Characterization of Lattices in Terms of (∈γ,∈γ∨qδ)-Fuzzy Ideals

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Characterization of Lattices in Terms of (∈γ,∈γ∨qδ)-Fuzzy Ideals

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Submitted:

26 October 2024

Posted:

28 October 2024

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Abstract

In this paper, $(\alpha,\beta)$-fuzzy sublattices and $(\alpha,\beta)$-fuzzy ideals of a lattice, where $\alpha\in\{ \in_{\gamma}, q_{\delta}, \in_{\gamma}\vee q_{\delta}\}$, $\beta\in\{ \in_{\gamma}, q_{\delta}, \in_{\gamma}\vee q_{\delta},\in_{\gamma}\wedge q_{\delta}\}$, are introduced and studied. In the special case, $(\in_{\gamma}, \in_{\gamma}\vee q_{\delta})$-fuzzy sublattices and $(\in_{\gamma}, \in_{\gamma}\vee q_{\delta})$-fuzzy ideals of a lattice are presented and their characterizations by three level sublattices and ideals are given.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

MSC: 06D72

1. Introduction

Zadeh introduced the concept of fuzzy sets for the first time in [1]. After that different fuzzy structures have been introduced and investigated. Rosenfeld introduced fuzzy groups and subgroups in [2]. Wang [3] introduced the notion of fuzzy ring, and Kuroki studied fuzzy semigroups in [4]. The concepts of fuzzy sublattices and fuzzy ideals of a lattice were presented in [5]. Bhakas and Das [6,7] introduced the concept of

(α, β)

-fuzzy subgroup. Shabir and Ali [8,9] studied

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy left (right, two-sided, interior) ideals of semigroups and characterized regular, intra-regular and semisimple semigroups by the properties of these fuzzy ideals.

In this paper,

(\in_{γ}, \in_{γ} \lor q_{δ}) -

fuzzy ideals of a lattice is defined, and sublattices are characterized by the properties of the fuzzy ideals.

2. Preliminaries

In this section we recall some notations and terminology which will be used in the sequel.

The notions of lattice, sublattice and ideal of a lattice are well known and can be found in the books [10,11]. Recall that a subset I of a lattice L is an ideal in L if

(i)

a, b \in I ⟹ a \lor b \in I

(ii)

a \in I

b \in L

and

b \leq a ⟹ b \in I

We denote by $(L, \lor, \land)$ (or simply by L) a lattice, by I an ideal in L, and by $I (L)$ the set of all ideals of L. Notice that $(I (L), \subseteq, \cup, \cap)$ is a distributive lattice. $([0, 1], \lor, \land)$ is a complete lattice, where [0,1] is the unit segment of real numbers, and $x \lor y = max (x, y)$ , $x \land y = min (x, y)$ .

2.1. Fuzzy Sublattices (Ideals) of a Lattice

A fuzzy subset

μ

of L is a function

μ : L ⟶ [0, 1]

. This function is called a membership function. The set of all fuzzy subsets of L is denoted by

P (L)

. Let

μ, λ \in P (L)

. The fuzzy subsets

μ \land λ

and

μ \lor λ

are defined as follows:

(μ \land λ) (x) = min {μ (x), λ (x)}; (μ \lor λ) (x) = max {μ (x), λ (x)} .

We say that

μ \leq λ

, if

μ (x) \leq λ (x),

for all

x \in L .

{μ_{s}}_{i \in S}

is a family of fuzzy subsets of L, then for all

x \in L

, we define:

(⋂_{s \in S} μ_{s}) (x) = ⋀_{i \in I} {μ_{i} (x)}; (⋃_{i \in I} μ_{i}) (x) = ⋁_{i \in I} {μ_{i} (x)} .

Definition 2.1

([12]). Let μ be a fuzzy subset in L. Then μ is called a fuzzy sublattice of L if for all

x, y \in L

(i)

μ (x \land y) \geq μ (x) \land μ (y)

(ii)

μ (x \lor y) \geq μ (x) \land μ (y)

or equivalently

μ (x \land y) \land μ (x \lor y) \geq μ (x) \land μ (y)

, for all

x, y \in L

Definition 2.2

([12]). Let μ be a fuzzy sublattice of L. Then μ is called a fuzzy ideal of L if

x \leq y \Rightarrow μ (x) \geq μ (y)

for all

x, y \in L

Let L be a lattice and

χ_{I}

be the characteristic function of a subset I of L. The

χ_{I}

is a fuzzy sublattice if and only if I is a sublattice.

Definition 2.3.

For

α \in [0, 1]

, the set

μ_{α} = {x \in L, μ (x) \geq α}

is called

α -

level subset of

μ

For any fuzzy subset

μ

of L, the set

{x \in L, μ (x) > 0}

is called the support of

μ

, and is denoted by

s u p p μ

Definition 2.4.

Let

t \in [0, 1]

and

x \in L

. A fuzzy set μ in L is defined by

μ (y) = \{\begin{matrix} t, & if x = y; \\ 0, & otherwise \end{matrix}

for all $y \in L$ , is called a fuzzy point and denoted by $x_{t}$ , where the point x is called its support point, and t is called its value.

2.2. $(\in, \in \lor q)$ -Fuzzy Sublattice and Ideal of a Lattice

A fuzzy point

x_{t}

is said to belong to (resp., be quasi-coincident with) a fuzzy set

μ

, written as

x_{t} \in μ

(resp.,

x_{t} q μ

) if

μ (x) \geq t

(resp.

μ (x) + t > 1

). If

μ (x) \geq t

x_{t} q μ

, then we write

x_{t} (\in \lor q) μ

. The symbol

\bar{\in \lor q}

means

\in \lor q

does not hold.

Definition 2.5

([13]). A fuzzy subset μ of a lattice L is said to be an

(\in, \in \lor q)

-fuzzy sublattice of L if for all

t, r \in (0, 1]

and

x, y \in L

(i)

x_{t}, y_{r} \in μ

implies

{(x \lor y)}_{t \land r} (\in \lor q) μ

(ii)

x_{t}, y_{r} \in μ

implies

{(x \land y)}_{t \land r} (\in \lor q) μ

μ is called an

(\in, \in \lor q)

-fuzzy idealof L if μ is an

(\in, \in \lor q)

-fuzzy sublattice of L and

(iii)

x_{t} \in μ

and

y \leq x

implies

y_{t} (\in \lor q) μ

([13]).Theorem 2.1 ( Conditions (i)-(iii) in the above definition are equivalent to the following conditions, respectively:

(1)

μ (x) \land μ (y) \land 0.5 \leq μ (x \lor y)

(2)

μ (x) \land μ (y) \land 0.5 \leq μ (x \land y)

(3)

y \leq x

implies

μ (x) \land 0.5 \leq μ (y)

for all

x, y \in L

Remark 2.1.

1) ([14]) Not every fuzzy subset of L need be

(\in, \in \lor q)

-fuzzy sublattice of L.

2) ([14]) Not every

(\in, \in \lor q)

-fuzzy sublattice of L need be

(\in, \in \lor q)

-fuzzy ideal of L.

3) ([14]) Any fuzzy ideal of L is an

(\in, \in \lor q)

-fuzzy ideal of L.

4) ([13]) There is an

\in, \in \lor q)

-fuzzy ideal of L, which is not a fuzzy ideal.

Theorem 2.2

([13]). Let μ is a fuzzy subset of L.

μ_{t} \neq Ø

is a sublattice (ideal) of L, for all

0 < t \leq 0.5

, if and only if μ is an

(\in, \in \lor q)

-fuzzy sublattice (ideal) of L.

Theorem 2.3

([14]). A non-empty subset I of L is an ideal of L if and only if the characteristic function

χ_{I}

is an

(\in, \in \lor q)

-fuzzy ideal of L.

3. $(α, β)$ -Fuzzy Sublattices and Ideals of a Lattice

This section gives information about

(α, β)

-fuzzy sublattices for

α, β \in {\in_{γ}, q_{δ}, \in_{γ} \lor q_{δ}, \in_{γ} \land q_{δ}}

and

α \neq \in_{γ} \land q_{δ}

(see the next definition).

Definition 3.1

([8]). Let

γ, δ, t \in [0, 1]

be such that

γ < δ

. For a fuzzy point

x_{t}

and a fuzzy subset μ of L, we define:

x_{t} \in_{γ} μ

μ (x) \geq t > γ

;

x_{t} q_{δ} μ

μ (x) + t > 2 δ

;

x_{t} (\in_{γ} \lor q_{δ}) μ

x_{t} \in_{γ} μ

x_{t} q_{δ} μ

;

x_{t} (\in_{γ} \land q_{δ}) μ

x_{t} \in_{γ} μ

and

x_{t} q_{δ} μ

;

x_{t} \bar{α}

x_{t} α μ

does not hold for

α \in {\in_{γ}, q_{δ}, \in_{γ} \lor q_{δ}, \in_{γ} \land q_{δ}}

In this paper, we will consider

γ, δ \in [0, 1]

, where

γ < δ

and

α, β \in {\in_{γ}, q_{δ}, \in_{γ} \lor q_{δ}, \in_{γ} \land q_{δ}}

and

α \neq \in_{γ} \land q_{δ} .

It is worth noting that if

x \in L

and

t \in [0, 1]

such that

x_{t} \in_{γ} \land q_{δ} μ,

then

μ (x) \geq t > γ

and

μ (x) + t > 2 δ

. Hence

2 δ < μ (x) + t \leq μ (x) + μ (x) = 2 μ (x) .

Consequently,

μ (x) > δ

Definition 3.2.

A fuzzy subset μ of a lattice L is said to be an

(α, β)

-fuzzy sublattice of L if for all

t, r \in (0, 1]

and

x, y \in L

the following hold;

(i)

x_{t}, y_{r} α μ

implies

{(x \lor y)}_{t \land r} β μ

(ii)

x_{t}, y_{r} α μ

implies

{(x \land y)}_{t \land r} β μ

μ is called an

(α, β)

-fuzzy idealof L if μ is an

(α, β)

-fuzzy sublattice of L and

(iii)

x_{t} α μ

and

y \leq x

implies

y_{t} β μ

Theorem 3.1.

2 δ = 1 + γ

and μ is an

(α, β)

-fuzzy sublattice (ideal) of L, then

μ_{γ} = {x \in L : μ (x) > γ}

is a sublattice (ideal) of L.

Proof.

Suppose

x, y \in μ_{γ}

and

x \land y \notin μ_{γ}

x \lor y \notin μ_{γ} .

Then

μ (x \land y) \leq γ or μ (x \lor y) \leq γ .

Case 1.

α \in {\in_{γ}, \in_{γ} \lor q_{δ}}

Define

t = μ (x)

and

r = μ (y) .

Hence

x_{μ (x)} α μ

and

y_{μ (y)} α μ

. It follows

μ (x \land y) \leq γ < μ (x) \land μ (y) ⟹ {(x \land y)}_{μ (x) \land μ (y)} \bar{\in_{γ}} μ or

μ (x \lor y) \leq γ < μ (x) \land μ (y) ⟹ {(x \lor y)}_{μ (x) \land μ (y)} \bar{\in_{γ}} μ,

and

μ (x \land y) + μ (x) \land μ (y) \leq γ + μ (x) \land μ (y) \leq γ + 1 = 2 δ ⟹ {(x \land y)}_{μ (x) \land μ (y)} \bar{q_{δ}} μ or

μ (x \lor y) + μ (x) \land μ (y) \leq γ + μ (x) \land μ (y) \leq γ + 1 = 2 δ ⟹ {(x \lor y)}_{μ (x) \land μ (y)} \bar{q_{δ}} μ .

So, we have

{(x \land y)}_{μ (x) \land μ (y)} \bar{β} μ or {(x \lor y)}_{μ (x) \land μ (y)} \bar{β} μ

This contradicts our assumptions.

Case 2.

α = q_{δ}

and

t = 1

Then then

μ (x) + 1 > γ + 1 = 2 δ ⟹ x_{1} q_{δ} μ and μ (y) + 1 > γ + 1 = 2 δ ⟹ y_{1} q_{δ} μ .

Hence,

μ (x \land y) + 1 \leq γ + 1 = 2 δ ⟹ {(x \land y)}_{1} \bar{q_{δ}} μ,

μ (x \lor y) + 1 \leq γ + 1 = 2 δ ⟹ {(x \lor y)}_{1} \bar{q_{δ}} μ .

Also

μ (x \land y) \leq γ < 1 ⟹ {(x \land y)}_{1} \bar{\in_{γ}} μ or

μ (x \lor y) \leq γ < 1 ⟹ {(x \lor y)}_{1} \bar{\in_{γ}} μ .

Therefore,

{(x \land y)}_{1} \bar{β} μ

{(x \lor y)}_{1} \bar{β} μ

. This is again a contradiction.

Thus

μ_{γ}

is a sublattice of L. □

The following theorem characterizes a sublattice (ideal) of a lattice.

Theorem 3.2.

Let

2 δ = 1 + γ

and A be a nonempty subset of L. Then A is a sublattice (ideal) of L if and only if the fuzzy subset μ of L defined by

μ (x) = \{\begin{matrix} t_{1} \in [δ, 1], & if x \in A; \\ t_{2} \in [0, γ], otherwise \end{matrix}

is an

(α, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

Proof. (⇒) If A is a sublattice of L, then the following cases can occur.

Case 1.

α = \in_{γ}

We have

x_{t} \in_{γ} μ ⟹ μ (x) \geq t > γ and y_{r} \in_{γ} μ ⟹ μ (y) \geq r > γ

Therefore, according to the definition of,

μ

, we conclude

x, y \in A

. Since A is a sublattice,

x \land y \in A

and

x \lor y \in A

, therefore,

μ (x \land y) \geq δ

and

μ (x \lor y) \geq δ

. Now we have two possibilities (i) and (ii) below.

(i)

t \land r \leq δ

Then

μ (x \land y) \geq δ \geq t \land r > γ and μ (x \lor y) \geq δ \geq t \land r > γ

which implies

{(x \land y)}_{t \land r} \in_{γ} μ and {(x \lor y)}_{t \land r} \in_{γ} μ

(ii)

t \land r > δ

Then

μ (x \land y) + t \land r > δ + δ = 2 δ and μ (x \lor y) + t \land r > δ + δ = 2 δ

which implies

{(x \land y)}_{t \land r} q_{δ} μ and {(x \lor y)}_{t \land r} q_{δ} μ .

We conclude from (i) and (ii) that

{(x \land y)}_{t \land r} \in_{γ} \lor q_{δ} μ and {(x \lor y)}_{t \land r} \in_{γ} \lor q_{δ} μ .

Therefore,

μ

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice.

Case 2.

α = q_{δ}

In this case we have

x_{t} q_{δ} μ ⟹ μ (x) + t > 2 δ ⟹ μ (x) > 2 δ - t > 2 δ - 1 = γ and

y_{r} q_{δ} μ ⟹ μ (y) + r > 2 δ ⟹ μ (y) > 2 δ - r > 2 δ - 1 = γ .

As in Case 1 we conclude

x, y \in A

. Since A is a sublattice,

x \land y \in A

and

x \lor y \in A

, hence

μ (x \land y) \geq δ

and

μ (x \lor y) \geq δ

. We have again two possibiities.

(i)

t \land r \leq δ

Then

μ (x \land y) \geq δ \geq t \land r > γ and μ (x \lor y) \geq δ \geq t \land r > γ

which implies

{(x \land y)}_{t \land r} \in_{γ} μ

and

{(x \lor y)}_{t \land r} \in_{γ} μ

(ii)

t \land r > δ

Now we have

μ (x \land y) + t \land r > δ + δ = 2 δ and μ (x \lor y) + t \land r > δ + δ = 2 δ

which implies

{(x \land y)}_{t \land r} q_{δ} μ

and

{(x \lor y)}_{t \land r} q_{δ} μ

From (i) and (ii) it follows

{(x \land y)}_{t \land r} \in_{γ} \lor q_{δ} μ

and

{(x \lor y)}_{t \land r} \in_{γ} \lor q_{δ} μ

, i.e., in Case 2 we also conclude ghat

μ

is a

(q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice.

Case 3.

α = \in_{γ} \lor q_{δ}

We have

x_{t} \in_{γ} μ ⟹ μ (x) \geq t > γ and

y_{r} q_{δ} μ ⟹ μ (y) + r > 2 δ ⟹ μ (y) > 2 δ - r > 2 δ - 1 = γ .

As in Cases 1 and 2 we conclude

x \land y \in A

and

x \lor y \in A

, hence

μ (x \land y) \geq δ

and

μ (x \lor y) \geq δ

. Then we obtain

{(x \land y)}_{t \land r} \in_{γ} \lor q_{δ} μ

and

{(x \lor y)}_{t \land r} \in_{γ} \lor q_{δ} μ

. Therefore, in this case we finally have that

μ

is a

(\in_{γ} \lor q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice.

(⇐) Let

μ

be an

(α, \in_{γ} \lor q_{δ})

-fuzzy sublattice. Then it follows from

A = μ_{γ} = {x \in L : μ (x) > γ}

and Theorems 2.3 and 3.1 that A is a sublattice of

L .

□

Corollary 3.1.

Let

2 δ = 1 + γ

, and let A be a nonempty subset of L . Then A is a sublattice (ideal) of L if and only if the fuzzy subset

χ_{I}

of L is an

(α, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

Theorem 3.3.

The following assertions are satisfied:

(1)

Every

(\in_{γ} \lor q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L;

(2)

Every

(\in_{γ} \lor q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy ideal of L is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy ideal of L.

Proof.

We can use the fact

x_{t} \in_{γ} μ

imploes

x_{t} \in_{γ} \lor q_{δ} μ

to prove this theorem. □

Theorem 3.4.

Let L be a lattice Then:

(1)

Every

(q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L;

(2)

Every

(q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy ideal of L is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy ideal of L.

Proof.

We prove only (1) because the proof of (2) is quite similar. Let

μ

be a

(q_{δ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L. Let

x, y \in L

and

t, r \in (γ, 1]

be such that

x_{t} \in_{γ} μ, y_{r} \in_{γ} μ ⟹ μ (x) \geq t > γ and μ (y) \geq r > γ .

Then we have

{(x \land y)}_{t \land r} \bar{\in_{γ} \lor q_{δ}} μ ⟹ μ (x \land y) < t \land r, μ (x \land y) + (t \land r) \leq 2 δ ⟹ μ (x \land y) < δ;

{(x \lor y)}_{t \land r} \bar{\in_{γ} \lor q_{δ}} μ ⟹ μ (x \lor y) < t \land r, μ (x \lor y) + (t \land r) \leq 2 δ ⟹ μ (x \lor y) < δ .

Further,

μ (x \land y) (μ (x \lor y)) < t \land r and γ < t \land r \leq μ (x) \land μ (y)

imply

μ (x \land y) (μ (x \lor y)) \lor γ < μ (x) \land μ (y) \land δ .

Choose

k \in (γ, 1]

such that

2 δ - μ (x \land y) \lor γ > k \geq 2 δ - μ (x) \land μ (y) \land δ .

Then we have

\begin{matrix} 2 δ & - μ (x \land y) \geq 2 δ - μ (x \land y) \lor γ > k \geq (2 δ - μ (x)) \land (2 δ - μ (y)) \land δ \\ ⟹ & μ (x) + k \geq 2 δ, μ (y) + k \geq 2 δ, μ (x \land y) + k < 2 δ a n d μ (x \land y) < δ \leq k \\ ⟹ & x_{k} q_{δ} μ, y_{k} q_{δ} μ, {(x \land y)}_{k} \bar{\in_{γ} \lor q_{δ}} μ . \end{matrix}

This contradicts our assumption. Therefore,

{(x \land y)}_{k} \in_{γ} \lor q_{δ} μ .

Similarly, we prove

{(x \lor y)}_{k} \in_{γ} \lor q_{δ} μ .

So, we conclude that

μ

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L. □

The above theorem shows that each

(α, β)

-fuzzy sublattice (ideal) of L is an

(α, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L, and each

(α, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

4. $(\in_{γ}, \in_{γ} \lor q_{δ})$ -Fuzzy Sublattice and Ideal of a Lattice

In this section we study

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattices and

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy ideals of a lattice L and characterize sublattices and ideals of L in terms of

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattices and ideals of L.

Theorem 4.1.

For any fuzzy sublattice μ of a lattice L and for all

x, y, z \in L

and

t, r \in (γ, 1]

the following hold:

$(1)$: μ is an $(\in_{γ}, \in_{γ} \lor q_{δ})$ -fuzzy sublattice of L if and only if $x_{t} \in_{γ} μ$ , $y_{r} \in_{γ} μ$ implies $μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ$ and $μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ$ .
$(2)$: μ is an $(\in_{γ}, \in_{γ} \lor q_{δ})$ [-fuzzy ideal of L if and only if $x_{t} \in_{γ} μ$ implies $μ (x \land y) \lor γ \geq μ (x) \land δ$ and $μ (x \lor y) \lor γ \geq μ (x) \land δ$ .

Proof.

We prove (1); the proof of (2) is similar.

(⇒) Let

μ

be a

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L and suppose, to the contrary, that there are

x, y \in L

such that

μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ or μ (x \lor y) \lor γ < μ (x) \land μ (y) \land δ .

μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ

, then we choose

t \in (γ, 1]

such that

μ (x \land y) \lor γ < t \leq μ (x) \land μ (y) \land δ

, which implies

μ (x) \geq t > γ, μ (y) \geq t > γ, μ (x \land y) < t and μ (x \land y) + t < δ + δ = 2 δ .

It follows from here

x_{t} \in_{γ} μ, y_{t} \in_{γ} μ, {(x \land y)}_{t} \bar{\in_{γ}} μ and {(x \land y)}_{t} \bar{q_{δ}} μ,

i.e.

{(x \land y)}_{t} \bar{\in_{γ} \lor q_{δ}} μ

Similarly, we prove

{(x \lor y)}_{t} \bar{\in_{γ} \lor q_{δ}} μ

. However, the last two conclusions are in contradiction with our assumption.

(⇐) Let

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ, μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ

and x_{t} \in_{γ} μ, y_{r} \in_{γ} μ .

Assume, to the contrary, that

{(x \land y)}_{t \land r} \bar{\in_{γ} \lor q_{δ}} μ or {(x \lor y)}_{t \land r} \bar{\in_{γ} \lor q_{δ}} μ .

Consider the first case,

{(x \land y)}_{t \land r} \bar{\in_{γ} \lor q_{δ}} μ

. Then

{(x \land y)}_{t \land r} \bar{\in_{γ}} μ and {(x \land y)}_{t \land r} \bar{q_{δ}} μ .

From here we have the following implications:

μ (x \land y) < t \land r and μ (x \land y) + t \land r \leq 2 δ

⟹ μ (x \land y) < δ and μ (x \land y) < t \land r \leq μ (x) \land μ (y)

⟹ μ (x \land y) < t \land r \leq μ (x) \land μ (y) \land δ, γ < δ

⟹ μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ .

Similarly one proves

μ (x \lor y) \lor γ < μ (x) \land μ (y) \land δ

. The last two conclusions contradict the assumption. □

Theorem 4.2.

Let μ be a fuzzy subset of a lattice L and

2 δ = 1 + γ

. Then

μ_{t} \neq Ø

is a sublattice (ideal) of L for all

t \in (γ, δ]

if and only if μ is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

Proof. (⇐) Let

μ

be an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L, and

x, y \in μ_{t}

. Then we have

μ (x) \geq t > γ and μ (y) \geq t > γ ⟹ μ (x) \land μ (y) \geq t .

Also, by the above theorem, we have

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ ⟹ μ (x \land y) \lor γ \geq t \land δ = t

⟹ μ (x \land y) \geq t > γ ⟹ x \land y \in μ_{t},

and

μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ ⟹ μ (x \lor y) \lor γ \geq t \land δ = t

⟹ μ (x \lor y) \geq t > γ ⟹ x \lor y \in μ_{t} .

Therefore,

μ_{t}

is a sublattice of L.

(⇒) Let

μ_{t}

be a sublattice of L for all

t \in (γ, δ]

. We will prove that

μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ and μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ .

Let

x_{t}, y_{t} \in_{γ} μ

and

μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ

. We choose

t \in (γ, δ]

such that

μ (x \land y) \lor γ < t \leq μ (x) \land μ (y) \land δ

. This implies

μ (x) \geq t > γ, μ (y) \geq t > γ and μ (x \land y) < t,

which finally implies

x, y \in μ_{t}

, but

x \land y \notin μ_{t}

. This is a contradiction showing that

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ

holds.

Similarly, we can prove that

μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ

, which mens that

μ

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L. □

Definition 4.1.

Let μ be a fuzzy subset of L. For all

r \in (γ, 1]

we define:

μ_{r}^{δ} = {x \in L : x_{r} q_{δ} μ} = {x \in L : μ (x) + r > 2 δ};

{[μ]}_{r}^{δ} = {x \in L : x_{r} \in_{γ} \lor q_{δ} μ} = μ_{r} \cup μ_{r}^{δ} .

Theorem 4.3.

Let μ be a fuzzy subset of L and

2 δ = 1 + γ

. If

μ_{r}^{δ} \neq Ø

is a sublattice (ideal) of L, for all

r \in (γ, 1]

, then μ is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

Also, for

r \in (δ, 1]

, if μ is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L, then

μ_{r}^{δ} \neq Ø

is a sublattice (ideal) of L.

Proof.

Let

μ_{r}^{δ} \neq Ø

be a sublattice of L. We prove that

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ and μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ .

We consider the following two cases:

Case 1.

r \in (γ, δ]

μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ

, then we define

r = μ (x) \land μ (y) \land δ

, and obtain

μ (x \land y) \lor γ < r ⟹ μ (x \land y) < r ⟹ μ (x \land y) + r < 2 r < 2 δ ⟹ \land y \notin μ_{r}^{δ} .

Similarly we get that if

μ (x \lor y) \lor γ < μ (x) \land μ (y) \land δ

, then

x \lor y \notin μ_{r}^{δ}

. Therefore, we have a contradiction.

Case 2.

r \in (δ, 1]

μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ

, then

2 δ - (μ (x \land y) \lor γ) > 2 δ - (μ (x) \land μ (y) \land δ) ⟹

(2 δ - μ (x \land y)) \land (2 δ - γ) > (2 δ - μ (x)) \lor (2 δ - μ (y)) \lor δ .

Take now

r \in (δ, 1]

such that

(2 δ - μ (x)) \lor (2 δ - μ (y)) \lor δ < r \leq (2 δ - μ (x \land y)) \land (2 δ - γ) .

Thus we have

2 δ - μ (x) < r ⟹ μ (x) + r > 2 δ ⟹ x \in μ_{r}^{δ},

2 δ - μ (y) < r ⟹ μ (y) + r > 2 δ ⟹ y \in μ_{r}^{δ},

2 δ - μ (x \land y) \geq r ⟹ μ (x \land y) + r \leq 2 δ ⟹ x \land y \notin μ_{r}^{δ} .

We have obtained a contradiction which shows

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ

Similarly, one obtains

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ

Conversely, let

r \in (δ, 1]

. Let

μ

be an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L, and

x, y \in μ_{r}^{δ}

. Then we have

μ (x) + r > 2 δ ⟹ μ (x) > 2 δ - r \geq 2 δ - 1 = γ and

μ (y) + r > 2 δ ⟹ μ (y) > 2 δ - r \geq 2 δ - 1 = γ .

It follows from here

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ > (2 δ - r) \land (2 δ - r) \land δ .

Since

δ \leq r \leq 1

we have

δ \geq 2 δ - r \geq 2 δ - 1 and (2 δ - r) \land (2 δ - r) \land δ = 2 δ - r,

which implies

μ (x \land y) > 2 δ - r ⟹ μ (x \land y) + r > 2 δ

which means

x \land y \in μ_{r}^{δ}

Similarly, we obtain

x \lor y \in μ_{r}^{δ}

. Thus

μ_{r}^{δ}

is a sublattice of L. □

Theorem 4.4.

Let μ be a fuzzy subset of L and

2 δ = 1 + γ

. Then

{[μ]}_{r}^{δ} \neq Ø

is a sublattice (ideal) of L, for all

t \in (γ, 1]

, if and only if μ is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

Proof. (⇐) Let

μ

be an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L, and

x, y \in {[μ]}_{r}^{δ}

. Then we have

μ (x) + r > 2 δ ⟹ μ (x) \geq 2 δ - r \geq 2 δ - 1 = γ or μ (x) \geq r > γ, and

μ (y) + r > 2 δ ⟹ μ (y) \geq 2 δ - r \geq 2 δ - 1 = γ or μ (y) \geq r > γ .

Since

μ

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L, we have

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ and

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ .

We consider the following two cases:

Case 1.

r \in (γ, δ]

In this case we have

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ \geq r \land r \land δ = r ⟹ μ (x \land y) \lor γ \geq r > γ

which means

{(x \land y)}_{r} \in_{γ} μ

, i.e.,

x \land y \in {[μ]}_{r}^{δ}

Similarly, we can prove

x \lor y \in {[μ]}_{r}^{δ}

. Thus

{[μ]}_{r}^{δ}

is a sublattice of L.

Case 2.

r \in (δ, 1]

Then

δ < r \leq 1

implies

δ > 2 δ - r \geq 2 δ - 1

. Let

x, y \in {[μ]}_{r}^{δ}

. We have

μ (x) + r > 2 δ ⟹ μ (x) > 2 δ - r \geq 2 δ - 1 = γ or μ (x) \geq r > γ, and

μ (y) + r > 2 δ ⟹ μ (y) > 2 δ - r \geq 2 δ - 1 = γ or μ (y) \geq r > γ .

We conclude from here

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ > (2 δ - r) \land (2 δ - r) \land δ = 2 δ - r,

i . e . μ (x \land y) > 2 δ - r ⟹ μ (x \land y) + r > 2 δ,

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ > r \land r \land δ = δ,

i . e . μ (x \land y) + r > δ + r > 2 δ .

So, we conclude that

{(x \land y)}_{r} q_{δ} μ

, which implies

x \land y \in {[μ]}_{r}^{δ}

Similarly, we prove that

x \lor y \in {[μ]}_{r}^{δ}

. Thus

{[μ]}_{r}^{δ}

is a sublattice of L.

(⇒) Let

{[μ]}_{r}^{δ} \neq Ø

be a sublattice of L. We prove that

μ (x \land y) \lor γ \geq μ (x) \land μ (y) \land δ and μ (x \lor y) \lor γ \geq μ (x) \land μ (y) \land δ .

Let, to the contrary,

μ (x \land y) \lor γ < μ (x) \land μ (y) \land δ

. Then choose

r \in (γ, 1]

such that

μ (x \land y) \lor γ < r \leq μ (x) \land μ (y) \land δ .

Then

μ (x) \geq r > γ, μ (y) \geq r > γ, μ (x \land y) < r and μ (x \land y) + r < δ + δ = 2 δ,

which implies

x_{r} \in_{γ} μ, y_{r} \in_{γ} μ, {(x \land y)}_{r} \bar{\in_{γ}} μ and {(x \land y)}_{r} \bar{q_{δ}} μ .

This means

{(x \land y)}_{r} \bar{\in_{γ} \lor q_{δ}} μ

Similarly, we get

{(x \lor y)}_{r} \bar{\in_{γ} \lor q_{δ}} μ

. The last two conclusions contradict the assumption. □

Theorem 4.5.

Let λ and μ be

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattices (ideals) of L. Then

λ \cap μ

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

Proof.

For any

x, y \in L

, we have

(λ \cap μ) (x \land y) \lor γ = (λ (x \land y) \land μ (x \land y)) \lor γ = (λ (x \land y) \lor γ) \land (μ (x \land y) \lor γ) .

By Theorems 2.3–3.3, we have

\begin{matrix} (λ (x \land y) \lor γ) \land (μ (x \land y) \lor γ) & \geq & (λ (x) \land λ (y) \land δ) \land (μ (x) \land μ (y) \land δ) \\ = & (λ (x) \land μ (x)) \land (λ (y) \land μ (y)) \land δ \\ = & (λ \cap μ (x)) \land (λ \cap μ (y)) \land δ, \end{matrix}

which means

(λ \cap μ) (x \land y) \lor γ \geq (λ \cap μ) (x) \land (λ \cap μ) (y) \land δ .

Similarly, we can prove

(λ \cap μ) (x \lor y) \lor γ \geq (λ \cap μ) (x) \land (λ \cap μ) (y) \land δ .

Hence,

λ \cap μ

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice of L. □

Remark 4.1.

{μ_{s}}_{s \in s}

, S is an index set, is a family of

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattices (ideals) of L, then

⋂_{s \in S} μ_{s}

is an

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattice (ideal) of L.

5. Conclusions

We introduced and studied

(α, β)

-fuzzy sublattices and

(α, β)

-fuzzy ideals of a lattice. In particular,

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy sublattices and

(\in_{γ}, \in_{γ} \lor q_{δ})

-fuzzy ideals are considered and characterized. we hope that this study can be extended to other mathematical structures.

Author Contributions

Conceptualization, A.F. and Lj. K.; methodology, A. F.; software, Lj.K.; formal analysis, A.F. and Lj.K.; investigation, A.F. and Lj.K.; resources, A.F.; writing—original draft preparation, A.F.; writing—review and editing, Lj.K.; visualization, Lj.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data are contained within the article and also are available from the authors.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Siripitukdet M.; Farajzadeh A.; Julatha P. Characterization of semigroup in terms of their γ-δ-fuzzy ideals, J. Alg. Num. Theo. Appl. 2017, 39, 815–842.
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Characterization of Lattices in Terms of (∈γ,∈γ∨qδ)-Fuzzy Ideals

Abstract

1. Introduction

2. Preliminaries

2.1. Fuzzy Sublattices (Ideals) of a Lattice

2.2. ( ∈ , ∈ ∨ q ) -Fuzzy Sublattice and Ideal of a Lattice

3. ( α , β ) -Fuzzy Sublattices and Ideals of a Lattice

4. ( ∈ γ , ∈ γ ∨ q δ ) -Fuzzy Sublattice and Ideal of a Lattice

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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2.2. $(\in, \in \lor q)$ -Fuzzy Sublattice and Ideal of a Lattice

3. $(α, β)$ -Fuzzy Sublattices and Ideals of a Lattice

4. $(\in_{γ}, \in_{γ} \lor q_{δ})$ -Fuzzy Sublattice and Ideal of a Lattice