EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1862-1877
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Dokdo filters and deductive systems of Sheffer stroke
Hilbert algebras

Sun Shin Ahn1,∗, Hee Sik Kim2, Seok-Zun Song3, Young Bae Jun4

1 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea
2 Research Institute for Natural Science, Department of Mathematics, Hanyang University,
Seoul 04763, Korea
3 Department of Mathematics, Jeju National University, Jeju 63243, Korea
4 Department of Mathematics Education, Gyeongsang National University, Jinju 52828,
Korea

Abstract. To investigate the filter and deductive system of the Schaefer stroke Hilbert algebra
using the Dokdo structure, the concept of Dokdo filter and Dokdo deductive system is defined,
examples are given, and various properties are investigated. The Dokdo filter is formed by attaching
appropriate conditions to the given Dokdo structure. Characterization of Dokdo filter is studied.
Dokdo filters related to filters are constructed. Dokdo filter and Dokdo deductive system turn out
to be the same concept.

2020 Mathematics Subject Classifications: 03B05, 03G25, 06F35, 08A72

Key Words and Phrases: Sheffer stroke Hilbert algebra, filter, deductive system, Dokdo filter,
Dokdo deductive system.

1. Introduction

The shaper stroke represented by the symbol ”|” is a logical operation for two inputs
that produces an invalid result only when both inputs are true, as shown in Table 1.
The Sheffer stroke has been applied to several algebraic structures, for example, Boolean
algebra, MV-algebra, BL-algebra, BCK-algebra, and ortholattices, etc., and it is also being
dealt with in the fuzzy environment (see [1, 4, 5, 10–14]). In 2021, Oner et al. [12] applied
the Sheffer stroke to Hilbert algebras. They introduced Sheffer stroke Hilbert algebra and
investigated several properties. In [11], Oner et al. introduced the notion of deductive
system and filter of Sheffer stroke Hilbert algebras, and dealt with their fuzzification. The
Dokdo structure, classified as a hybrid structure, was introduced by Jun [3], and it consists

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4872

Email addresses: sunshine@dongguk.edu (S. S. Ahn), heekim@hanyang.ac.kr (H. S. Kim),
szsong@jejunu.ac.kr (S. Z. Song), skywine@gmail.com (Y. B. Jun)

https://www.ejpam.com 1862 © 2023 EJPAM All rights reserved.



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1863

Table 1: The truth table for the Sheffer stroke “|”

P Q P |Q
F F T
F T T
T F T
T T F

of a combination of soft set, bipolar fuzzy set, and interval-value fuzzy set. Here, “Dokdo”
is the name of Korea’s most beautiful island.

Focusing on examining the filter and deductive system of the Sheffer stroke Hilbert
algebra using the Dokdo structure, we define the concept of Dokdo filter and Dokdo
deductive system, give examples, and then investigate various properties. We form a
Dokdo filter by attaching appropriate conditions to a given Dokdo structure. We study
characterizations of Dokdo filters. We construct Dokdo filters that are associated with
filters. Ultimately, we show that Dokdo filter and Dokdo deductive system are a matching
concept.

2. Preliminaries

2.1. Preliminaries on Sheffer stroke Hilbert algebras

Definition 1 ([9]). Let A := (A, |) be a groupoid. Then the operation “|” is said to be
Sheffer stroke or Sheffer operation if it satisfies:

(s1) (∀a, b ∈ A) (a|b = b|a),

(s2) (∀a, b ∈ A) ((a|a)|(a|b) = a),

(s3) (∀a, b, c ∈ A) (a|((b|c)|(b|c)) = ((a|b)|(a|b))|c),

(s4) (∀a, b, c ∈ A) ((a|((a|a)|(b|b)))|(a|((a|a)|(b|b))) = a).

Definition 2 ([12]). A Sheffer stroke Hilbert algebra is a groupoid X := (X, |) with a
Sheffer stroke “|” that satisfies:

(sH1) (a|((A)|(A)))|(((B)|((C)|(C)))|((B)|((C)|(C)))) = a|(a|a),
where A := b|(c|c), B := a|(b|b) and C := a|(c|c),

(sH2) a|(b|b) = b|(a|a) = a|(a|a) ⇒ a = b

for all a, b, c ∈ X.

Let X := (X, |) be a Sheffer stroke Hilbert algebra. Then the order relation “ ≤X ” on
X is defined as follows:

(∀a, b ∈ X)(a ≤X b ⇔ a|(b|b) = 1). (1)



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1864

We observe that the relation “ ≤X ” is a partial order in a Sheffer stroke Hilbert
algebra X := (X, |) (see [12]).

Proposition 1 ([12]). Every Sheffer stroke Hilbert algebra X := (X, |) satisfies:

(∀a ∈ X)(a|(a|a) = 1), (2)

(∀a ∈ X)(a|(1|1) = 1), (3)

(∀a ∈ X)(1|(a|a) = a), (4)

(∀a, b ∈ X)(a ≤X b|(a|a)), (5)

(∀a, b ∈ X)((a|(b|b))|(b|b) = (b|(a|a))|(a|a)), (6)

(∀a, b ∈ X) (((a|(b|b))|(b|b))|(b|b) = a|(b|b)) , (7)

(∀a, b, c ∈ X) (a|((b|(c|c))|(b|(c|c))) = b|((a|(c|c))|(a|(c|c)))) , (8)

Definition 3 ([11]). Let (X, |) be a Sheffer stroke Hilbert algebra. A subset F of X is
called

• a deductive system of (X, |) if it satisfies:

1 ∈ F, (9)

(∀a, b ∈ X)(a ∈ F, a|(b|b) ∈ F ⇒ b ∈ F ), (10)

• a filter of (X, |) if it satisfies (9) and

(∀a, b ∈ X)(b ∈ F ⇒ a|(b|b) ∈ F ), (11)

(∀a, b, c ∈ X)(b, c ∈ F ⇒ (a|(b|c))|(b|c) ∈ F ). (12)

2.2. Basic concepts about Dokdo structure

Let X be a set. A bipolar fuzzy set in X (see [6]) is an object of the following type

f̊ = {(a, f̊−(a), f̊+(a)) | a ∈ X} (13)

where f̊− : X → [−1, 0] and f̊+ : X → [0, 1] are mappings. The bipolar fuzzy set which is
described in (13) is simply denoted by f̊ := (X; f̊−, f̊+).

A bipolar fuzzy set can be reinterpreted as a function:

f̊ : X → [−1, 0]× [0, 1], a 7→ (f̊−(a), f̊+(a)).

Denote by BF (X) the set of all bipolar fuzzy sets in X. We define a binary relation
“≤b” on BF (X) as follows:

(∀f̊ , g̊ ∈ BF (X))

(
f̊ ≤b g̊ ⇔

{
f̊−(a) ≥ g̊−(a)

f̊+(a) ≤ g̊+(a)
for all a ∈ X

)
. (14)

Then (BF(X),≤b) is a poset.



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1865

Let U be an initial universe set and X be a set of parameters. For any subset A of X,
a pair (fs, A) is called a soft set over U (see [7, 8]), where fs is a mapping described as
follows:

f s : A → 2U

where 2U is the power set of U . If A = X, the soft set (fs, A) over U is simply denoted
by fs only.

A mapping f̃ : X → [[0, 1]] is called an interval-valued fuzzy set (briefly, an IVF set)
in X (see [2, 15]) where [[0, 1]] is the set of all closed subintervals of [0, 1], and members of
[[0, 1]] are called interval numbers and are denoted by ã, b̃, c̃, etc., where ã = [al, ar] with
0 ≤ al ≤ ar ≤ 1.

For every two interval numbers ã and b̃, we define

ã ⊴ b̃ (or b̃ ⊵ ã) ⇔ al ≤ bl, ar ≤ br, (15)

ã = b̃ ⇔ ã ⊴ b̃, b̃ ⊴ ã, (16)

rmin{ã, b̃} = [min{al, bl},min{ar, br}]. (17)

Let U be an initial universe set and X a set of parameters. A triple Dokf := (f̊ , fs,

f̃) is called a Dokdo structure in (U,X) (see [3]) if f̊ : X → [−1, 0] × [0, 1] is a bipolar
fuzzy set in X, fs : X → 2U is a soft set over U and f̃ : X → [[0, 1]] is an interval-valued
fuzzy set in X.

The Dokdo structure Dokf := (f̊ , fs, f̃) in (U,X) can be represented as follows:

Dokf := (f̊ , f s, f̃) : X → ([−1, 0]× [0, 1])× 2U × [[0, 1]],

x 7→
(
f̊(x), fs(x), f̃(x)

) (18)

where f̊(x) = (f̊−(x), f̊+(x)) and f̃(x) = [f̃L(x), f̃R(x)].
Given a Dokdo structure Dokf := (f̊ , f s, f̃) in a Dokdo universe (U,X), we consider

the following sets:

f̊(M,m) :=

{
x

(y,z) ∈
X

X×X

∣∣∣ f̊−(x) ≤ max{f̊−(y), f̊−(z)}
f̊+(x) ≥ min{f̊+(y), f̊+(z)}

}
and

f̊(t−) := {x ∈ X | f̊−(x) ≤ t−},
f̊(t+) := {x ∈ X | f̊+(x) ≥ t+},
f̊(t−, t+) := f̊(t−) ∩ f̊(t+),

fs
α := {x ∈ X | fs(x) ⊇ α},
f̃ã := {x ∈ X | f̃(x) ⊵ ã},

where (t−, t+) ∈ [−1, 0]× [0, 1], α ∈ 2U and ã = [al, ar].



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1866

3. Dokdo filters

Let U be an initial universe set and X a set of parameters. We say that the pair
(U,X ) is the SSH-Dokdo universe if X := (X, |) is a Sheffer stroke Hilbert algebra. In
what follows, let (U,X ) denote the SSH-Dokdo universe unless otherwise specified.

Definition 4. A Dokdo structure Dokf := (f̊ , fs, f̃) is called a Dokdo filter of (U,X ) if
it satisfies:

(∀x ∈ X)

(
1

(x,x) ∈ f̊(M,m)

fs(1) ⊇ fs(x), f̃(1) ⊵ f̃(x)

)
, (19)

(∀x, y ∈ X)

 x|(y|y)
(y,y) ∈ f̊(M,m)

fs(x|(y|y)) ⊇ f s(y)

f̃(x|(y|y)) ⊵ f̃(y)

 , (20)

(∀x, y, z ∈ X)

 (x|(y|z))|(y|z)
(y,z) ∈ f̊(M,m)

fs((x|(y|z))|(y|z)) ⊇ f s(y) ∩ fs(z)

f̃((x|(y|z))|(y|z)) ⊵ rmin{f̃(y), f̃(z)}

 . (21)

Example 1. Consider a set X = {0, 1, 2, 3, 4, 5, 6, 7}. The Hasse diagram and the Sheffer
stroke “|” on X are given by Figure 1 and Table 2, respectively.

Figure 1: Hasse Diagram

r
r r r
r r r

r

0

2

5

3

6

4

7

1

��
���

HH
HHH

���
��

HHH
HH

���
���

���
�

HHH
HH H

HHH
H

Then X := (X, |) is a Sheffer stroke Hilbert algebra (see [12]). Let Dokf := (f̊ , fs, f̃) be
a Dokdo structure in (X,U = Z) which is given by Table 3.
It is routine to verify that Dokf := (f̊ , f s, f̃) is a Dokdo filter of (U = Z,X ).

Proposition 2. Every Dokdo filter Dokf := (f̊ , f s, f̃) of (U,X ) satisfies:

(∀x, y ∈ X)

 (x|(y|y))|(y|y)
(x,x) ∈ f̊(M,m)

fs((x|(y|y))|(y|y)) ⊇ f s(x)

f̃((x|(y|y))|(y|y)) ⊵ f̃(x)

 , (22)

(∀x, y ∈ X)

 x ≤X y ⇒


y

(x,x) ∈ f̊(M,m)

fs(x) ⊆ fs(y)

f̃(x) ⊴ f̃(y)

 . (23)



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1867

Table 2: Cayley table for the Sheffer stroke “|”

| 0 2 3 4 5 6 7 1

0 1 1 1 1 1 1 1 1
2 1 7 1 1 7 7 1 7
3 1 1 6 1 6 1 6 6
4 1 1 1 5 1 5 5 5
5 1 7 6 1 4 7 6 4
6 1 7 1 5 7 3 5 3
7 1 1 6 5 6 5 2 2
1 1 7 6 5 4 3 2 0

Table 3: Tabular representation of Dokf := (f̊ , fs, f̃)

X f̊(x) fs(x) f̃(x)

0 (−0.41, 0.48) 16N [0.28, 0.65]
2 (−0.55, 0.67) 16N [0.28, 0.65]
3 (−0.41, 0.48) 8N [0.28, 0.65]
4 (−0.41, 0.48) 16N [0.32, 0.73]
5 (−0.63, 0.78) 8N [0.28, 0.65]
6 (−0.55, 0.67) 16N [0.38, 0.76]
7 (−0.41, 0.48) 4N [0.32, 0.73]
1 (−0.71, 0.82) 2N [0.42, 0.91]

Proof. Let Dokf := (f̊ , fs, f̃) be a Dokdo filter of (U,X ). Then

f̊−((x|(y|y))|(y|y)) = f̊−((y|(x|x))|(x|x)) ≤ max{f̊−(x), f̊−(x)} = f̊−(x),

and

f̊+((x|(y|y))|(y|y)) = f̊+((y|(x|x))|(x|x)) ≥ min{f̊+(x), f̊+(x)} = f̊+(x)

by (6) and (21), that is, (x|(y|y))|(y|y)
(x,x) ∈ f̊(M,m) for all x, y ∈ X. Also, we have

fs((x|(y|y))|(y|y)) = f s((y|(x|x))|(x|x)) ⊇ fs(x) ∩ fs(x) = f s(x)

and
f̃((x|(y|y))|(y|y)) = f̃((y|(x|x))|(x|x)) ⊵ rmin{f̃(x), f̃(x)} = f̃(x)

for all x, y ∈ X. Therefore (22) is valid. Let x, y ∈ X be such that x ≤X y. Then
x|(y|y) = 1 by (1). Using (4) and (22), we have

f̊−(y) = f̊−(1|(y|y)) = f̊−((x|(y|y))|(y|y)) ≤ f̊−(x),



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1868

f̊+(y) = f̊+(1|(y|y)) = f̊+((x|(y|y))|(y|y)) ≥ f̊+(x),

which shows that y
(x,x) ∈ f̊(M,m). Also we get

fs(x) ⊆ fs((x|(y|y))|(y|y)) = fs(1|(y|y)) = fs(y)

and f̃(x) ⊴ f̃((x|(y|y))|(y|y)) = f̃(1|(y|y)) = f̃(y).

We have a question: If a Dokdo structure Dokf := (f̊ , fs, f̃) in (U,X ) satisfies the
condition (23) then is it a Dokdo filter of (U,X )? The example below provides a negative
answer to the question.

Example 2. Consider a set X = {0, 1, 2, 3}. The Hasse diagram and the Sheffer stroke
“|” on X are given by Figure 2 and Table 4, respectively.

Figure 2: Hasse Diagram

rr rr

0

2 3

1

�
�

A
A
�
�

A
A

Table 4: Cayley table for the Sheffer stroke “|”

| 1 2 3 0

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

Then X := (X, |) is a Sheffer stroke Hilbert algebra (see [12]). Let Dokf := (f̊ , fs, f̃) be
a Dokdo structure in (U = Z, X) which is given by Table 5.

Table 5: Tabular representation of Dokf := (f̊ , fs, f̃)

X f̊(x) fs(x) f̃(x)

0 (−0.13, 0.10) 8N [0.29, 0.63]
2 (−0.38, 0.17) 4N [0.32, 0.67]
3 (−0.55, 0.29) 4Z [0.36, 0.75]
1 (−0.82, 0.63) 2Z [0.47, 0.89]



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1869

It is routine to check that Dokf := (f̊ , fs, f̃) in (U,X ) satisfies the condition (23). But
it is not a Dokdo filter of (U = Z, X) since

(0|(2|3))|(2|3)
(2,3) = 0

(2,3) /∈ f̊(M,m),

fs((0|(2|3))|(2|3)) = fs(0) = 8N ⊊ 4N = fs(2) ∩ f s(3) or

f̃((0|(2|3))|(2|3)) = f̃(0) = [0.29, 0.63] ̸⊵ [0.32, 0.67] = rmin{f̃(2), f̃(3)}.

We provide conditions for the Dokdo structure to be the Dokdo filter.

Theorem 1. Let Dokf := (f̊ , f s, f̃) be a Dokdo structure in (U,X). Then it is Dokdo
filter of (U,X) if and only if it satisfies the condition (23) and

(∀x, y ∈ X)

 (x|y)|(x|y)
(x,y) ∈ f̊(M,m),

fs((x|y)|(x|y)) ⊇ f s(x) ∩ fs(y),

f̃((x|y)|(x|y)) ⊵ rmin{f̃(x), f̃(y)}

 . (24)

Proof. Let Dokf := (f̊ , fs, f̃) be a Dokdo filter of (U,X). The condition (23) is valid
by Proposition 2. Since

((1|1)|(x|y))|(x|y) (s1)
= ((x|y)|(1|1))|(x|y) (3)

= 1|(x|y)
(s2)
= 1|(((x|y)|(x|y))|((x|y)|(x|y)))
(4)
= (x|y)|(x|y)

for all x, y ∈ X, it follows from (21) that

(x|y)|(x|y)
(x,y) = ((1|1)|(x|y))|(x|y)

(x,y) ∈ f̊(M,m),

fs((x|y)|(x|y)) = f s(((1|1)|(x|y))|(x|y)) ⊇ fs(x) ∩ fs(y)}

and f̃((x|y)|(x|y)) = f̃(((1|1)|(x|y))|(x|y)) ⊵ rmin{f̃(x), f̃(y)} for all x, y ∈ X.
Conversely, suppose that a Dokdo structure Dokf := (f̊ , fs, f̃) satisfies the conditions

(23) and (24). Since x ≤X 1 for all x ∈ X, we have 1
(x,x) ∈ f̊(M,m), fs(x) ⊆ fs(1), and

f̃(x) ⊴ f̃(1) by (23). Since y ≤X x|(y|y) for all x, y ∈ X, we have x|(y|y)
(y,y) ∈ f̊(M,m),

fs(y) ⊆ fs(x|(y|y)), and f̃(y) ⊴ f̃(x|(y|y)) by (23). In (5), if we replace a and b with
(y|z)|(y|z) and x|(y|z), respectively, and use (s2), then

(y|z)|(y|z) ≤X (x|(y|z))|(((y|z)|(y|z))|((y|z)|(y|z))) = (x|(y|z))|(y|z)

for all x, y, z ∈ X. Using (23) and (24), we have

(x|(y|z))|(y|z)
((y|z)|(y|z),(y|z)|(y|z)) ∈ f̊(M,m),



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1870

and so max{f−(y), f−(z)} ≥ f−((y|z)|(y|z)) ≥ f−((x|(y|z))|(y|z)) and

min{f+(y), f+(z)} ≤ f+((y|z)|(y|z)) ≤ f+((x|(y|z))|(y|z)).

Hence (x|(y|z))|(y|z)
(y,z) ∈ f̊(M,m). Also, we have

fs(y) ∩ fs(z) ⊆ fs((y|z)|(y|z)) ⊆ fs((x|(y|z))|(y|z))

and rmin{f̃(y), f̃(z)} ⊴ f̃((y|z)|(y|z)) ⊴ f̃((x|(y|z))|(y|z)). Therefore, Dokf := (f̊ , fs, f̃)
is a Dokdo filter of (U,X).

Theorem 2. If Dokf := (f̊ , f s, f̃) is a Dokdo filter of (U,X), then the sets f̊(t−, t+), fs
α

and f̃ã are filters of X := (X, |) whenever they are nonempty for all (t−, t+) ∈ [−1, 0] ×
[0, 1], α ∈ 2U and ã = [al, ar].

Proof. Assume that Dokf := (f̊ , f s, f̃) is a Dokdo filter of (U,X) and let (t−, t+) ∈
[−1, 0]× [0, 1], α ∈ 2U and ã = [al, ar] be such that f̊(t−, t+), fs

α and f̃ã are nonempty. It
is clear that 1 ∈ f̊(t−, t+) ∩ fs

α ∩ f̃ã by (19). Let x ∈ X and y ∈ f̊(t−, t+) ∩ fs
α ∩ f̃ã. Then

f̊−(y) ≤ t−, f̊+(y) ≥ t+, fs(y) ⊇ α, and f̃(y) ⊵ ã. Using (20), we have f̊−(x|(y|y)) ≤
f̊−(y) ≤ t− and f̊+(x|(y|y)) ≥ f̊+(y) ≥ t+, that is, x|(y|y) ∈ f̊(t−, t+). Also, we obtain
fs(x|(y|y)) ⊇ fs(y) ⊇ α and f̃(x|(y|y)) ⊵ f̃(y) ⊵ ã. Hence x|(y|y) ∈ f s

α ∩ f̃ã. Let x ∈ X
and y, z ∈ f̊(t−, t+) ∩ fs

α ∩ f̃ã. Then f̊−(y) ≤ t−, f̊+(y) ≥ t+, f s(y) ⊇ α, f̃(y) ⊵ ã,
f̊−(z) ≤ t−, f̊+(z) ≥ t+, f s(z) ⊇ α, and f̃(z) ⊵ ã. It follows from (21) that

f̊−((x|(y|z))|(y|z)) ≤ max{f̊−(y), f̊−(z)} ≤ t−,

f̊+((x|(y|z))|(y|z)) ≥ min{f̊+(y), f̊+(z)} ≥ t+,

i.e., (x|(y|z))|(y|z) ∈ f̊(t−, t+). Also, f s((x|(y|z))|(y|z)) ⊇ fs(y) ∩ fs(z) ⊇ α and

f̃((x|(y|z))|(y|z)) ⊵ rmin{f̃(y), f̃(z)} ⊵ ã.

Thus (x|(y|z))|(y|z) ∈ fs
α ∩ f̃ã. Therefore, f̊(t

−, t+), fs
α and f̃ã are filters of X := (X, |).

The example below shows that the converse of Theorem 2 may not be true.

Example 3. Consider the Sheffer stroke Hilbert algebra X := (X, |) in Example 1 and let
Dokf := (f̊ , f s, f̃) be a Dokdo structure in (X,U = Z) which is given by Table 6.

It is routine to verify that the nonempty sets f̊(t−, t+), fs
α and f̃ã are filters of X := (X, |)

for all (t−, t+) ∈ [−1, 0]× [0, 1], α ∈ 2U and ã = [al, ar]. But Dokf := (f̊ , f s, f̃) is not a

Dokdo filter of (U,X) since 2|(4|4)
(4,4) = 7

(4,4) /∈ f̊(M,m).

We provide conditions for a Dokdo structure to be a Dokdo filter.

Theorem 3. Given a Dokdo stucture Dokf := (f̊ , fs, f̃) in (U,X), If the nonempty sets

f̊(t−), f̊(t+), fs
α and f̃ã are filters of X := (X, |) for all (t−, t+) ∈ [−1, 0]× [0, 1], α ∈ 2U

and ã = [al, ar], then Dokf := (f̊ , f s, f̃) is a Dokdo filter of (U,X).



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1871

Table 6: Tabular representation of Dokf := (f̊ , fs, f̃)

X f̊(x) fs(x) f̃(x)

0 (−0.37, 0.48) 8N [0.26, 0.62]
2 (−0.37, 0.67) 8N [0.26, 0.62]
3 (−0.37, 0.48) 8Z [0.26, 0.62]
4 (−0.55, 0.48) 8N [0.31, 0.70]
5 (−0.63, 0.78) 8Z [0.26, 0.62]
6 (−0.55, 0.48) 8N [0.36, 0.73]
7 (−0.37, 0.48) 4Z [0.32, 0.70]
1 (−0.71, 0.82) 2Z [0.41, 0.88]

Proof. Assume that f̊(t−), f̊(t+), fs
α and f̃ã are nonempty filters of X := (X, |) for

all (t−, t+) ∈ [−1, 0] × [0, 1], α ∈ 2U and ã = [al, ar]. If there is a ∈ X such that
1

(a,a) /∈ f̊(M,m), then f̊−(1) > f̊−(a) or f̊+(1) < f̊+(a). Hence a ∈ f̊(f̊−(a)) ∩ f̊(f̊+(a))

and 1 /∈ f̊(f̊−(a)) ∩ f̊(f̊+(a)), a contradiction. Thus 1
(x,x) ∈ f̊(M,m) for all x ∈ X. Let

x, a ∈ X be such that fs(x) = α and f̃(a) = ã. Then (x, a) ∈ fs
α × f̃ã, i.e., f

s
α ̸= ∅ ̸= f̃ã,

and so 1 ∈ fs
α ∩ f̃ã. Hence f s(1) ⊇ α = fs(x) and f̃(1) ⊵ ã = f̃(a). If there are a, b ∈ X

such that a|(b|b)
(b,b) /∈ f̊(M,m), then f̊−(a|(b|b)) > f̊−(b) or f̊+(a|(b|b)) < f̊+(b). It follows

that b ∈ f̊(f̊−(b)) ∩ f̊(f̊+(b)) and a|(b|b) /∈ f̊(f̊−(b)) ∩ f̊(f̊+(b)), a contradiction. Hence
x|(y|y)
(y,y) ∈ f̊(M,m) for all x, y ∈ X. Let y, b ∈ X be such that f s(y) = α and f̃(b) = ã.

Then (y, b) ∈ f s
α × f̃ã, which implies that (x|(y|y), a|(b|b)) ∈ fs

α × f̃ã for all x, a ∈ X.
Hence fs(x|(y|y)) ⊇ α = fs(y) and f̃(a|(b|b)) ⊵ ã = f̃(b). If there are a, b, c ∈ X such

that (a|(b|c))|(b|c)
(b,c) /∈ f̊(M,m), then

f̊−((a|(b|c))|(b|c)) > max{f̊−(b), f̊−(c)}

or f̊+((a|(b|c))|(b|c)) < min{f̊+(b), f̊+(c)}. If we take t− := max{f̊−(b), f̊−(c)} and t+ :=
min{f̊+(b), f̊+(c)}, then b, c ∈ f̊(t−) ∩ f̊(t+) and (a|(b|c))|(b|c) /∈ f̊(t−) ∩ f̊(t+). This is a

contradiction, and thus (x|(y|z))|(y|z)
(y,z) ∈ f̊(M,m) for all x, y, z ∈ X. Let (x, a), (y, b), (z, c) ∈

X × X be such that fs(y) ∩ fs(z) = α and rmin{f̃(b), f̃(c)} = ã. Then y, z ∈ fs
α

and b, c ∈ f̃ã. It follows that (x|(y|z))|(y|z) ∈ f s
α and (a|(b|c))|(b|c) ∈ f̃ã. Therefore

fs((x|(y|z))|(y|z)) ⊇ α = fs(y) ∩ fs(z) and f̃((a|(b|c))|(b|c)) ⊵ ã = rmin{f̃(b), f̃(c)}.
Consequently, Dokf := (f̊ , fs, f̃) is a Dokdo filter of (U,X).

Theorem 4. Given a nonempty subset F of X, let DokfF := (f̊F , f
s
F , f̃F ) be a Dokdo

structure in (U,X) defined by

DokfF := (f̊F , f
s
F , f̃F ) : X → ([−1, 0]× [0, 1])× 2U × [[0, 1]],

x 7→
{

((t−, t+), α, ã) if x ∈ F,(
(0, 0), ∅, 0̃

)
otherwise,

(25)



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1872

where t− ̸= 0 ̸= t+, α ̸= ∅ and ã ̸= 0̃ := [0, 0]. Then DokfF := (f̊F , f
s
F , f̃F ) is a Dokdo

filter of (U,X) if and only if F is a filter of X := (X, |). Moreover, we have

F = XfF := {x ∈ X | f̊F (x) = f̊F (1), f
s
F (x) = fs

F (1), f̃F (x) = f̃F (1)}.

Proof. Assume that DokfF := (f̊F , f s
F , f̃F ) is a Dokdo filter of (U,X). Then

DokfF (1) = ((t−, t+), α, ã) by (19), and so 1 ∈ F . Let x, y ∈ X be such that y ∈ F .

Then f̊−
F (y) = t−, f̊+

F (y) = t+, f s
F (y) = α, and f̃F (y) = ã. It follows from (20) that

f̊−
F (x|(y|y)) ≤ f̊−

F (y) = t−, f̊+
F (x|(y|y)) ≥ f̊+

F (y) = t+, fs
F (x|(y|y)) ⊇ fs

F (y) = α and

f̃F (x|(y|y)) ⊵ f̃F (y) = ã. Hence f̊F (x|(y|y)) = (t−, t+), f s
F (x|(y|y)) = α and f̃F (x|(y|y)) =

ã. This shows that x|(y|y) ∈ F . Let y, z ∈ F . Then f̊−
F (y) = t− = f̊−

F (z), f̊+
F (y) = t+ =

f̊+
F (z), f s

F (y) = α = fs
F (z), and f̃F (y) = ã = f̃F (z). Using (21), we have

f̊−
F ((x|(y|z))|(y|z)) ≤ max{f̊−

F (y), f̊−
F (z)} = t−,

f̊+
F ((x|(y|z))|(y|z)) ≥ min{f̊+

F (y), f̊+
F (z)} = t+,

fs
F ((x|(y|z))|(y|z)) ⊇ fs

F (y)∩fs
F (z) = α and f̃F ((x|(y|z))|(y|z)) ⊵ rmin{f̃F (y), f̃F (z)} = ã.

It follows that f̊F ((x|(y|z))|(y|z)) = (t−, t+), f s
F ((x|(y|z))|(y|z)) = α and

f̃F ((x|(y|z))|(y|z)) = ã.

Hence (x|(y|z))|(y|z) ∈ F . Therefore F is a filter of X := (X, |).
Conversely, let F be a filter of X := (X, |). Since 1 ∈ F , we have f̊−(1) = t− ≤ f̊−(x)

and f̊+(1) = t+ ≥ f̊+(x), and so 1
(x,x) ∈ f̊(M,m) for all x ∈ X. Also fs(1) = α ⊇ fs(x)

and f̃(1) = ã ⊵ f̃(x) for all x ∈ X. Let x, y ∈ X. If y ∈ F , then x|(y|y) ∈ F , and thus

f̊−(x|(y|y)) = t− = f̊−(y) and f̊+(x|(y|y)) = t+ = f̊+(y). Hence x|(y|y)
(y,y) ∈ f̊(M,m). Also

we get fs(x|(y|y)) = α = fs(y) and f̃(x|(y|y)) = ã = f̃(y). If y /∈ F , then it is clear

that x|(y|y)
(y,y) ∈ f̊(M,m), fs(x|(y|y)) ⊇ f s(y) and f̃(x|(y|y)) ⊵ f̃(y). Let x, y, z ∈ X. It is

obvious that if y /∈ F or z /∈ F , (x|(y|z))|(y|z)
(y,z) ∈ f̊(M,m), fs((x|(y|z))|(y|z)) ⊇ fs(y)∩ fs(z)

and f̃((x|(y|z))|(y|z)) ⊵ rmin{f̃(y), f̃(z). Suppose that y, z ∈ F . Then (x|(y|z))|(y|z) ∈
F . Thus f̊−((x|(y|z))|(y|z)) = t− = max{f̊−(y), f̊−(z)} and f̊+((x|(y|z))|(y|z)) = t+ =

min{f̊+(y), f̊+(z)}. Hence (x|(y|z))|(y|z)
(y,z) ∈ f̊(M,m). Also we get fs((x|(y|z))|(y|z)) = α =

fs(y)∩ fs(z) and f̃((x|(y|z))|(y|z)) = ã = rmin{f̃(y), f̃(z)}. Consequently, DokfF := (f̊F ,
fs
F , f̃F ) is a Dokdo filter of (U,X). Since F is a filter of X := (X, |), we get

XfF = {x ∈ X | f̊F (x) = f̊F (1), f
s
F (x) = fs

F (1), f̃F (x) = f̃F (1)}
= {x ∈ X | f̊F (x) = (t−, t+), f s

F (x) = α, f̃F (x) = ã}
= {x ∈ X | x ∈ F} = F.

This completes the proof.



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1873

Definition 5. A Dokdo structure Dokf := (f̊ , fs, f̃) is called a Dokdo deductive system
of (U,X ) if it satisfies (19) and

(∀x, y ∈ X)

 y
(x,x|(y|y)) ∈ f̊(M,m),

fs(y) ⊇ fs(x) ∩ fs(x|(y|y)),
f̃(y) ⊵ rmin{f̃(x), f̃(x|(y|y))}

 . (26)

Example 4. Consider the Sheffer stroke Hilbert algebra X := (X, |) in Example 1 and let
Dokf := (f̊ , f s, f̃) be a Dokdo structure in (X,U = Z) which is given by Table 7.

Table 7: Tabular representation of Dokf := (f̊ , fs, f̃)

X f̊(x) fs(x) f̃(x)

0 (−0.37, 0.48) 8N [0.19, 0.54]
2 (−0.57, 0.67) 4N [0.24, 0.62]
3 (−0.37, 0.48) 8Z [0.19, 0.54]
4 (−0.37, 0.48) 8N [0.19, 0.54]
5 (−0.61, 0.72) 4Z [0.26, 0.63]
6 (−0.57, 0.67) 4N [0.24, 0.62]
7 (−0.37, 0.48) 8N [0.19, 0.54]
1 (−0.69, 0.79) 2Z [0.41, 0.87]

It is routine to verify that Dokf := (f̊ , f s, f̃) is a Dokdo deductive system of (X,U = Z).

Theorem 5. A Dokdo structure Dokf := (f̊ , fs, f̃) in (X,U) is a Dokdo deductive system
of (X,U) if and only if it is a Dokdo filter of (X,U).

Proof. Assume that Dokf := (f̊ , fs, f̃) is a Dokdo deductive system of (X,U) and let
x, y, z ∈ X. Using (1) and (5) induces y|((x|(y|y))|(x|(y|y))) = 1. It follows from (19) and
(26) that

f̊−(x|(y|y)) ≤ max{f̊−(y), f̊−(y|((x|(y|y))|(x|(y|y))))}
= max{f̊−(y), f̊−(1)} = f̊−(y),

f̊+(x|(y|y)) ≥ min{f̊+(y), f̊+(y|((x|(y|y))|(x|(y|y))))}
= min{f̊+(y), f̊+(1)} = f̊+(y),

that is, x|(y|y)
(y,y) ∈ f̊(M,m), and

fs(x|(y|y)) ⊇ fs(y) ∩ f s(y|((x|(y|y))|(x|(y|y)))) = f s(y) ∩ fs(1)} = fs(y)



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1874

and f̃(x|(y|y)) ⊵ rmin{f̃(y), f̃(y|((x|(y|y))|(x|(y|y))))} = rmin{f̃(y), f̃(1)} = f̃(y). Note
that

y|(((y|z)|z)|((y|z)|z)) = y|(((y|z)|((z|z)|(z|z)))|((y|z)|((z|z)|(z|z))))
= (y|z)|((y|((z|z)|(z|z)))|(y|((z|z)|(z|z))))
= (y|z)|((y|z)|(y|z)) = 1

by (s2), (2) and (8). Using (19) and (26), we have

f̊−((y|z)|z) ≤ max{f̊−(y), f̊−(y|(((y|z)|z)|((y|z)|z)))}
= max{f̊−(y), f̊−(1)} = f̊−(y),

f̊+((y|z)|z) ≥ min{f̊+(y), f̊+(y|(((y|z)|z)|((y|z)|z)))}
= min{f̊+(y), f̊+(1)} = f̊+(y),

fs((y|z)|z) ⊇ fs(y) ∩ fs(y|(((y|z)|z)|((y|z)|z))) = f s(y) ∩ fs(1)} = fs(y), and

f̃((y|z)|z) ⊵ rmin{f̃(y), f̃(y|(((y|z)|z)|((y|z)|z)))} = rmin{f̃(y), f̃(1)} = f̃(y).

Since z|(((y|z)|(y|z))|((y|z)|(y|z))) (s2)
= z|(y|z) (s1)

= (y|z)|z, we obtain

f̊−((y|z)|(y|z)) ≤ max{f̊−(z), f̊−(z|(((y|z)|(y|z))|((y|z)|(y|z))))}
= max{f̊−(z), f̊−((y|z)|z)}
≤ max{f̊−(z), f̊−(y)},

f̊+((y|z)|(y|z)) ≥ min{f̊+(z), f̊+(z|(((y|z)|(y|z))|((y|z)|(y|z))))}
= min{f̊+(z), f̊+((y|z)|z)}
≥ min{f̊+(z), f̊+(y)},

fs((y|z)|(y|z)) ⊇ fs(z) ∩ fs(z|(((y|z)|(y|z))|((y|z)|(y|z))))
= fs(z) ∩ fs((y|z)|z)} ⊇ f s(z) ∩ fs(y)},

and

f̃((y|z)|(y|z)) ⊵ rmin{f̃(z), f̃(z|(((y|z)|(y|z))|((y|z)|(y|z))))}
= rmin{f̃(z), f̃((y|z)|z)}
⊵ rmin{f̃(z), f̃(y)}.

Therefore

f̊−((x|(y|z))|(y|z))
= f̊−((x|(((y|z)|(y|z))|((y|z)|(y|z))))|(((y|z)|(y|z))|((y|z)|(y|z))))
≤ f̊−((y|z)|(y|z)) ≤ max{f̊−(z), f̊−(y)}



S. S. Ahn et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1862-1877 1875

and

f̊+((x|(y|z))|(y|z))
= f̊+((x|(((y|z)|(y|z))|((y|z)|(y|z))))|(((y|z)|(y|z))|((y|z)|(y|z))))
≥ f̊+((y|z)|(y|z)) ≥ min{f̊+(z), f̊+(y)},

that is, (x|(y|z))|(y|z)
(y,z) ∈ f̊(M,m), and

fs((x|(y|z))|(y|z))
= fs((x|(((y|z)|(y|z))|((y|z)|(y|z))))|(((y|z)|(y|z))|((y|z)|(y|z))))
⊇ fs((y|z)|(y|z)) ⊇ fs(z) ∩ fs(y),

and

f̃((x|(y|z))|(y|z))
= f̃((x|(((y|z)|(y|z))|((y|z)|(y|z))))|(((y|z)|(y|z))|((y|z)|(y|z))))
⊵ f̃((y|z)|(y|z)) ⊵ rmin{f̃(z), f̃(y)}.

Consequently, Dokf := (f̊ , fs, f̃) is a Dokdo filter of (X,U).

Conversely, suppose that Dokf := (f̊ , f s, f̃) is a Dokdo filter of (X,U). For every
x, y ∈ X, we have

y = ((x|x)|(1|1))|(y|y) = ((x|x)|((y|(y|y))|(y|(y|y))))|(y|y)
= ((((x|x)|y)|((x|x)|y))|(y|y))|(y|y)
= (y|((x|x)|y))|((x|x)|y)
= ((((x|x)|y)|y)|y)|(((x|x)|y)|y)
= (y|(x|(x|(y|y))))|(x|(x|(y|y)))

by (s1), (s2), (s3), (2), (3), (4) (6) and (7). It follows from (21) that

f̊−(y) = f̊−((y|(x|(x|(y|y))))|(x|(x|(y|y)))) ≤ max{f̊−(x), f̊−(x|(y|y))},
f̊+(y) = f̊+((y|(x|(x|(y|y))))|(x|(x|(y|y)))) ≥ min{f̊+(x), f̊+(x|(y|y))},

that is, y
(x,x|(y|y)) ∈ f̊(M,m), and

fs(y) = fs((y|(x|(x|(y|y))))|(x|(x|(y|y)))) ⊇ fs(x) ∩ fs(x|(y|y)),

and f̃(y) = f̃((y|(x|(x|(y|y))))|(x|(x|(y|y)))) ⊵ rmin{f̃(x), f̃(x|(y|y))}. Therefore Dokf :=

(f̊ , fs, f̃) is a Dokdo deductive system of (X,U).

Remark 1. By Theorem 5, it can be seen that all the results for the Dokdo filter covered
above can be handled in the same way using the Dokdo deductive system.



REFERENCES 1876

4. Conclusion

The Dokdo structure Dokf := (f̊ , f s, f̃) in a set X consists of a combination of soft
set, bipolar fuzzy set, and interval-value fuzzy set, and it can be shaped into a pentagon
as shown in the figure below.

r
r r

r r

2X

[−1, 0]

X

[0, 1]

�
�

�
�

�
�

Q
Q

Q
Q
Q

Q
B
B
B
B
B
BB �

�
�
�
�
��

[[0, 1]]

����1f+

PPPPi f−

?

f̃

6
fs

where fs is a soft set of X, f̊ := (X; f−, f+) is a bipolar fuzzy set in X, and f̃ : X → [[0, 1]]
is an interval-valued fuzzy set in X. What this paper intends to do is look at filters
and deductive systems in Sheffer stroke Hilbert algebras using the Dokdo structure. We
defined the concept of Dokdo filter and Dokdo deductive system, and investigated several
properties. We formed a Dokdo filter by attaching appropriate conditions to a given Dokdo
structure. We studied characterizations of Dokdo filters. We constructed Dokdo filters
that are associated with filters. Finally, we showed that the Dokdo filter is consistent with
the Dokdo deductive system, which means that all the results covered using the Dokdo
filter can be treated using Dokdo deductive system. As an application of the Dokdo
structure in the future, we will apply it to other logical algebras such as BL-algebras,
MTL-algebras, hoops, Sheffer stroke Hilbert algebras, Sheffer stroke BE-algebras, etc.
Based on these studies, we think the possibility of application to decision-making theory,
pattern recognition, and medical diagnosis systems, etc. will open up.

Acknowledgements

The authors wish to thank the anonymous reviewers for their valuable suggestions.
The 3rd author (S. Z. Song) was partially supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry of
Education [grant number 2016R1D1A1B02006812].

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