{a},\mathit{b}{)}^{\ast }=(\mathit{a}{)}^{\ast }=\mathit{b}{)}^{\ast }$.

Theorem 3.20.

Let A₁ and A₂ be two C-algebras. If ${\eta }_1\in \mathit{F}{I}^{\mathit{\alpha }}(A_1)$ and ${\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(A_2)$, then ${\eta }_1\times {\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Proof.

If ${\eta }_1\in \mathit{F}{I}^{\mathit{\alpha }}(A_1)$ and ${\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(A_2)$, then ${\eta }_1\times {\eta }_2\in \mathit{FI}(\mathit{A})$. To show ${\eta }_1\times {\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, let $\mathit{a},\mathit{c}\in A_1$ and $\mathit{b},\mathit{d}\in A_2$ such that $(\mathit{a},\mathit{b}{)}^{\ast }=(\mathit{c},\mathit{d}{)}^{\ast }$. Then $(\mathit{a}{)}^{\ast }=(\mathit{b}{)}^{\ast }$ and $(\mathit{c}{)}^{\ast }=(\mathit{d}{)}^{\ast }$. Now, let $({\eta }_1\times {\eta }_2)(\mathit{a},\mathit{b})=\mathit{\beta }$. Then ${\eta }_1(\mathit{a})\ge \mathit{\beta }$ and ${\eta }_2(\mathit{b})\ge \mathit{\beta }$. Since ${\eta }_1\in \mathit{F}{I}^{\mathit{\alpha }}(A_1)$ and $(\mathit{a}{)}^{\ast }=(\mathit{c}{)}^{\ast }$, ${\eta }_1(\mathit{a})={\eta }_1(\mathit{c})\ge \mathit{\beta }$. Similarly, ${\eta }_2(\mathit{b})={\eta }_2(\mathit{d})\ge \mathit{\beta }$. Which implies that $({\eta }_1\times {\eta }_2)(\mathit{c},\mathit{d})\ge ({\eta }_1\times {\eta }_2)(\mathit{a},\mathit{b})$. Similarly, we easily verify that $({\eta }_1\times {\eta }_2)(\mathit{a},\mathit{b})\ge ({\eta }_1\times {\eta }_2)(\mathit{c},\mathit{d})$. Thus $({\eta }_1\times {\eta }_2)(\mathit{a},\mathit{b})=({\eta }_1\times {\eta }_2)(\mathit{c},\mathit{d})$ and hence ${\eta }_1\times {\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$.

Theorem 3.21.

Let A₁ and A₂ be two C-algebras and assume that each of the C-algebras has the meet identity T. Then every α-fuzzy ideal of $A_1\times A_2$ can be expressed as $\mathit{\eta }={\eta }_1\times {\eta }_2$ where η₁ and η₂ are α-fuzzy ideals of A₁ and A₂, respectively.

Proof.

Suppose $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(A_1\times A_2)$. Now, we proceed to show $\mathit{\eta }={\eta }_1\times {\eta }_2$ where ${\eta }_1\in \mathit{F}{I}^{\mathit{\alpha }}(A_1)$ and ${\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(A_2)$.

Let $\mathit{a}\in A_1$ and $\mathit{c}\in A_2$. Consider the following:

$\begin{array}{ll}{\eta }_1(\mathit{a})& =\bigvee \{\mathit{\eta }(\mathit{a},\mathit{b}):\mathit{b}\in A_2\}\text{ and}\\ {\eta }_2(\mathit{c})& =\bigvee \{\mathit{\eta }(\mathit{d},\mathit{c}):\mathit{d}\in A_1\}.\end{array}$

Then η₁ and η₂ are fuzzy ideals and $\mathit{\eta }={\eta }_1\times {\eta }_2$. Now, we proceed to show that η₁ and η₂ are α-fuzzy ideals. Let $\mathit{i},\mathit{j}\in A_1$ and $(\mathit{i}{)}^{\ast }=(\mathit{j}{)}^{\ast }$. Then ${\eta }_1(\mathit{i})=\bigvee \{\mathit{\eta }(\mathit{i},\mathit{b}):\mathit{b}\in A_2\}$. Let $\mathit{a}\in A_2$. Then $(\mathit{i},\mathit{a}{)}^{\ast }=(\mathit{j},\mathit{a}{)}^{\ast }$. Since $\mathit{\eta }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$ and $(\mathit{i},\mathit{a}{)}^{\ast }=(\mathit{j},\mathit{a}{)}^{\ast }$, $\mathit{\eta }(\mathit{x},\mathit{a})=\mathit{\eta }(\mathit{y},\mathit{a})$. Now,

$\begin{array}{ll}{\eta }_1(\mathit{j})& =\bigvee \{\mathit{\eta }(\mathit{j},\mathit{c}):\mathit{c}\in A_2\}\\ & =\bigvee \{\mathit{\eta }(\mathit{i},\mathit{c}):\mathit{c}\in A_2\}\\ & ={\eta }_1(\mathit{i}).\end{array}$

Thus ${\eta }_1(\mathit{i})={\eta }_1(\mathit{j})$ for each $(\mathit{i}{)}^{\ast }=(\mathit{j}{)}^{\ast },\mathit{i},\mathit{j}\in A_1$ and hence ${\eta }_1\in \mathit{F}{I}^{\mathit{\alpha }}(A_1)$. Similarly, we can easily verify that ${\eta }_2\in \mathit{F}{I}^{\mathit{\alpha }}(A_2)$.

Theorem 3.22.

The following are equivalent.

1. Every fuzzy ideal is an α-fuzzy ideal.

2. Every prime fuzzy ideal is an α-fuzzy ideal.

3. For $\mathit{a},\mathit{b}\in \mathit{A}$, $(\mathit{a}{)}^{\ast }=(\mathit{b}{)}^{\ast }$ implies $\mathit{\eta }(\mathit{a})=\mathit{\eta }(\mathit{b})$ for all fuzzy ideal η of A.

4. Space of prime α-fuzzy ideals

The purpose this section is to study the space of prime α-fuzzy ideals of C-algebras. We prove that the collection $\{\mathit{\kappa }(\mathit{\eta }):\mathit{\eta }\in \mathit{FI}(\mathit{A})\}$ is a topology on Ω and $\mathcal{B}=\{\mathit{\kappa }(c_{\mathit{\beta }}):\mathit{c}\in \mathit{A},\mathit{\beta }\in \mathit{L}-\{0\}\}$ is a basis for a topology on Ω. We also show that the space Ω is a T₀ space. Moreover, we established a necessary and sufficient condition for the space Ω is a T₁ space.

Let Ω be the set of all prime α-fuzzy ideals of A. For any fuzzy subset η of A, $\mathit{\tau }(\mathit{\eta })=\{\mathit{\lambda }\in \mathrm{\Omega }:\mathit{\eta }\subseteq \mathit{\lambda }\}$ and $\mathit{\kappa }(\mathit{\eta })=\{\mathit{\lambda }\in \mathrm{\Omega }:\mathit{\eta }⊈\mathit{\lambda }\}=\mathrm{\Omega }-\mathit{\tau }(\mathit{\eta })$. We let ${\lambda }_{\ast }=\{\mathit{a}\in \mathit{A}:\mathit{\lambda }(\mathit{a})=1\}$.

Lemma 4.1.

If $\mathit{\eta },\mathit{\lambda }\in \mathit{FI}(\mathit{A})$, then

1. $\mathit{\eta }\subseteq \mathit{\lambda }\Rightarrow \mathit{\kappa }(\mathit{\eta })\subseteq \mathit{\kappa }(\mathit{\lambda })$.

2. $\mathit{\kappa }(\mathit{\eta }\cap \mathit{\lambda })=\mathit{\kappa }(\mathit{\eta })\cap \mathit{\kappa }(\mathit{\lambda })$.

3. $\mathit{\kappa }(\mathit{\eta }\vee \mathit{\lambda })=\mathit{\kappa }(\mathit{\eta })\cup \mathit{\kappa }(\mathit{\lambda })$.

Proof.

Let $\mathit{\eta },\mathit{\lambda }\in \mathit{FI}(\mathit{A})$

1. If $\mathit{\eta }\subseteq \mathit{\lambda }$ and $\mathit{\gamma }\in \mathit{\kappa }(\mathit{\eta })$, then $\mathit{\eta }⊈\mathit{\gamma }$. Which implies $\mathit{\lambda }⊈\mathit{\gamma }$.

2. Since $\mathit{\eta }\cap \mathit{\lambda }\subseteq \mathit{\eta },\mathit{\lambda }$, we have $\mathit{\kappa }(\mathit{\eta }\cap \mathit{\lambda })\subseteq \mathit{\kappa }(\mathit{\eta })\cap \mathit{\kappa }(\mathit{\lambda })$. Conversely, let $\mathit{\gamma }\in \mathit{\kappa }(\mathit{\eta })\cap \mathit{\kappa }(\mathit{\lambda })$. Then $\mathit{\eta }⊈\mathit{\gamma }$ and $\mathit{\lambda }⊈\mathit{\gamma }$. Since $\mathit{\gamma }\in \mathrm{\Omega }$, we have $\mathit{\eta }\cap \mathit{\lambda }⊈\mathit{\gamma }$.

3. Clearly, $\mathit{\kappa }(\mathit{\eta })\cup \mathit{\kappa }(\mathit{\lambda })\subseteq \mathit{\kappa }(\mathit{\eta }\vee \mathit{\lambda })$. If $\mathit{\gamma }\in \mathit{\kappa }(\mathit{\eta }\vee \mathit{\lambda })$, then $\mathit{\eta }\vee \mathit{\lambda }⊈\mathit{\gamma }$. Thus, either $\mathit{\eta }⊈\mathit{\gamma }$ or $\mathit{\lambda }⊈\mathit{\gamma }$. Therefore, $\mathit{\kappa }(\mathit{\eta })\cup \mathit{\kappa }(\mathit{\lambda })=\mathit{\kappa }(\mathit{\eta }\vee \mathit{\lambda })$.

Lemma 4.2.

for any fuzzy subset η of A, $\mathit{\kappa }(\mathit{\eta })=\mathit{\kappa }(⟨\mathit{\eta }⟩)$.

Proof.

Since $\mathit{\eta }\subseteq ⟨\mathit{\eta }⟩$, we have $\mathit{\kappa }(\mathit{\eta })\subseteq \mathit{\kappa }(⟨\mathit{\eta }⟩)$. Conversely, let $\mathit{\gamma }\in ⟨\mathit{\eta }⟩$. Then $⟨\mathit{\eta }⟩⊈\mathit{\gamma }$. Suppose that $\mathit{\eta }\subseteq \mathit{\gamma }$. Then $⟨\mathit{\eta }⟩\subseteq \mathit{\gamma }$. Which is impossible. Thus $\mathit{\eta }⊈\mathit{\gamma }$ and hence $\mathit{\kappa }(\mathit{\eta })=\mathit{\kappa }(⟨\mathit{\eta }⟩)$.

Lemma 4.3.

Let $\mathit{c},\mathit{d}\in \mathit{A}$ and $\mathit{\beta }\in \mathit{L}-\{0\}$. Then

1. $\mathit{\kappa }((\mathit{c}\wedge \mathit{d}{)}_{\beta })=\mathit{\kappa }(c_{\beta })\cap \mathit{\kappa }(d_{\beta }).$

2. $\mathrm{\Omega }=\bigcup _{\mathit{c}\in \mathit{A},\mathit{\beta }\in \mathit{L}-\{0\}}\mathit{\kappa }(c_{\beta }).$

Proof.

1. Let $\mathit{\eta }\in \mathit{\kappa }((\mathit{c}\wedge \mathit{d}{)}_{\mathit{\beta }})$. Then $(\mathit{c}\wedge \mathit{d}{)}_{\mathit{\beta }}⊈\mathit{\eta }$. Then $\mathit{\beta }>\mathit{\eta }(\mathit{c}\wedge \mathit{d})$ and $\mathit{\beta }>\mathit{\eta }(\mathit{c})$, $\mathit{\beta }>\mathit{\eta }(\mathit{d})$. Which implies that $c_{\mathit{\beta }}⊈\mathit{\eta }$ and $d_{\mathit{\beta }}⊈\mathit{\eta }$. Thus $\mathit{\eta }\in \mathit{\kappa }(c_{\mathit{\beta }})\cap \mathit{\kappa }(d_{\mathit{\beta }})$. To show the other inclusion, let $\mathit{\gamma }\in \mathit{\kappa }(c_{\mathit{\beta }})\cap \mathit{\kappa }(d_{\mathit{\beta }})$. Then $c_{\mathit{\beta }}⊈\mathit{\gamma }$ and $d_{\mathit{\beta }}⊈\mathit{\gamma }$. Which implies that $\mathit{\beta }>\mathit{\gamma }(\mathit{c})$ and $\mathit{\beta }>\mathit{\gamma }(\mathit{d})$. This shows that $\mathit{c},\mathit{d}\notin {\gamma }_{\ast }$. Since γ is prime fuzzy ideal, we have ${\gamma }_{\ast }$ prime ideal and $\mathit{c}\wedge \mathit{d}\notin {\gamma }_{\ast }$. Thus $\mathit{\beta }>\mathit{\gamma }(\mathit{c}\wedge \mathit{d})$ and hence $\mathit{\gamma }\in \mathit{\kappa }((\mathit{c}\wedge \mathit{d}{)}_{\mathit{\beta }})$.

2. Clearly, $\bigcup _{\mathit{c}\in \mathit{A},\mathit{\beta }\in \mathit{L}-\{0\}}\mathit{\kappa }(c_{\mathit{\beta }})\subseteq \mathrm{\Omega }$. Let $\mathit{\eta }\in \mathrm{\Omega }$. Then $\mathit{Im\eta }=\{\mathit{\beta },1\}$, $\mathit{\beta }\in \mathit{L}-\{1\}$. Which implies that there is c in A such that $\mathit{\eta }(\mathit{c})=\mathit{\beta }$. Let $\mathit{\gamma }\in \mathit{L}-\{0\}$ such that $\mathit{\gamma }>\mathit{\beta }$. Then $c_{\mathit{\gamma }}⊈\mathit{\eta }$. Thus $\mathit{\eta }\in \bigcup _{\mathit{c}\in \mathit{A},\mathit{\beta }\in \mathit{L}-\{0\}}\mathit{\kappa }(c_{\mathit{\beta }})$.

Lemma 4.4.

If $\mathit{\eta }\in \mathit{FI}(\mathit{A})$, then $\mathit{\kappa }(\mathit{\eta })=\mathit{\kappa }({\eta }^{\mathit{\alpha }})$.

Proof.

Since $\mathit{\eta }\subseteq {\eta }^{\mathit{\alpha }}$, we have $\mathit{\kappa }(\mathit{\eta })\subseteq \mathit{\kappa }({\eta }^{\mathit{\alpha }})$. Conversely, suppose $\mathit{\lambda }\in \mathit{\kappa }({\eta }^{\mathit{\alpha }})$. Then ${\eta }^{\mathit{\alpha }}⊈\mathit{\lambda }$. Assume that $\mathit{\lambda }\notin \mathit{\kappa }(\mathit{\eta })$. Then $\mathit{\eta }\subseteq \mathit{\lambda }$. Since $\mathit{\lambda }\in \mathit{F}{I}^{\mathit{\alpha }}(\mathit{A})$, we have ${\eta }^{\mathit{\alpha }}\subseteq \mathit{\lambda }$. Which is a contradiction. Thus $\mathit{\kappa }(\mathit{\eta })=\mathit{\kappa }({\eta }^{\mathit{\alpha }})$.

Lemma 4.5.

Let $\{{\eta }_{\mathit{i}}:\mathit{i}\in \mathrm{\Delta }\}\subseteq \mathit{FI}(\mathit{A})$. Then

$\begin{array}{l}\bigcap _{\mathit{i}\in \mathrm{\Delta }}\mathit{\tau }({\eta }_{\mathit{i}})=\mathit{\tau }(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}⟩).\end{array}$

Proof.

Clearly, $\mathit{\tau }(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}⟩)\subseteq \bigcap _{\mathit{i}\in \mathrm{\Delta }}\mathit{\tau }({\eta }_{\mathit{i}})$. Conversely, let, $\mathit{\lambda }\in \bigcap _{\mathit{i}\in \mathrm{\Delta }}\mathit{\tau }({\eta }_{\mathit{i}})$. Then ${\eta }_{\mathit{i}}\subseteq \mathit{\lambda }$ for all $\mathit{i}\in \mathrm{\Delta }$. Which implies that $⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}⟩\subseteq \mathit{\lambda }$ and $\mathit{\lambda }\in \mathit{\tau }(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}⟩)$. Thus $\bigcap _{\mathit{i}\in \mathrm{\Delta }}\mathit{\tau }({\eta }_{\mathit{i}})\subseteq \mathit{\tau }(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}⟩)$ and hence $\bigcap _{\mathit{i}\in \mathrm{\Delta }}\mathit{\tau }({\eta }_{\mathit{i}})=\mathit{\tau }(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\eta }_{\mathit{i}}⟩)$.

Theorem 4.6.

The collection $\mathcal{F}=\{\mathit{\kappa }(\mathit{\eta }):\mathit{\eta }\in \mathit{FI}(\mathit{A})\}$ is a topology on Ω.

Proof.

Let λ₁ and λ₂ be fuzzy subsets of A such that ${\lambda }_1(\mathit{c})=0$ and ${\lambda }_2(\mathit{c})=1$ for all $\mathit{c}\in \mathit{A}$. Then $⟨{\lambda }_1⟩,{\lambda }_2\in \mathit{FI}(\mathit{A})$ and $\mathit{\kappa }({\lambda }_1)=\mathit{\varphi }$ and $\mathit{\kappa }({\lambda }_2)=\mathrm{\Omega }$. Thus $\mathit{\varphi },\mathrm{\Omega }\in \mathcal{F}$. For any fuzzy ideals λ₁ and λ₂ of A, we have $\mathit{\kappa }({\lambda }_1)\cap \mathit{\kappa }({\lambda }_2)=\mathit{\kappa }({\lambda }_1\cap {\lambda }_2)$. Which implies that $\mathcal{F}$ is closed under finite intersections.

Let $\{\mathit{a}{\lambda }_{\mathit{i}}:\mathit{i}\in \mathrm{\Delta }\}\subseteq \mathit{FI}(\mathit{A})$. Then $\bigcap _{\mathit{i}\in \mathrm{\Delta }}\mathit{\tau }({\lambda }_{\mathit{i}})=(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\lambda }_{\mathit{i}}⟩)$. This shows that $\bigcup _{\mathit{i}\in \mathrm{\Delta }}\mathit{\kappa }({\lambda }_{\mathit{i}})=\mathit{\kappa }(⟨\bigcup _{\mathit{i}\in \mathrm{\Delta }}{\lambda }_{\mathit{i}}⟩)$. Hence, $\mathcal{F}$ is closed under arbitrary union and hence it is a topology on Ω.

Definition 4.7.

The topological space $(\mathrm{\Omega },\mathcal{F})$ is called the space of prime α-fuzzy ideals of A.

Lemma 4.8.

The subfamily $\mathcal{B}=\{\mathit{\kappa }(c_{\mathit{\beta }}):\mathit{c}\in \mathit{A},\mathit{\beta }\in \mathit{L}-\{0\}\}$ of $\mathcal{F}$ is a basis of $\mathcal{F}$.

Proof.

Let $\mathit{\kappa }(\mathit{\eta })\in \mathcal{F})$ and $\mathit{\lambda }\in \mathit{\kappa }(\mathit{\eta })$. Then $\mathit{\eta }⊈\mathit{\lambda }$. Which implies that there is $\mathit{c}\in \mathit{A}$ such that $\mathit{\eta }(\mathit{c})>\mathit{\lambda }(\mathit{c})$. Let $\mathit{\eta }(\mathit{c})=\mathit{\beta }$. Then $\mathit{\lambda }\in \mathit{\kappa }(c_{\mathit{\beta }})$. To show $\mathit{\kappa }(c_{\mathit{\beta }})\subseteq \mathit{\kappa }(\mathit{\eta })$, let $\mathit{\theta }\in \mathit{\kappa }(c_{\mathit{\beta }})$. Then $\mathit{\eta }(\mathit{c})>\mathit{\theta }(\mathit{c})$. This shows that $\mathit{\theta }\in \mathit{\kappa }(\mathit{\eta })$. Thus $\mathit{\kappa }(c_{\mathit{\beta }})\subseteq \mathit{\kappa }(\mathit{\eta })$ and hence $\mathcal{B}$ forms a basis for $\mathcal{F}$.

Lemma 4.9.

Let $\mathcal{P}\subseteq \mathrm{\Omega }$. The closure of $\mathcal{P}$ is given by $\bar{\mathcal{P}}=\mathit{\tau }(\bigcap _{\mathit{\eta }\in \mathcal{P}}\mathit{\eta })$.

Definition 4.10.

A proper α-fuzzy ideal η of A is called maximal if there is no proper α-fuzzy ideal λ such that $\mathit{\eta }\subseteq \mathit{\lambda }$.

Lemma 4.11.

The space Ω is a T₀-space.

Theorem 4.12.

The space Ω is a T₁-space if and only if every prime α-fuzzy ideal is maximal.

Proof.

Let the space Ω is a T₁-space and $\mathit{\eta }\in \mathrm{\Omega }$. Assume that η is not maximal. Then, there is a maximal α-fuzzy ideal λ such that $\mathit{\eta }\subset \mathit{\lambda }$. Since Ω is a T₁ space, there exist two open sets $\mathit{\kappa }(c_{\mathit{\beta }})$ and $\mathit{\kappa }(d_{\mathit{\gamma }})$ such that $\mathit{\eta }\in \mathit{\kappa }(c_{\mathit{\beta }})$, $\mathit{\lambda }\in \mathit{\kappa }(d_{\mathit{\gamma }})$ and $\mathit{\lambda }\notin \mathit{\kappa }(c_{\mathit{\beta }})$, $\mathit{\eta }\notin \mathit{\kappa }(d_{\mathit{\gamma }})$. This implies that $d_{\mathit{\gamma }}\subseteq \mathit{\eta }$ and $d_{\mathit{\gamma }}⊈\mathit{\lambda }$. Which is a contradiction. Therefore, η is a maximal α-fuzzy ideal. Conversely, suppose that every prime α-fuzzy ideal is maximal. Let ${\eta }_1,{\eta }_2\in \mathrm{\Omega }$. By the assumption, ${\eta }_1\ne {\eta }_2$. Which implies $\mathit{c}\in \mathit{A}$ such that ${\eta }_1(\mathit{c})>{\eta }_2(\mathit{c})$. Put ${\eta }_1(\mathit{c})=\mathit{\beta }$. Then $c_{\mathit{\beta }}\subseteq {\eta }_1$ and $c_{\mathit{\beta }}⊈{\eta }_2$. Thus $\mathit{\kappa }(c_{\mathit{\beta }})$ is an open set containing η₂ but not η₁ and hence Ω is a T₁-space.

5. Conclusion

The concept of annulates and α-ideals of C-algebra was introduced by M. S. Rao Rao (2013a). They characterized α-ideals in terms of annulates and minimal prime ideals. Motivated by this work in this paper we study the fuzzy version of α-ideals of C-algebras.

In this paper, we have explored the concept of α-fuzzy ideals in C-algebras. Our study has provided valuable insights into the properties and characteristics of α-fuzzy ideals, contributing to the broader field of fuzzy algebraic structures. We have presented several characterizations for fuzzy ideals to be α-fuzzy ideals in C-algebras. These characterizations allow us to identify the conditions under which a fuzzy ideal can be considered an α-fuzzy ideal. By establishing these characterizations, we have enhanced our understanding of the relationship between fuzzy ideals and α-fuzzy ideals in C-algebras. Furthermore, we have investigated the smallest α-fuzzy ideal containing a given fuzzy ideal of a C-algebra A. Moreover, we have provided a set of equivalent conditions for a given C-algebra, under which every fuzzy ideal becomes an α-fuzzy ideal. Finally, we have studied the space of prime α-fuzzy ideals in C-algebras and derived a necessary and sufficient condition for this space to be a T₁ space. Our future work will focus on fuzzy semiprime ideals in general lattices.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

No data were used to support this study.

Supplementary Material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2024.2352918