EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1498-1511
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Positive implicative makgeolli ideals of BCK-algebras

Seok-Zun Song1,∗, Mehmet Ali Öztürk2, Young Bae Jun3

1 Department of Mathematics, Jeju National University, Jeju 63243, Korea
2 Department of Mathematics, Faculty of Arts and Sciences, Adıyaman University, 02040
Adıyaman, Turkiye
3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828,
Korea

Abstract. The concept of a positive implicative makgeolli ideal in BCK-algebras is introduced,
and its properties are investigated. The relationship between a makgeolli ideal and a positive
implicative makgeolli ideal is established. The conditions under which a makgeolli ideal can be
a positive implicative makgeolli ideal are explored. Characterizations of a positive implicative
makgeolli ideal are discussed, and the extension property for a positive implicative makgeolli ideal
is established.

2020 Mathematics Subject Classifications: 03G25, 06F35, 08A72

Key Words and Phrases: BCK-soft universe, makgeolli structure, makgeolli ideal, positive
implicative makgeolli ideal.

1. Introduction

Many problems that need to be solved in the real world often involve inherently uncer-
tain, inaccurate, and ambiguous factors. Zadeh [24] pointed out Various problems in sys-
tem identification involve characteristics which are essentially non-probabilistic in nature,
and he introduced fuzzy set theory as an alternative to probability theory. Uncertainty is
an attribute of information. In order to suggest a more general framework, the approach
to uncertainty is outlined by Zadeh [25]. Uncertainties can’t be handled using traditional
mathematical tools but may be dealt with using a wide range of existing theories such
as probability theory, theory of (intuitionistic) fuzzy sets, theory of interval mathematics,
theory of vague sets, and theory of rough sets. But, Molodtsov [21] pointed out all of
these theories have their own difficulties. Maji et al. [18] and Molodtsov [21] suggested
that one reason for these difficulties may be due to the inadequacy of the parametrization
tool of the theory. To overcome these difficulties, Molodtsov [21] introduced the concept

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4522

Email addresses: szsong@jejunu.ac.kr (S. Z. Song),
mehaliozturk@gmail.com (M. A. Öztürk), skywine@gmail.com (Y. B. Jun)

https://www.ejpam.com 1498 © 2022 EJPAM All rights reserved.



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1499

of soft set as a new mathematical tool for dealing with uncertainties that is free from the
difficulties that have troubled the usual theoretical approaches, and he pointed out several
directions for the applications of soft sets. Globally, interest in soft set theory and its
application has been growing rapidly in recent years. Soft set theory has been applied to
algebraic structures, for example, groups, rings, fields and modules (see [1, 3–5, 14]), and
BCK/BCI-algebras etc. (see [9–13, 15–17, 22, 23]). In 2019, Ahn et al. [2] introduced the
notion of makgeolli structures as a hybrid structure based on fuzzy set and soft set theory,
and applied it to BCK/BCI-algebras.

In this paper, we introduce the notion of a positive implicative makgeolli ideal in
BCK-algebras, and investigate its properties. We establish the relationship between a
makgeolli ideal and a positive implicative makgeolli ideal. We explore the conditions
under which a makgeolli ideal can be a positive implicative makgeolli ideal. We discusse
the characterization of positive implicative makgeolli ideal, and construct the extension
property for a positive implicative makgeolli ideal.

2. Preliminaries

2.1. Preliminaries on BCK-algebras

BCI/BCK-algebra is an important type of logical algebra introduced by K. Iséki (see
[7] and [8]), and it has been extensively investigated by several researchers. See the books
[6, 20] for further information regarding BCI-algebras and BCK-algebras.

In this section, we recall the definitions and basic results required in this paper.
Let X be a set with a special element “0” and a binary operation “ ∗ ”. If it satisfies

the following conditions:

(I1) (∀a, b, c ∈ X) (((a ∗ b) ∗ (a ∗ c)) ∗ (c ∗ b) = 0),

(I2) (∀a, b ∈ X) ((a ∗ (a ∗ b)) ∗ b = 0),

(I3) (∀a ∈ X) (a ∗ a = 0),

(I4) (∀a, b ∈ X) (a ∗ b = 0, b ∗ a = 0 ⇒ a = b),

(K) (∀a ∈ X) (0 ∗ a = 0),

then it is called a BCK-algebra, and it is denoted by (X, ∗)0.
The order relation “ ≤ ” in a BCK-algebra (X, ∗)0 is defined as follows:

(∀a, b ∈ X)(a ≤ b ⇔ a ∗ b = 0). (1)

Every BCK-algebra (X, ∗)0 satisfies the following conditions (see [19, 20]):

(∀a ∈ X) (a ∗ 0 = a) , (2)

(∀a, b, c ∈ X) (a ≤ b ⇒ a ∗ c ≤ b ∗ c, c ∗ b ≤ c ∗ a) , (3)

(∀a, b, c ∈ X) ((a ∗ b) ∗ c = (a ∗ c) ∗ b) . (4)



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1500

A BCK-algebra (X, ∗)0 is said to be positive implicative (see [20]) if (a ∗ c) ∗ (b ∗ c) =
(a ∗ b) ∗ c for all a, b, c ∈ X.

A subset C of a BCK-algebra (X, ∗)0 is called an ideal of (X, ∗)0 (see [6, 20]) if it
satisfies:

0 ∈ C, (5)

(∀a, b ∈ X)(a ∗ b ∈ C, b ∈ C ⇒ a ∈ C). (6)

A subset C of a BCK-algebra (X, ∗)0 is called a positive implicative ideal of (X, ∗)0
(see [20]) if it satisfies (5) and

(∀a, b, c ∈ X)((a ∗ b) ∗ c ∈ C, b ∗ c ∈ C ⇒ a ∗ c ∈ C). (7)

2.2. Preliminaries on makgeolli structures

Let X be a universal set and E a set of parameters. We say that the pair (X,E) is a
soft universe.

Definition 1 ([2]). Let (X,E) be a soft universe and let C and D be subsets of E. A
makgeolli structure on X (related to C and D) is a structure of the form:

M(C,D,X) := {⟨(a, b, x);MC(a), GD(b), ξ(x)⟩ | (a, b, x) ∈ C ×D ×X} (8)

where MC := (M,C) and GD := (G,D) are soft sets over X and ξ is a fuzzy set in X.

For the sake of simplicity, the makgeolli structure in (8) will be denoted by M(C,D,X) =
(MC , GD, ξ). The makgeolli structure M(C,C,X) = (MC , GC , ξ) on X related to a subset
C of E is simply denoted by M(C,X) = (MC , GC , ξ). If C = D = E, we use the notation
M(X,E) := (ME , GE , ξ) as the makgeolli structure of (X,E).

By a BCK/BCI-soft universe, we mean a soft universe (X,E) in which X and E are
BCK/BCI-algebras with binary operations “∗” and “↬”, respectively.

Definition 2 ([2]). Let (X,E) be a BCK/BCI-soft universe. A makgeolli structure
M(X,E) := (ME , GE , ξ) is called a makgeolli ideal of (X,E) if it satisfies:{

(∀a ∈ E) (ME(0) ⊇ ME(a), GE(0) ⊆ GE(a)) .
(∀x ∈ X) (0/ξ(x) ∈ ξ) .

(9)
(∀a, b ∈ E)

(
ME(a) ⊇ ME(a ↬ b) ∩ME(b)

GE(a) ⊆ GE(a ↬ b) ∪GE(b)

)
.

(∀x, y, z ∈ X)(∀t, r ∈ (0, 1])

(
{x ∗ y}/t ∈ ξ, y/r ∈ ξ
⇒ x/min{t, r} ∈ ξ

)
.

(10)

Lemma 1 ([2]). Let (X,E) be a BCK/BCI-soft universe. Every makgeolli ideal M(X,E) :=
(ME , GE , ξ) of (X,E) satisfies the following assertions.



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1501

(i)

 (∀a, b ∈ E)

(
a ≤ b ⇒

{
ME(a) ⊇ ME(b)
GE(a) ⊆ GE(b)

)
.

(∀x, y ∈ X) (x ≤ y ⇒ ξ(x) ≥ ξ(y ∗ z)) .

(ii)

 (∀a, b, c ∈ E)

(
a ↬ b ≤ c ⇒

{
ME(a) ⊇ ME(b) ∩ME(c)
GE(a) ⊆ GE(b) ∪GE(c)

)
.

(∀x, y, z ∈ X) (x ∗ y ≤ z ⇒ ξ(x) ≥ min{ξ(y ∗ z), ξ(z)}) .
Let (X,E) be a BCK/BCI-soft universe. Given a makgeolli structure M(X,E) := (ME ,

GE , ξ) on (X,E), consider the following sets:

E(ME ;α) := {a ∈ E | ME(a) ⊇ α},
E(GE ;β) := {b ∈ E | GE(b) ⊆ β},
X (ξ; t) := {x ∈ X | ξ(x) ≥ t}

where α and β are subsets of X and t ∈ [0, 1].

3. Positive implicative makgeolli ideals

In what follows, let (X,E) be a BCK-soft universe unless otherwise specified.

Definition 3. A makgeolli structure M(X,E) := (ME , GE , ξ) is called a positive implica-
tive makgeolli ideal of (X,E) if it satisfies (9) and

(∀a, b, c ∈ E)

(
ME(a ↬ c) ⊇ ME((a ↬ b) ↬ c) ∩ME(b ↬ c)

GE(a ↬ c) ⊆ GE((a ↬ b) ↬ c) ∪GE(b ↬ c)

)
. (11)

(∀x, y, z ∈ X)(∀t, r ∈ (0, 1])

(
{(x ∗ y) ∗ z}/t ∈ ξ, {y ∗ z}/r ∈ ξ
⇒ {x ∗ z}/min{t, r} ∈ ξ

)
. (12)

Note that the condition (12) is equivalent to the following assertion.

(∀x, y, z ∈ X) (ξ(x ∗ z) ≥ min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)}) . (13)

In fact, suppose that the condition (12) is valid. Since {(x ∗ y) ∗ z}/ξ((x ∗ y) ∗ z) ∈ ξ and
{y ∗ z}/ξ(y ∗ z) ∈ ξ, it follows from (12) that {x ∗ z}/min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)} ∈ ξ.
Hence ξ(x ∗ z) ≥ min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)}. Conversely, assume that the condition
(13) is valid. Let x, y, z ∈ X and t, r ∈ (0, 1] be such that {(x ∗ y) ∗ z}/t ∈ ξ and
{y ∗ z}/r ∈ ξ. Then ξ((x ∗ y) ∗ z) ≥ t and ξ(y ∗ z) ≥ r. It follows from (13) that
ξ(x ∗ z) ≥ min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)} ≥ min{t, r}, that is, {x ∗ z}/min{t, r} ∈ ξ.

Example 1. Consider a BCK-soft universe (X,E) in which X := {0, 1, 2, 3, 4} and E :=
{0, 1, 2, 3} with binary operations “∗” and “↬”, respectively, given by Table 1.
Let M(X,E) := (ME , GE , ξ) be a makgeolli structure on (X,E) defined as follows:

(ME , GE) : E → P(X)× P(X), x 7→


(X, {2}) if x = 0,
({0, 1, 3, 4}, {0, 2}) if x = 1,
({1, 3, 4}, X) if x = 2,
({1, 4}, {0, 2, 3}) if x = 3,



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1502

Table 1: Cayley table for the binary operations “↬”

∗ 0 1 2 3 4

0 0 0 0 0 0
1 1 0 0 0 0
2 2 2 0 2 2
3 3 3 3 0 3
4 4 4 4 4 0

↬ 0 1 2 3

0 0 0 0 0
1 1 0 0 0
2 2 2 0 2
3 3 3 3 0

ξ : X → [0, 1], y 7→


0.78 if y = 0,
0.63 if y = 1,
0.51 if y = 2,
0.44 if y = 3,
0.59 if y = 4.

It is routine to verify that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal
of (X,E).

Proposition 1. Every positive implicative makgeolli ideal M(X,E) := (ME , GE , ξ) of
(X,E) satisfies:  (∀a, b ∈ E)

(
ME(a ↬ b) ⊇ ME((a ↬ b) ↬ b)

GE(a ↬ b) ⊆ GE((a ↬ b) ↬ b)

)
,

(∀x, y ∈ X) (ξ((x ∗ y) ≥ ξ((x ∗ y) ∗ y)) .
(14)

Proof. If we replace c and z with b and y in (11) and (12), respectively, and use (I3)
and (9), then we have (14).

We discuss the relationship between positive implicative makgeolli ideal and makgeolli
ideal.

Theorem 1. Every positive implicative makgeolli ideal is a makgeolli ideal.

Proof. Let M(X,E) := (ME , GE , ξ) be a positive implicative makgeolli ideal of (X,E).
If we replace c and z with 0 in (11) and (12), then we can get (10). Hence M(X,E) := (ME ,
GE , ξ) is a makgeolli ideal of (X,E).

The following example shows that the converse of Theorem 1 may not be true.

Example 2. Consider a BCK-soft universe (X,E) in which X := {0, 1, 2, 3, 4} and E :=
{0, 1, 2, 3} with binary operations “∗” and “↬”, respectively, given by Table 2.
Let M(X,E) := (ME , GE , ξ) be a makgeolli structure on (X,E) defined as follows:

(ME , GE) : E → P(X)× P(X), x 7→


(X, {2, 4}) if x = 0,
({1, 3}, {0, 1, 2, 4}) if x = 1,
({1, 3}, {0, 1, 2, 4}) if x = 2,
({0, 1, 3, 4}, {0, 2, 4}) if x = 3,



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1503

Table 2: Cayley table for the binary operations “↬”

∗ 0 1 2 3 4

0 0 0 0 0 0
1 1 0 1 0 1
2 2 2 0 2 0
3 3 1 3 0 3
4 4 4 4 4 0

↬ 0 1 2 3

0 0 0 0 0
1 1 0 0 1
2 2 1 0 2
3 3 3 3 0

ξ : X → [0, 1], y 7→


0.83 if y = 0,
0.52 if y = 1,
0.77 if y = 2,
0.52 if y = 3,
0.69 if y = 4.

It is routine to check that M(X,E) := (ME , GE , ξ) is a makgeolli ideal of (X,E). But
it is not a positive implicative makgeolli ideal of (X,E) since GE(2 ↬ 1) = GE(1) =
{0, 1, 2, 4} ⊈ {2, 4} = GE((2 ↬ 1) ↬ 1) ∪ GE(1 ↬ 1) or {(3 ∗ 1) ∗ 1}/0.76 = 0/0.76 ∈ ξ
and {1 ∗ 1}/0.79 = 0/0.79 ∈ ξ, but {3 ∗ 1}/min{0.76, 0.79} = 1/0.76 /∈ ξ.

We explore the conditions under which the converse of Theorem 1 can be established.

Theorem 2. In a BCK-soft universe (X,E) in which X and E are positive implicative
BCK-algebras, every makgeolli ideal is a positive implicative makgeolli ideal.

Proof. Straightforward.

Theorem 3. If a makgeolli ideal M(X,E) := (ME , GE , ξ) of (X,E) satisfies the condition
(14), then it is a positive implicative makgeolli ideal of (X,E).

Proof. Let M(X,E) := (ME , GE , ξ) be a makgeolli ideal of (X,E) that satisfies the
condition (14). The combination of (I1) and (4) derive

((a ↬ c) ↬ c) ↬ (b ↬ c) ≤ (a ↬ c) ↬ b = (a ↬ b) ↬ c

and ((x ∗ z) ∗ z) ∗ (y ∗ z) ≤ (x ∗ z) ∗ y = (x ∗ y) ∗ z for all a, b, c ∈ E and x, y, z ∈ X, and so

ME(((a ↬ c) ↬ c) ↬ (b ↬ c)) ⊇ ME((a ↬ b) ↬ c),

GE(((a ↬ c) ↬ c) ↬ (b ↬ c)) ⊆ GE((a ↬ b) ↬ c),

ξ(((x ∗ z) ∗ z) ∗ (y ∗ z)) ≥ ξ((x ∗ y) ∗ z)

by Lemma 1(i). It follows from (10) and (14) that

ME(a ↬ c) ⊇ ME((a ↬ c) ↬ c)

⊇ ME(((a ↬ c) ↬ c) ↬ (b ↬ c)) ∩ME(b ↬ c)



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1504

⊇ ME((a ↬ b) ↬ c) ∩ME(b ↬ c),

GE(a ↬ c) ⊆ GE((a ↬ c) ↬ c)

⊆ GE(((a ↬ c) ↬ c) ↬ (b ↬ c)) ∪GE(b ↬ c)

⊆ GE((a ↬ b) ↬ c) ∪GE(b ↬ c),

and

ξ(x ∗ z) ≥ ξ((x ∗ z) ∗ z)
≥ min{ξ(((x ∗ z) ∗ z) ∗ (y ∗ z)), ξ(y ∗ z)}
≥ min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)}.

Hence M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal of (X,E).

Theorem 4. A makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) is a positive
implicative makgeolli ideal of (X,E) if and only if it is a makgeolli ideal of (X,E) that
satisfies the following condition. (∀a, b, c ∈ E)

(
ME((a ↬ c) ↬ (b ↬ c)) ⊇ ME((a ↬ b) ↬ c)

GE((a ↬ c) ↬ (b ↬ c)) ⊆ GE((a ↬ b) ↬ c)

)
,

(∀x, y, z ∈ X) (ξ((x ∗ z) ∗ (y ∗ z)) ≥ ξ(((x ∗ y) ∗ z)) .
(15)

Proof. Let M(X,E) := (ME , GE , ξ) be a makgeolli ideal of (X,E) that satisfies (15).
If we put b := c and y := z in (15), then

ME(a ↬ c) = ME((a ↬ c) ↬ 0) = ME((a ↬ c) ↬ (c ↬ c)) ⊇ ME((a ↬ c) ↬ c),

GE(a ↬ c) = GE((a ↬ c) ↬ 0) = GE((a ↬ c) ↬ (c ↬ c)) ⊆ GE((a ↬ c) ↬ c),

ξ(x ∗ z) = ξ((x ∗ z) ∗ 0) = ξ((x ∗ z) ∗ (z ∗ z)) ≥ ξ((x ∗ z) ∗ z)

for all a, c ∈ E and x, z ∈ X by (I3) and (2). It follows from Theorem 3 that M(X,E) :=
(ME , GE , ξ) is a positive implicative makgeolli ideal of (X,E).

Conversely, assume that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli
ideal of (X,E). Then it is a makgeolli ideal of (X,E) by Theorem 1. Since

((a ↬ (b ↬ c)) ↬ c) ↬ c = ((a ↬ c) ↬ (b ↬ c)) ↬ c ≤ (a ↬ b) ↬ c

and ((x∗ (y ∗z))∗z)∗z = ((x∗z)∗ (y ∗z))∗z ≤ (x∗y)∗z for all a, b, c ∈ E and x, y, z ∈ X,
we have

ME((a ↬ c) ↬ (b ↬ c)) = ME((a ↬ (b ↬ c)) ↬ c)

⊇ ME(((a ↬ (b ↬ c)) ↬ c) ↬ c) ⊇ ME((a ↬ b) ↬ c),

GE((a ↬ c) ↬ (b ↬ c)) = GE((a ↬ (b ↬ c)) ↬ c)



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1505

⊆ GE(((a ↬ (b ↬ c)) ↬ c) ↬ c) ⊆ GE((a ↬ b) ↬ c),

and

ξ((x ∗ z) ∗ (y ∗ z)) = ξ((x ∗ (y ∗ z)) ∗ z)
≥ ξ(((x ∗ (y ∗ z)) ∗ z) ∗ z) ≥ ξ((x ∗ y) ∗ z)

by (4), Lemma 1(i) and Proposition 1. Hence (15) is valid.

Theorem 5. A makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) is a positive
implicative makgeolli ideal of (X,E) if and only if it satisfies (9) and (∀a, b, c ∈ E)

(
ME(a ↬ b) ⊇ ME(((a ↬ b) ↬ b) ↬ c) ∩ME(c)

GE(a ↬ b) ⊆ GE(((a ↬ b) ↬ b) ↬ c) ∪GE(c)

)
,

(∀x, y, z ∈ X) (ξ(x ∗ y) ≥ min{ξ(((x ∗ y) ∗ y) ∗ z), ξ(z)}) .
(16)

Proof. Let M(X,E) := (ME , GE , ξ) be a positive implicative makgeolli ideal of (X,E).
Then it is a makgeolli ideal of (X,E) by Theorem 1, and so the condition (9) is valid.
Using (I3), (2), (4), (10) and (15), we get

ME(a ↬ b) ⊇ ME((a ↬ b) ↬ c) ∩ME(c) = ME(((a ↬ c) ↬ b) ↬ (b ↬ b)) ∩ME(c)

⊇ ME(((a ↬ c) ↬ b) ↬ b) ∩ME(c) = ME(((a ↬ b) ↬ b) ↬ c) ∩ME(c),

GE(a ↬ b) ⊆ GE((a ↬ b) ↬ c) ∪ME(c) = GE(((a ↬ c) ↬ b) ↬ (b ↬ b)) ∪GE(c)

⊆ GE(((a ↬ c) ↬ b) ↬ b) ∪GE(c) = GE(((a ↬ b) ↬ b) ↬ c) ∪GE(c)

and

ξ(x ∗ y) ≥ min{ξ((x ∗ y) ∗ z), ξ(z)} = min{ξ(((x ∗ z) ∗ y) ∗ (y ∗ y)), ξ(z)}
≥ min{ξ(((x ∗ z) ∗ y) ∗ y), ξ(z)} = min{ξ(((x ∗ y) ∗ y) ∗ z), ξ(z)}

for all a, b, c ∈ E and x, y, z ∈ X. Therefore (16) is valid.
Conversely, assume that M(X,E) := (ME , GE , ξ) satisfies (9) and (16). Then

ME(a) = ME(a ↬ 0) ⊇ ME(((a ↬ 0) ↬ 0) ↬ c) ∩ME(c) = ME(a ↬ c) ∩ME(c),

GE(a) = GE(a ↬ 0) ⊆ GE(((a ↬ 0) ↬ 0) ↬ c) ∪GE(c) = GE(a ↬ c) ∪GE(c),

and

ξ(x) = ξ(x ∗ 0) ≥ min{ξ(((x ∗ 0) ∗ 0) ∗ z), ξ(z)} = min{ξ(x ∗ z), ξ(z)}

for all a, c ∈ E and x, z ∈ X. Hence M(X,E) := (ME , GE , ξ) is a makgeolli ideal of (X,E).
If we put c = 0 = z in (16) and use (2), then

ME(a ↬ b) ⊇ ME(((a ↬ b) ↬ b) ↬ 0) ∩ME(0)



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1506

= ME((a ↬ b) ↬ b) ∩ME(0) = ME((a ↬ b) ↬ b),

GE(a ↬ b) ⊆ GE(((a ↬ b) ↬ b) ↬ 0) ∪GE(0)

= GE((a ↬ b) ↬ b) ∪GE(0) = GE((a ↬ b) ↬ b)

and ξ(x ∗ y) ≥ min{ξ(((x ∗ y) ∗ y) ∗ 0), ξ(0)} = min{ξ((x ∗ y) ∗ y), ξ(0)} = ξ((x ∗ y) ∗ y)
for all a, b ∈ E and x, y ∈ X. Therefore M(X,E) := (ME , GE , ξ) is a positive implicative
makgeolli ideal of (X,E) by Theorem 3.

Lemma 2. If a makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) satisfies the as-
sertion (ii) in Lemma 1, then it is a makgeolli ideal of (X,E).

Proof. Since 0 ↬ a ≤ a and 0 ∗ x ≤ x for all a ∈ E and x ∈ X, we have ME(0) ⊇
ME(a)∩ME(a) = ME(a), GE(0) ⊆ GE(a)∪GE(a) = GE(a), and ξ(0) ≥ min{ξ(x), ξ(x)} =
ξ(x), i.e., 0/ξ(x) ∈ ξ by the condition (ii) in Lemma 1. Since a ↬ (a ↬ b) ≤ b and
x∗(x∗y) ≤ y for all a, b ∈ E and x, y ∈ X, it follows from the condition (ii) in Lemma 1 that
ME(a) ⊇ ME(a ↬ b)∩ME(b), GE(a) ⊆ GE(a ↬ b)∪GE(b), and ξ(x) ≥ min{ξ(x∗y), ξ(y)}
for all a, b ∈ E and x, y ∈ X. Consequently, M(X,E) := (ME , GE , ξ) is a makgeolli ideal
of (X,E).

The next corollary is derived by the combination of Theorem 4 and Lemma 2.

Corollary 1. If a makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) satisfies (15)
and the assertion (ii) in Lemma 1, then M(X,E) := (ME , GE , ξ) is a positive implicative
makgeolli ideal of (X,E).

Theorem 6. A makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) is a positive
implicative makgeolli ideal of (X,E) if and only if it satisfies:

(∀a, b, x, y ∈ E)

 ((a ↬ b) ↬ b) ↬ x ≤ y

⇒
{

ME(a ↬ b) ⊇ ME(x) ∩ME(y)
GE(a ↬ b) ⊆ GE(x) ∪GE(y)

 .

(∀x, y, a, b ∈ X) (((x ∗ y) ∗ y) ∗ a ≤ b ⇒ ξ(x ∗ y) ≥ min{ξ(a), ξ(b)}) .

(17)

Proof. Assume that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal
of (X,E). Then it is a makgeolli ideal of (X,E). Let a, b, x, y ∈ E and x, y, a, b ∈ X be
such that ((a ↬ b) ↬ b) ↬ x ≤ y and ((x ∗ y) ∗ y) ∗ a ≤ b. Then

ME(a ↬ b) ⊇ ME((a ↬ b) ↬ b) ⊇ ME(x) ∩ME(y),

GE(a ↬ b) ⊆ GE((a ↬ b) ↬ b) ⊆ GE(x) ∪GE(y),

ξ(x ∗ y) ≥ ξ((x ∗ y) ∗ y) ≥ min{ξ(a), ξ(b)}

by Proposition 1 and the assertion (ii) in Lemma 1.
Conversely, letM(X,E) := (ME , GE , ξ) be a makgeolli structure on (X,E) that satisfies

(17). Let a, b, c ∈ E and x, y, z ∈ X be such that a ↬ b ≤ c and x ∗ y ≤ z. Then ((a ↬



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1507

0) ↬ 0) ↬ b ≤ c and ((x ∗ 0) ∗ 0) ∗ y ≤ z, and so ME(a) = ME(a ↬ 0) ⊇ ME(b) ∩ME(c),
GE(a) = GE(a ↬ 0) ⊆ GE(b) ∪ GE(c) and ξ(x) = ξ(x ∗ 0) ≥ min{ξ(y), ξ(z)} by (2) and
(17). Hence M(X,E) := (ME , GE , ξ) is a makgeolli ideal of (X,E) by Lemma 2. Since
(((a ↬ b) ↬ b) ↬ ((a ↬ b) ↬ b)) ↬ 0 = 0 and (((x ∗ y) ∗ y) ∗ ((x ∗ y) ∗ y)) ∗ 0 = 0 for all
a, b ∈ E and x, y ∈ X, It follows from (9) and (17) that

ME(a ↬ b) ⊇ ME((a ↬ b) ↬ b) ∩ME(0) = ME((a ↬ b) ↬ b),

GE(a ↬ b) ⊆ GE((a ↬ b) ↬ b) ∪GE(0) = GE((a ↬ b) ↬ b),

ξ(x ∗ y) ≥ min{ξ((x ∗ y) ∗ y), ξ(0)} = ξ((x ∗ y) ∗ y).

Consequently, M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal of (X,E)
by Theorem 3.

Theorem 7. A makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) is a positive
implicative makgeolli ideal of (X,E) if and only if it satisfies:

(∀a, b, c, x, y ∈ E)

 ((a ↬ b) ↬ c) ↬ x ≤ y

⇒
{

ME((a ↬ c) ↬ (b ↬ c)) ⊇ ME(x) ∩ME(y)
GE((a ↬ c) ↬ (b ↬ c)) ⊆ GE(x) ∪GE(y)

 .

(∀x, y, z, a, b ∈ X)

(
((x ∗ y) ∗ z) ∗ a ≤ b
⇒ ξ((x ∗ z) ∗ (y ∗ z)) ≥ min{ξ(a), ξ(b)}

)
.

(18)

Proof. Assume that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal of
(X,E). Then it is a makgeolli ideal of (X,E) (see Theorem 1). Let ((a ↬ b) ↬ c) ↬ x ≤ y
for all a, b, c, x, y ∈ E, and let ((x ∗ y) ∗ z) ∗ a ≤ b for all x, y, z, a, b ∈ X. The combination
of the assertion (ii) in Lemma 1 and Theorem 4 leads to

ME((a ↬ c) ↬ (b ↬ c)) ⊇ ME((a ↬ b) ↬ c) ⊇ ME(x) ∩ME(y),

GE((a ↬ c) ↬ (b ↬ c)) ⊆ GE((a ↬ b) ↬ c) ⊆ GE(x) ∪GE(y),

ξ((x ∗ z) ∗ (y ∗ z)) ≥ ξ((x ∗ y) ∗ z) ≥ min{ξ(a), ξ(b)}.

Conversely, letM(X,E) := (ME , GE , ξ) be a makgeolli structure on (X,E) that satisfies
(18). Let a, b, x, y ∈ E be such that ((a ↬ b) ↬ b) ↬ x ≤ y, and let x, y, a, b ∈ X be such
that ((x ∗ y) ∗ y) ∗ a ≤ b. Using (I3), (2) and (18), we have

ME(a ↬ b) = ME((a ↬ b) ↬ (b ↬ b)) ⊇ ME(x) ∩ME(y),

GE(a ↬ b) = GE((a ↬ b) ↬ (b ↬ b)) ⊆ GE(x) ∪GE(y),

ξ(x ∗ y) = ξ((x ∗ y) ∗ (y ∗ y)) ≥ min{ξ(a), ξ(b)}.

It follows from Theorem 6 that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli
ideal of (X,E).

Theorem 8. A makgeolli structure M(X,E) := (ME , GE , ξ) on (X,E) is a positive
implicative makgeolli ideal of (X,E) if and only if the sets E(ME ;α) and E(GE ;β) are
positive implicative ideals of E, and the set X (ξ; t) is a positive implicative ideal of X for
all α, β ∈ P(X) and t ∈ [0, 1].



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1508

Proof. Assume that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal
of (X,E). It is clear that 0 ∈ E(ME ;α) ∩ E(GE ;β) ∩ X (ξ; t) for all α, β ∈ P(X) and
t ∈ [0, 1]. Let a, b, c ∈ E be such that (a ↬ b) ↬ c ∈ E(ME ;α) ∩ E(GE ;β) and b ↬ c ∈
E(ME ;α) ∩ E(GE ;β). Then

ME(a ↬ c) ⊇ ME((a ↬ b) ↬ c) ∩ME(b ↬ c) ⊇ α,

GE(a ↬ c) ⊆ GE((a ↬ b) ↬ c) ∪GE(b ↬ c) ⊆ β,

and so a ↬ c ∈ E(ME ;α)∩E(GE ;β). Let x, y, z ∈ X be such that (x ∗ y) ∗ z ∈ X (ξ; t) and
y ∗ z ∈ X (ξ; t). Then ξ(x ∗ z) ≥ min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)} ≥ t, and thus x ∗ z ∈ X (ξ; t).
Therefore E(ME ;α) and E(GE ;β) are positive implicative ideals of E, and X (ξ; t) is a
positive implicative ideal of X for all α, β ∈ P(X) and t ∈ [0, 1].

Conversely, suppose that E(ME ;α) and E(GE ;β) are positive implicative ideals of E,
and X (ξ; t) is a positive implicative ideal of X for all α, β ∈ P(X) and t ∈ [0, 1]. Then
E(ME ;α) and E(GE ;β) are subalgebras of E, and X (ξ; t) is a subalgebra of X. Let
(a1, b1, x1), (a2, b2, x2) ∈ E × E ×X be such that M(X,E)(a1, b1, x1) := (ME(a1), GE(b1),
ξ(x1)) = (α1, β1, t1) and M(X,E)(a2, b2, x2) := (ME(a2), GE(b2), ξ(x2)) = (α2, β2, t2). If
we take (α, β, t) := (α1∩α2, β1∪β2,min{t1, t2}), then a1, a2 ∈ E(ME ;α), b1, b2 ∈ E(GE ;β)
and x1, x2 ∈ X (ξ; t). Hence a1 ↬ a2 ∈ E(ME ;α), b1 ↬ b2 ∈ E(GE ;β) and x1∗x2 ∈ X (ξ; t).
If we put a1 = a2, b1 = b2, and x1 = x2, then 0 ∈ E(ME ;α) ∩ E(GE ;β) ∩ X (ξ; t), and
so ME(0) ⊇ α = ME(a), GE(0) ⊆ GE(b) and ξ(0) ≥ ξ(x) for all (a, b, x) ∈ E × E × X.
Let a, b, c ∈ E and x, y, z ∈ X be such that ME((a ↬ b) ↬ c) = α1, ME(b ↬ c) = α2

GE((a ↬ b) ↬ c) = β1, GE(b ↬ c) = β2, ξ((x ∗ y) ∗ z) = t1, and ξ(y ∗ z) = t2. If
we take α = α1 ∩ α2, β = β1 ∪ β2 and t = min{t1, t2}, then (a ↬ b) ↬ c ∈ E(ME ;α),
b ↬ c ∈ E(ME ;α), (a ↬ b) ↬ c ∈ E(GE ;α), b ↬ c ∈ E(GE ;α), (x ∗ y) ∗ z ∈ X (ξ; t), and
y ∗ z ∈ X (ξ; t). It follows that a ↬ c ∈ E(ME ;α) ∩ E(GE ;α) and x ∗ z ∈ X (ξ; t). Hence

ME(a ↬ c) ⊇ α = α1 ∩ α2 = ME((a ↬ b) ↬ c) ∩ME(b ↬ c),

GE(a ↬ c) ⊆ β = β1 ∪ β2 = GE((a ↬ b) ↬ c) ∪GE(b ↬ c),

ξ(x ∗ z) ≥ t = min{t1, t2} = min{ξ((x ∗ y) ∗ z), ξ(y ∗ z)}.

Therefore M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal of (X,E).

Note that a makgeolli ideal might not be a positive implicative makgeolli ideal (see
Example 2). But we have the following extension property for a positive implicative
makgeolli ideal.

Theorem 9. Let M(X,E) := (ME , GE , ξ) and N(X,E) := (NE , HE , η) be makgeolli ideals
of (X,E) such that ME(0) = NE(0), GE(0) = HE(0), ξ(0) = η(0), ME(a) ⊆ NE(a),
GE(b) ⊇ HE(b) and ξ(x) ≤ η(x) for all (a, b, x) ∈ E ×E ×X. If M(X,E) := (ME , GE , ξ)
is a positive implicative makgeolli ideal of (X,E), then so is N(X,E) := (NE , HE , η).

Proof. Assume that M(X,E) := (ME , GE , ξ) is a positive implicative makgeolli ideal
of (X,E). Using (I3), (4), Theorem 4 and the given assumption, we have

NE(0) = ME(0) = ME(((a ↬ b) ↬ c) ↬ ((a ↬ b) ↬ c))



S. Z. Song, M. A. Öztürk, Y. B. Jun / Eur. J. Pure Appl. Math, 15 (4) (2022), 1498-1511 1509

= ME(((a ↬ b) ↬ ((a ↬ b) ↬ c)) ↬ c)

= ME(((a ↬ ((a ↬ b) ↬ c)) ↬ b) ↬ c)

⊆ ME(((a ↬ ((a ↬ b) ↬ c)) ↬ c) ↬ (b ↬ c))

⊆ NE(((a ↬ ((a ↬ b) ↬ c)) ↬ c) ↬ (b ↬ c))

= NE(((a ↬ c) ↬ ((a ↬ b) ↬ c)) ↬ (b ↬ c))

= NE(((a ↬ c) ↬ (b ↬ c)) ↬ ((a ↬ b) ↬ c)),

HE(0) = GE(0) = GE(((a ↬ b) ↬ c) ↬ ((a ↬ b) ↬ c))

= GE(((a ↬ b) ↬ ((a ↬ b) ↬ c)) ↬ c)

= GE(((a ↬ ((a ↬ b) ↬ c)) ↬ b) ↬ c)

⊇ GE(((a ↬ ((a ↬ b) ↬ c)) ↬ c) ↬ (b ↬ c))

⊇ HE(((a ↬ ((a ↬ b) ↬ c)) ↬ c) ↬ (b ↬ c))

= HE(((a ↬ c) ↬ ((a ↬ b) ↬ c)) ↬ (b ↬ c))

= HE(((a ↬ c) ↬ (b ↬ c)) ↬ ((a ↬ b) ↬ c))

and

η(0) = ξ(0) = ξ(((x ∗ y) ∗ z) ∗ ((x ∗ y) ∗ z))
= ξ(((x ∗ y) ∗ ((x ∗ y) ∗ z)) ∗ z)
= ξ(((x ∗ ((x ∗ y) ∗ z)) ∗ y) ∗ z)
≤ ξ(((x ∗ ((x ∗ y) ∗ z)) ∗ z) ∗ (y ∗ z))
≤ η(((x ∗ ((x ∗ y) ∗ z)) ∗ z) ∗ (y ∗ z))
= η(((x ∗ z) ∗ ((x ∗ y) ∗ z)) ∗ (y ∗ z))
= η(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)).

It follows from (9) and (10) that

NE((a ↬ c) ↬ (b ↬ c))

⊇ NE(((a ↬ c) ↬ (b ↬ c)) ↬ ((a ↬ b) ↬ c)) ∩NE((a ↬ b) ↬ c)

⊇ NE(0) ∩NE((a ↬ b) ↬ c) = NE((a ↬ b) ↬ c),

HE((a ↬ c) ↬ (b ↬ c))

⊆ HE(((a ↬ c) ↬ (b ↬ c)) ↬ ((a ↬ b) ↬ c)) ∪HE((a ↬ b) ↬ c)

⊆ HE(0) ∪HE((a ↬ b) ↬ c) = HE((a ↬ b) ↬ c),

and

η((x ∗ z) ∗ (y ∗ z))
≥ min{η(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)), η((x ∗ y) ∗ z)}



REFERENCES 1510

≥ min{η(0), η((x ∗ y) ∗ z)} = η((x ∗ y) ∗ z)

for all a, b, c ∈ E and x, y, z ∈ X. Therefore N(X,E) := (NE , HE , η) is a positive implicative
makgeolli ideal of (X,E) by Theorem 4.

4. Conclusions

A fuzzy set is an extension of an existing set using fuzzy logic. Soft set theory is a
generalization of fuzzy set theory. Fuzzy and soft set theory are good mathematical tools
for dealing with uncertainty in a parametric manner. Ahn et al. [2] introduced the concept
of makgeolli structures as a hybrid structure using fuzzy and soft set theory, and applied
it to BCK/BCI-algebras. In this article, we introduced the notion of a positive implicative
makgeolli ideal in BCK-algebras, and investigated its properties. We established the
relationship between makgeolli ideal and positive implicative makgeolli ideal, and explored
the conditions under which makgeolli ideal can be positive implicative makgeolli ideal. We
discussed the characterization of positive implicative makgeolli ideal, and constructed the
extension property for a positive implicative makgeolli ideal.

Acknowledgements

The author, Seok-Zun Song, was supported by Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of Education
(No. 2016R1D1A1B02006812). The authors wish to thank the anonymous reviewers for
their valuable suggestions.

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