EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5876
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Semidetached SUP-Subalgebras of Sheffer Stroke
UP-Algebras

Tahsin Oner1, Neelamegarajan Rajesh2, Aiyared Iampan3,∗, Ibrahim Senturk1

1 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey
2 Department of Mathematics, Rajah Serfoji Government College, Thanjavur-613005,
Tamil Nadu, India
3 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang,
Phayao 56000, Thailand

Abstract. The notion of semidetached Sheffer stroke UP-algebras is introduced, and their proper-
ties are investigated. Several conditions for a semidetached structure in Sheffer stroke UP-algebras
to be a semidetached SUP-subalgebra are provided. The concepts of (∈,∈ ∨ qk)-fuzzy SUP-
subalgebra, k-left (k-right) (qk,∈ ∨ qk)-fuzzy SUP-subalgebra, (qk,∈ ∨ qk)-fuzzy SUP-subalgebra
and (∈∨ qk,∈∨ qk)-fuzzy SUP-subalgebra are introduced, and relative relations and properties are
discussed.

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

Key Words and Phrases: Sheffer stroke UP-algebra, (∈,∈ ∨ qk)-fuzzy SUP-subalgebra, k-left
(k-right) (qk,∈ ∨ qk)-fuzzy SUP-subalgebra, (∈ ∨ qk,∈ ∨ qk)-fuzzy SUP-subalgebra, semidetached
SUP-subalgebra

1. Introduction

The Sheffer operation, commonly referred to as the Sheffer stroke or NAND operator,
was first introduced by Sheffer [1]. This operation is particularly notable for its ability to
form a complete logical system on its own, without relying on any other logical connectives.
In fact, any logical axiom can be expressed using only the Sheffer stroke, which simplifies
the manipulation and analysis of logical systems. Moreover, all the axioms of Boolean
algebra, the algebraic foundation of classical propositional logic, can also be expressed
exclusively with the Sheffer operation. This underscores the fundamental role of the Sheffer
stroke in both logic and algebra, showcasing its power and flexibility in constructing and
understanding logical frameworks.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5876

Email addresses: tahsin.oner@ege.edu.tr (T. Oner),
nrajesh topology@yahoo.co.in (N. Rajesh), aiyared.ia@up.ac.th (A. Iampan),
ibrahim.senturk@ege.edu.tr (I. Senturk)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 2 of 16

Building upon this foundational framework, Sheffer stroke UP-algebras establish a dis-
tinctive algebraic structure that seamlessly integrates logical and algebraic principles. As
elaborated by Iampan [2] in 2017, UP-algebras mark a significant advancement in the
field of logical algebra, introducing a versatile paradigm for analyzing algebraic systems
enriched with sophisticated logical constructs. This innovative approach not only deepens
our comprehension of the interplay between logical operations and their algebraic counter-
parts but also unlocks potential applications across diverse disciplines, including decision
theory and computational logic. By bridging these domains, Sheffer stroke UP-algebras
demonstrate their capacity to contribute to both theoretical exploration and practical
problem-solving.

Sheffer stroke UP-algebras lie at the crossroads of logic, algebra, and analysis, provid-
ing a unique framework for exploration. These algebras are pre-normed structures where
the primary operation is the Sheffer stroke logic, and they possess a multiplicative identity
element. They serve as a useful tool for investigating logical systems within an algebraic
context, with potential applications in areas such as functional analysis, operator theory,
and the algebraic study of logic. By incorporating the Sheffer stroke as a core opera-
tion, these algebras extend the concept of Boolean algebras, offering a broader perspective
on logical and algebraic interactions. Recent works have highlighted this richness: Ra-
jesh et al. [3] introduced the notion of intuitionistic fuzzy subalgebras in Sheffer stroke
UP-algebras and established important structural properties of their level sets, while Vid-
hya et al. [4] further extended this framework by exploring neutrosophic N-subalgebras
and their corresponding lattice-theoretic characterizations. These studies emphasize the
growing relevance of fuzzy and neutrosophic perspectives in the theory of Sheffer stroke
UP-algebras, paving the way for deeper investigations into generalized substructures, such
as the semidetached forms explored in this paper.

The concept of the quasi-coincidence of a fuzzy point with a fuzzy set, as discussed in
Bhakat and Das’s pioneering work [5], has been instrumental in shaping the development
of various classifications of fuzzy subgroups. This innovative approach extends traditional
notions, enabling the formulation of new types of fuzzy subgroups that have broadened
the scope of algebraic studies in this domain. Notably, the (∈,∈ ∨q)-fuzzy subgroup
represents a significant and practical generalization of Rosenfeld’s foundational concept
of fuzzy subgroups [6], thereby offering a more flexible framework for understanding the
structural relationships in fuzzy algebra.

The motivation for studying semidetached SUP-subalgebras arises from the need to
extend the algebraic understanding of Sheffer stroke-based logical systems under uncer-
tainty. By incorporating fuzzy sets and semidetached structures into SUP-algebras, the
framework becomes more flexible and applicable to real-world scenarios involving par-
tial truth or threshold reasoning. These structures have potential applications in areas
such as fuzzy decision-making, knowledge representation in AI, and logical circuit design,
particularly where NAND logic and graded membership play a central role.

In this paper, we introduce the concepts of (∈,∈ ∨ qk)-fuzzy SUP-subalgebras, k-
left (k-right) (qk,∈ ∨ qk)-fuzzy SUP-subalgebras, (qk,∈ ∨ qk)-fuzzy SUP-subalgebras, and
(∈ ∨ qk,∈ ∨ qk)-fuzzy SUP-subalgebras, and investigate relative relations and properties.



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 3 of 16

We provide several conditions for a semidetached structure in SUP-algebras to be a semide-
tached SUP-subalgebra.

2. Preliminaries

Sheffer Stroke UP-algebras epitomize a fascinating convergence of algebraic theory and
logical principles, distinguished by the Sheffer stroke operation, which serves as a funda-
mental connective within the realm of propositional calculus. This particular algebraic
construct not only augments our comprehension of logical operations but also bears con-
siderable ramifications in disciplines such as computer science and decision theory. The
present article will explore the definitions and foundational elements of Sheffer Stroke
UP-algebras, underscoring their significance within the context of algebraic theory.

Definition 1. [1] Let ⟨X, |⟩ be a groupoid. The operation | is said to be a Sheffer stroke
operation if it satisfies the following conditions: for all x, y, z ∈ X,

(S1) x|y = y|x
(S2) (x|x)|(x|y) = x
(S3) x|((y|z)|(y|z)) = ((x|y)|(x|y))|z
(S4) (x|((x|x)|(y|y)))|(x|((x|x)|(y|y))) = x.

Definition 2. [7] A Sheffer stroke UP-algebra (briefly, SUP-algebra) is a structure ⟨X, |, 0⟩
of type (2, 0) such that 0 is the fixed element in X and the following conditions are satisfied
for all x, y, z ∈ X,

(SUP-1) (((z|(x|x))|(z|(x|x)))|(((y|(x|x))|(z|(y|y)))|((y|(x|x))|
(z|(y|y)))))|(((z|(x|x))|(z|(x|x)))|(((y|(x|x))|(z|(y|y)))|
((y|(x|x))|(z|(y|y))))) = 0

(SUP-2) x|x = x|(0|0)
(SUP-3) (x|(y|y))|(x|(y|y)) = 0 and (y|(x|x))|(y|(x|x)) = 0 ⇒ x = y.

Proposition 1. [7] Let ⟨X, |, 0⟩ be an SUP-algebra. Then the binary relation x ≤ y if and
only if (y|(x|x))|(y|(x|x)) = 0 is a partial order on X.

Definition 3. [7] A nonempty subset G of an SUP-algebra ⟨X, |, 0⟩ is called an SUP-
subalgebra of X if (x|(y|y))|(x|(y|y)) ∈ G for all x, y ∈ G.

Lemma 1. [7] Let ⟨X, |, 0⟩ be an SUP-algebra. Then for all x, y, z ∈ X, we have

(1) x ≤ y ⇒ y|(z|z) ≤ x|(z|z) and z|(x|x) ≤ z|(y|y)

(2) x ≤ y ⇔ y|y ≤ x|x

(3) y|(x|x) ≤ x

(4) y ≤ (y|(x|x))|(y|(x|x))



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 4 of 16

(5) x ≤ y ⇒ x ≤ (y|(z|z))|(y|(z|z))

(6) z|(y|y) ≤ z|(y|(x|x))

(7) ((z|(y|y))|(z|(y|y)))|(x|x) ≤ z|(y|(x|x))

(8) x|((y|(z|z))|(y|(z|z))) ≤ (x|(y|y))|((x|(z|z))|(x|(z|z))).

Definition 4. A fuzzy set µ in an SUP-algebra ⟨X, |, 0⟩ is called a fuzzy SUP-subalgebra
of X if it satisfies the following:

(∀x, y ∈ X)(µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y)}).

Definition 5. A fuzzy set µ in an SUP-algebra ⟨X, |, 0⟩ is called an (∈,∈ ∨qk)-fuzzy
SUP-subalgebra of X if it satisfies the following:

(∀x, y ∈ X)(∀t, r ∈ (0, 1])(xt ∈ µ, yr ∈ µ ⇒ ((x|(y|y))|(x|(y|y)))min{t,r} ∈ ∨qkµ). (1)

Definition 6. [8] A fuzzy set µ in a set X of the form

µ(y) =

{
t ∈ (0, 1] if y = x

0 otherwise

is said to be a fuzzy point with support x and value t and is denoted by xt.

The general form of the symbol xtqµ as follows: for an arbitrary element k ∈ [0, 1), we
say that

• xtqkµ if µ(x) + t+ k > 1.

• xt ∈ ∨qkµ if xt ∈ µ or xtqkµ.

Definition 7. For any fuzzy set µ in a set X and any t ∈ [0, 1], the set U(µ, t) = {x ∈
X : µ(x) ≥ t} is called a level subset of µ.

3. Foundational Results on Semidetached SUP-Subalgebras

Before delving into the concept of semidetached SUP-subalgebras, it is essential to
recognize the foundational framework of SUP-algebras as a unique algebraic structure
that integrates logical operations through the Sheffer stroke. This section explores how
semidetached structures can emerge within SUP-algebras, emphasizing their significance
in the broader context of fuzzy subalgebra theory. By establishing a robust theoretical
basis, we aim to illustrate the intricate relationships and conditions that govern these
semidetached structures.

In what follows, let X = ⟨X, |, 0⟩ denote an SUP-algebra unless otherwise specified.
Given a set X and a subinterval Ω of [0, 1], a semidetached structure over Ω is defined

to be a pair (X, f), where f : Ω → P (X) is a mapping when P (X) is represented as the
power set of X.



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 5 of 16

Definition 8. A semidetached structure (X, f) over Ω is called a semidetached SUP-
subalgebra over Ω with respect to t ∈ Ω (briefly, t-semidetached SUP-subalgebra) if f(t) is
an SUP-subalgebra of X.

We say that (X, f) is a semidetached SUP-subalgebra over Ω if it is a t-semidetached
SUP-subalgebra with respect to all t ∈ Ω.

Given a fuzzy set µ in X, consider the following mappings:

ℓµU : Ω → P (X); t 7→ U(µ, t) (2)

ℓµQk
: Ω → P (X); t 7→ Qk(µ, t) (3)

ℓµEk
: Ω → P (X); t 7→ Ek(µ, t) (4)

where Qk(µ, t) = {x ∈ X : xtqkµ} and Ek(µ, t) = {x ∈ X : xt ∈ ∨qkµ}, which are called
the qk-set and ∈ ∨qk-set with respect to t (briefly, t-qk-set and t-∈ ∨qk-set), respectively,
of µ. A t-qk-set with k = 0 is called a t-q-set and is denoted by Q(µ, t). A t-∈ ∨qk-set
with k = 0 is called a t-∈ ∨q-set and is denoted by E (µ, t). Note that, for any t, r ∈ (0, 1],
if t ≥ r, then every r-qk-set is contained in the t-qk-set, that is, Qk(µ, r) ⊆ Qk(µ, t).
Obviously, Ek(µ, t) = U(µ, t) ∪Qk(µ, t).

Lemma 2. [9] A fuzzy set µ is a fuzzy SUP-subalgebra of X if and only if U(µ, t) is a
SUP-subalgebra of X for all t ∈ (0, 1].

Theorem 1. A semidetached structure (X, ℓµU ) is a semidetached SUP-subalgebra over
Ω = (0, 1] if and only if µ is a fuzzy SUP-subalgebra of X.

Proof. Straightforward from Lemma 2.

Theorem 2. If µ is an (∈,∈)-fuzzy SUP-subalgebra (or equivalently, µ is a fuzzy SUP-
subalgebra) of X, then a semidetached structure (X, ℓµQk

) is a semidetached SUP-subalgebra
over Ω = (0, 1].

Proof. Let x, y ∈ ℓµQk
(t) for t ∈ Ω = (0, 1]. Then xtqkµ and ytqkµ, that is, µ(x)+t+k >

1 and µ(y) + t + k > 1. Then µ((x|(y|y))|(x|(y|y))) + t + k ≥ min{µ(x), µ(y)} + t +
k = min{µ(x) + t + k, µ(y) + t + k} > 1. Hence, ((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ, and so
(x|(y|y))|(x|(y|y)) ∈ ℓµQk

(t). Therefore, ℓµQk
(t) is an SUP-subalgebra of X. Consequently,

(X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (0, 1].

Corollary 1 is a direct consequence of Theorem 1. Specifically, Theorem 1 states
that a semidetached structure (X, ℓUµ ) over Ω = (0, 1] exists if and only if µ is a fuzzy
SUP-subalgebra of X. Since an (∈,∈)-fuzzy SUP-subalgebra is equivalent to a fuzzy
SUP-subalgebra, the condition in Corollary 1 is satisfied, and the semidetached property
follows immediately.

Corollary 1. If µ is an (∈,∈)-fuzzy SUP-subalgebra (or equivalently, µ is a fuzzy SUP-
subalgebra) of X, then a semidetached structure (X, ℓµU ) is a semidetached SUP-subalgebra
over Ω = (0, 1].



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 6 of 16

Given a fuzzy set µ in X and k ∈ [0, 1), we consider the following condition:

(∀x, y ∈ X)(∀t, r ∈ [0, 1])(xtqkµ, yrqkµ ⇒ ((x|(y|y))|(x|(y|y)))min{t,r} ∈ ∨qkµ). (5)

Definition 9. A fuzzy set µ in X is called a k-left (resp., k-right) (qk,∈ ∨qk)-fuzzy
subalgebra of X if it satisfies the condition (5) for all x, y ∈ X and t, r ∈ (0, 1−k

2 ] (resp.,

t, r ∈ (1−k
2 , 1]).

Theorem 3. Every k-right (qk,∈ ∨qk)-fuzzy SUP-subalgebra is an (∈,∈ ∨qk)-fuzzy SUP-
subalgebra.

Proof. Let µ be a k-right (qk,∈ ∨qk)-fuzzy SUP-subalgebra of X. Let x, y ∈ X
and t, r ∈ (0, 1] be such that xt ∈ µ and yr ∈ µ. Then µ(x) ≥ t and µ(y) ≥ r. Sup-
pose that ((x|(y|y))|(x|(y|y)))min{t,r}∈ qkµ. Then µ((x|(y|y))|(x|(y|y))) < min{t, r} and

µ((x|(y|y))|(x|(y|y))) + min{t, r} + k ≤ 1. It follows that µ((x|(y|y))|(x|(y|y))) < 1−k
2 ,

and so that µ((x|(y|y))|(x|(y|y))) < min{t, r, 1−k
2 }. Hence, 1−k−µ((x|(y|y))|(x|(y|y))) >

1 − k −min{t, r, 1−k
2 } = max{1 − k − t, 1 − k − r, 1 − k − 1−k

2 } ≥ max{1 − k − µ(x), 1 −
k − µ(y), 1−k

2 }, and so there exists δ ∈ (0, 1] such that 1 − k − µ((x|(y|y))|(x|(y|y))) ≥
δ > max{1 − k − µ(x), 1 − k − µ(y), 1−k

2 }. Then δ ∈ (1−k
2 , 1], µ(x) + δ + k > 1 and

µ(y) + δ + k > 1, that is, xδqkµ and yδqkµ. Since µ is a k-right (qk,∈ ∨qk)-fuzzy
SUP-subalgebra of X, it follows that ((x|(y|y))|(x|(y|y)), δ) ∈ ∨qkµ. On the other hand,
1 − k − µ((x|(y|y))|(x|(y|y))) ≥ δ implies that µ((x|(y|y))|(x|(y|y))) + δ + k ≤ 1, that is,
((x|(y|y))|(x|(y|y)), δ)qkµ, and µ((x|(y|y))|(x|(y|y))) ≤ 1− δ− k < 1− k− 1−k

2 = 1−k
2 < δ,

that is, ((x|(y|y))|(x|(y|y)), δ)∈µ. Hence, ((x|(y|y))|(x|(y|y)), δ)∈ ∨qkµ, which is a contra-
diction. Therefore, ((x|(y|y))|(x|(y|y)))min{t,r} ∈ ∨qkµ, and thus µ is an (∈,∈ ∨qk)-fuzzy
SUP-subalgebra of X.

Corollary 2 is an immediate consequence of Theorem 3 by setting k = 0. Specifically,
a 0-right (q,∈ ∨q)-fuzzy SUP-subalgebra satisfies all the conditions required by Theorem
3, and hence it is also an (∈,∈ ∨q)-fuzzy SUP-subalgebra.

Corollary 2. Every 0-right (q,∈ ∨q)-fuzzy SUP-subalgebra is an (∈,∈ ∨q)-fuzzy subalge-
bra.

Theorem 4. If every fuzzy point has the value t in (0, 1−k
2 ], then every (∈,∈ ∨qk)-fuzzy

subalgebra is a k-left (qk,∈ ∨qk)-fuzzy SUP-subalgebra.

Proof. Let µ be an (∈,∈ ∨qk)-fuzzy SUP-subalgebra of X. Let x, y ∈ X and t, r ∈
(0, 1−k

2 ] be such that xtqkµ and yrqkµ. Then µ(x) + t+ k > 1 and µ(y) + r+ k > 1. Since

t, r ∈ (0, 1−k
2 ], we have µ(x) > 1 − t − k ≥ 1−k

2 ≥ t and µ(y) > 1 − r − k ≥ 1−k
2 ≥ r,

that is, xt ∈ µ and yr ∈ µ. Then ((x|(y|y))|(x|(y|y)))min{t,r} ∈ ∨qkµ. Hence, µ is a k-left
(qk,∈ ∨qk)-fuzzy SUP-subalgebra of X.

Corollary 3 is a direct consequence of Theorem 4 by setting k = 0. In this case,
the interval

(
0, 1−k

2

]
becomes (0, 0.5], and the (∈,∈ ∨qk)-fuzzy SUP-subalgebra reduces

to the (∈,∈ ∨q)-fuzzy case. Theorem 4 guarantees that under these conditions, every
(∈,∈ ∨q)-fuzzy SUP-subalgebra is a 0-left (q,∈ ∨q)-fuzzy SUP-subalgebra, which proves
the corollary.



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 7 of 16

Corollary 3. If every fuzzy point has the value t in (0, 0.5], then every (∈,∈ ∨q)-fuzzy
SUP-subalgebra is a 0-left (q,∈ ∨q)-fuzzy SUP-subalgebra.

Proposition 2. If (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1], then µ
satisfies:

(∀x, y ∈ X)(∀t, r ∈ Ω)(xt ∈ µ, yr ∈ µ ⇒ ((x|(y|y))|(x|(y|y)))max{t,r}qkµ). (6)

Proof. Let x, y ∈ X and t, r ∈ Ω = (1−k
2 , 1] be such that xt ∈ µ and yr ∈ µ. Then

µ(x) ≥ t > 1−k
2 and µ(y) ≥ r > 1−k

2 , which imply that µ(x) + t + k > 1 and µ(y) + r +
k > 1, that is, xtqkµ and yrqkµ. It follows that x, y ∈ ℓµQk

(max{t, r}) and max{t, r} ∈
(1−k

2 , 1]. Since ℓµQk
(max{t, r}) is an SUP-subalgebra of X, we have (x|(y|y))|(x|(y|y)) ∈

ℓµQk
(max{t, r}), and so ((x|(y|y))|(x|(y|y)))max{t,r}qkµ.

Corollary 4 is a direct consequence of Proposition 2 by taking k = 0, which leads
to Ω = (0.5, 1] and the standard quasi-coincidence operator q. The proposition ensures
that under the semidetached structure condition, the image of the Sheffer stroke operation
remains within the fuzzy quasi-coincidence set, establishing the desired inclusion.

Corollary 4. If (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (0.5, 1], then µ

satisfies:

(∀x, y ∈ X)(∀t, r ∈ Ω)(xt ∈ µ, yr ∈ µ ⇒ ((x|(y|y))|(x|(y|y)))max{t,r}qµ). (7)

Proposition 3. If (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (0, 1−k

2 ], then µ
satisfies:

(∀x, y ∈ X)(∀t, r ∈ Ω)(xtqkµ, yrqkµ) ⇒ ((x|(y|y))|(x|(y|y)))max{t,r} ∈ µ). (8)

Proof. Let x, y ∈ X and t, r ∈ Ω = (0, 1−k
2 ] be such that xtqkµ and yrqkµ. Then

x ∈ ℓµQk
(t) and y ∈ ℓµQk

(r). It follows that x, y ∈ ℓµQk
(max{t, r}) and max{t, r} ∈ (0, 1−k

2 ].

Thus, (x|(y|y))|(x|(y|y)) ∈ ℓµQk
(max{t, r}) since ℓµQk

(max{t, r}) is an SUP-subalgebra of
X. Hence, µ((x|(y|y))|(x|(y|y))) + k + max{t, r} > 1, and so µ((x|(y|y))|(x|(y|y))) >
1 − k − max{t, r} ≥ 1−k

2 ≥ max{t, r}. Thus, ((x|(y|y))|(x|(y|y)))max{t,r} ∈ µ and (8) is
valid.

Corollary 5 follows directly from Proposition 3 by setting k = 0, which implies Ω =
(0, 0.5] and uses the standard quasi-coincidence operator q. Proposition 3 establishes that
when the fuzzy elements xt and yr quasi-coincide with µ, their Sheffer stroke combination
also satisfies the membership condition in µ, as required.

Corollary 5. If (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (0, 0.5], then µ

satisfies:

(∀x, y ∈ X)(∀t, r ∈)(xtqµ, yrqµ ⇒ ((x|(y|y))|(x|(y|y)))max{t,r} ∈ µ). (9)



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 8 of 16

Theorem 5. If µ is a k-right (qk,∈ ∨qk)-fuzzy SUP-subalgebra of X, then (X, ℓµQk
) is a

semidetached SUP-subalgebra over Ω = (1−k
2 , 1].

Proof. Let x, y ∈ ℓµQk
(t) for t ∈ (1−k

2 , 1]. Then xtqkµ and ytqkµ. Since µ is a k-right
(qk,∈ ∨qk)-fuzzy SUP-subalgebra of X, we have ((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ, that is,
((x|(y|y))|(x|(y|y)))t ∈ µ or ((x|(y|y))|(x|(y|y)))tqkµ. If ((x|(y|y))|(x|(y|y)))t ∈ µ, then
µ((x|(y|y))|(x|(y|y))) ≥ t > 1−k

2 > 1− t− k, and so µ((x|(y|y))|(x|(y|y)))+ t+ k > 1, that
is, ((x|(y|y))|(x|(y|y)))tqkµ. Hence, (x|(y|y))|(x|(y|y)) ∈ ℓµQk

. If ((x|(y|y))|(x|(y|y)))tqkµ,
then (x|(y|y))|(x|(y|y)) ∈ ℓµQk

(t). Therefore, ℓµQk
(t) is an SUP-subalgebra of X, and con-

sequently, (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1].

Corollary 6 is a direct consequence of Theorem 5 by taking k = 0, which yields the
interval Ω = (0.5, 1] and replaces qk with the standard q. Since Theorem 5 ensures that a
k-right (qk,∈ ∨qk)-fuzzy SUP-subalgebra induces a semidetached structure, it follows that
a 0-right (q,∈ ∨q)-fuzzy SUP-subalgebra also generates a semidetached SUP-subalgebra
over this interval.

Corollary 6. If µ is a 0-right (q,∈ ∨q)-fuzzy SUP-subalgebra of X, then (X, ℓµQk
) is a

semidetached SUP-subalgebra over Ω = (0.5, 1].

4. Equivalences and Characterizations of Fuzzy SUP-Subalgebras

This section is devoted to a deeper exploration of the relationships between various
classes of fuzzy SUP-subalgebras and their role in generating semidetached structures
within Sheffer stroke UP-algebras. Building upon the foundational results established in
the previous section, we present a series of theorems and corollaries that provide necessary
and sufficient conditions for a fuzzy set to induce a semidetached SUP-subalgebra over
specified subintervals of (0, 1]. Particular emphasis is placed on the structural implications
of (∈,∈ ∨qk)-fuzzy, k-left, k-right, and (qk,∈ ∨qk)-fuzzy SUP-subalgebras, along with their
interrelationships and equivalences. These characterizations not only unify various fuzzy
concepts under a common algebraic framework but also demonstrate how logical fuzziness
can precisely determine the formation of algebraic substructures. The results in this
section contribute to a comprehensive theoretical foundation for fuzzy logic integration in
algebraic systems based on Sheffer stroke operations.

Theorem 6. For an SUP-subalgebra A of X, let µ be a fuzzy set in X such that

(1) µ(x) ≥ 1−k
2 for all x ∈ A,

(2) µ(x) = 0 for all x ∈ X\A.

Then µ is a k-left (qk,∈ ∨qk)-fuzzy SUP-subalgebra of X.

Proof. Let x, y ∈ X and t, r ∈ (0, 1−k
2 ] be such that xtqkµ and yrqkµ. Then µ(x) +

t + k > 1 and µ(y) + r + k > 1, which imply that µ(x) > 1 − t − k ≥ 1−k
2 and µ(y) >

1 − r − k ≥ 1−k
2 . Hence, x ∈ A and y ∈ A. Since A is an SUP-subalgebra of X,



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 9 of 16

we get (x|(y|y))|(x|(y|y)) ∈ A, and so µ((x|(y|y))|(x|(y|y))) ≥ 1−k
2 ≥ max{t, r}. Thus,

((x|(y|y))|(x|(y|y)))max{t,r} ∈ µ, and so ((x|(y|y))|(x|(y|y)))max{t,r} ∈ ∨qkµ. Therefore, µ
is a k-left (qk,∈ ∨qk)-fuzzy SUP-subalgebra of X.

Corollary 7 is a special case of Theorem 6 by setting k = 0. In this case, the threshold
1−k
2 becomes 0.5, and the conditions in Corollary 7 match exactly the assumptions of

Theorem 6. Therefore, the fuzzy set µ constructed as described satisfies the definition of
a 0-left (q,∈ ∨q)-fuzzy SUP-subalgebra of X.

Corollary 7. For an SUP-subalgebra A of X, let µ be a fuzzy set in X such that

(1) µ(x) ≥ 0.5 for all x ∈ A,

(2) µ(x) = 0 for all x ∈ X\A.

Then µ is a 0-left (q,∈ ∨q)-fuzzy SUP-subalgebra of X.

Proposition 4. If (X, ℓµEk
) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1], then µ
satisfies:

(∀x, y ∈ X)(∀t, r ∈ Ω)(xtqkµ, yrqkµ ⇒ ((x|(y|y))|(x|(y|y)))max{t,r} ∈ ∨qkµ). (10)

Proof. Let x, y ∈ X and t, r ∈ Ω = (0, 1] be such that xtqkµ and yrqkµ. Then x ∈
ℓµQk

(t) ⊆ ℓµEk
(t) and y ∈ ℓµQk

(r) ⊆ ℓµEk
(r). It follows that x, y ∈ ℓµEk

(max{t, r}), and so from

the hypothesis that (x|(y|y))|(x|(y|y)) ∈ ℓµEk
(max{t, r}). Hence, ((x|(y|y))|(x|(y|y)))max{t,r} ∈

∨qkµ, and consequently, (10) is valid.

Corollary 8 follows directly from Proposition 4 by taking k = 0, which implies that
Ω = (0.5, 1] and the fuzzy quasi-coincidence operator qk becomes the standard q.

Corollary 8. If (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1], then µ
satisfies:

(∀x, y ∈ X)(∀t, r ∈ Ω)(xtqµ, yrqµ ⇒ ((x|(y|y))|(x|(y|y)))max{t,r} ∈ ∨qµ). (11)

The following lemma is directly proved by Definition 5.

Lemma 3. A fuzzy set µ in X is an (∈,∈ ∨qk)-fuzzy SUP-subalgebra of X if and only if
it satisfies the following:

(∀x, y ∈ X)(µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1− k

2
}). (12)

Theorem 7. If µ is an (∈,∈ ∨qk)-fuzzy SUP-subalgebra of X, then (X, ℓµQk
) is a semide-

tached SUP-subalgebra over Ω = (1−k
2 , 1].



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 10 of 16

Proof. Let x, y ∈ ℓµQk
(t) for t ∈ Ω = (1−k

2 , 1]. Then xtqkµ and ytqkµ, that is, µ(x)+ t+
k > 1 and µ(y) + t+ k > 1. It follows from Lemma 3 that µ((x|(y|y))|(x|(y|y))) + t+ k ≥
min{µ(x), µ(y), 1−k

2 } + t + k = min{µ(x) + t + k, µ(y) + t + k, 1−k
2 + t + k} > 1. Hence,

((x|(y|y))|(x|(y|y)))tqkµ, and so (x|(y|y))|(x|(y|y)) ∈ ℓµQk
(t). Therefore, ℓµQk

(t) is an SUP-

subalgebra of X for all t ∈ (1−k
2 , 1], and consequently, (X, ℓµQk

) is a semidetached SUP-

subalgebra over Ω = (1−k
2 , 1].

Corollary 9 is a special case of Theorem 7 by setting k = 0, which leads to the interval
Ω = (0.5, 1]. The assumption that µ is an (∈,∈ ∨q)-fuzzy SUP-subalgebra guarantees, via
Theorem 7, that the level sets ℓQµ (t) are SUP-subalgebras for all t ∈ Ω, and thus (X, ℓQµ )
is a semidetached SUP-subalgebra over this interval.

Corollary 9. If µ is an (∈,∈ ∨q)-fuzzy SUP-subalgebra of X, then µ is a semidetached
SUP-subalgebra over Ω = (0.5, 1].

Theorem 8. If µ is a semidetached SUP-subalgebra over Ω = (1−k
2 , 1], then µ is an

(∈,∈ ∨qk)-fuzzy SUP-subalgebra of X.

Proof. For a semidetached SUP-subalgebra µ over Ω = (1−k
2 , 1], assume that there

exist a, b ∈ X such that µ((a|(b|b))|(a|(b|b))) < min{µ(a), µ(b), 1−k
2 } = t0. Then t0 ∈

(0, 1−k
2 ], a, b ∈ U(µ, t0) ⊆ ℓµEk

(t0), which implies that (a|(b|b))|(a|(b|b)) ∈ ℓµEk
(t0). Hence

µ((a|(b|b))|(a|(b|b))) ≥ t0 or µ((a|(b|b))|(a|(b|b)))+t0+k > 1. This is a contradiction. Thus,
µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−k

2 } for all x, y ∈ X. It follows from Lemma 3
that µ is an (∈,∈ ∨qk)-fuzzy SUP-subalgebra of X.

Corollary 10 follows immediately from Theorem 8, which shows that if (X, ℓQk
µ ) is a

semidetached SUP-subalgebra over Ω =
(
1−k
2 , 1

]
, then µ must be an (∈,∈ ∨qk)-fuzzy

SUP-subalgebra. By applying Theorem 7, it follows that (X, ℓQk
µ ) is again a semidetached

SUP-subalgebra over the same interval. Thus, the conclusion reconfirms the consistency
of the structure.

Corollary 10. If (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1], then

(X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1].

Theorem 9. If µ is an (∈,∈ ∨qk)-fuzzy SUP-subalgebra of X, then (X, ℓµQk
) is a semide-

tached SUP-subalgebra over Ω = (1−k
2 , 1].

Proof. Let x, y ∈ ℓµEk
(t) for t ∈ Ω = (0, 1−k

2 ]. Then xt ∈ ∨qkµ and yt ∈ ∨qkµ. Hence,
we have the following four cases:

(1) xt ∈ µ and yt ∈ µ,

(2) xt ∈ µ and ytqkµ,

(3) xtqkµ and yt ∈ µ,

(4) xtqkµ and ytqkµ.



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 11 of 16

The first case implies that ((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ, and so (x|(y|y))|(x|(y|y)) ∈ ℓµEk
(t).

For the second case, ytqkµ induces µ(y) > 1 − t − k ≥ t, that is, yt ∈ µ. Hence,
((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ, and so (x|(y|y))|(x|(y|y)) ∈ ℓµEk

(t). Similarly, the third

case implies (x|(y|y))|(x|(y|y)) ∈ ℓµEk
(t). The last case induces µ(x) > 1 − t − k ≥ t and

µ(y) > 1−t−k ≥ t, that is, xt ∈ µ and yt ∈ µ. It follows that ((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ
and so that (x|(y|y))|(x|(y|y)) ∈ ℓµEk

(t). Therefore, ℓµEk
(t) is an SUP-subalgebra of X for

all t ∈ 0, 1−k
2 . Hence, (X, ℓµQk

) is a semidetached SUP-subalgebra over Ω = (1−k
2 , 1].

Corollary 11 is a direct consequence of Theorem 9 by setting k = 0. When k = 0, the
interval Ω =

(
0, 1−k

2

]
becomes (0, 0.5], and the operator qk becomes the standard quasi-

coincidence operator q. Theorem 9 ensures that if µ is an (∈,∈ ∨q)-fuzzy SUP-subalgebra,
then the structure (X, ℓEµ ) forms a semidetached SUP-subalgebra over this interval.

Corollary 11. If µ is an (∈,∈ ∨q)-fuzzy SUP-subalgebra of X, then (X, ℓµEk
) is a semide-

tached SUP-subalgebra over Ω = (0, 0.5].

Theorem 10. If µ is a (qk,∈ ∨qk)-fuzzy SUP-subalgebra of X, then (X, ℓµEk
) is a semide-

tached SUP-subalgebra over Ω = (1−k
2 , 1].

Proof. Let x, y ∈ ℓµEk
(t) for t ∈ Ω = (1−k

2 , 1]. Then xt ∈ ∨qkµ and yt ∈ ∨qkµ. Hence,
we have the following four cases:

(1) xt ∈ µ and yt ∈ µ,

(2) xt ∈ µ and ytqkµ,

(3) xtqkµ and yt ∈ µ,

(4) xtqkµ and ytqkµ.

For the first case, we have µ(x)+ t+ k ≥ 2t+ k > 1 and µ(y)+ t+ k ≥ 2t+ k > 1, that is,
xtqkµ and ytqkµ. Hence, ((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ, and so (x|(y|y))|(x|(y|y)) ∈ ℓµEk

(t).
In the second case, xt ∈ µ implies µ(x) + t + k ≥ 2t + k > 1, that is, xtqkµ. Hence,
((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ, and so (x|(y|y))|(x|(y|y)) ∈ ℓµEk

(t). Similarly, the third case

implies (x|(y|y))|(x|(y|y)) ∈ ℓµEk
(t). For the last case, we have ((x|(y|y))|(x|(y|y)))t ∈ ∨qkµ,

and so (x|(y|y))|(x|(y|y)) ∈ ℓµEk
(t). Consequently, ℓµEk

(t) is an SUP-subalgebra of X for all

t ∈ Ω = (1−k
2 , 1]. Therefore, (X, ℓµEk

) is a semidetached SUP-subalgebra over Ω = (1−k
2 , 1].

Corollary 12 is derived from Theorem 10 by taking k = 0, which yields the interval
Ω = (0.5, 1] and converts the generalized operator qk into the standard quasi-coincidence
operator q. Theorem 10 proves that if µ is a (qk,∈ ∨qk)-fuzzy SUP-subalgebra, then the
structure (X, ℓEµ ) forms a semidetached SUP-subalgebra over Ω =

(
1−k
2 , 1

]
. Substituting

k = 0 confirms the conclusion for the (q,∈ ∨q)-fuzzy case.

Corollary 12. If µ is a (q,∈ ∨q)-fuzzy SUP-subalgebra of X, then (X, ℓµEk
) is a semide-

tached SUP-subalgebra over Ω = (0.5, 1].



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 12 of 16

For α ∈ {∈, qk} and t ∈ (0, 1], we say that xtαµ if xtαµ does not hold.

Definition 10. A fuzzy set µ in X is called an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X if
it satisfies the following:

(∀x, y ∈ X)(∀t, r ∈ (0, 1])(((x|(y|y))|(x|(y|y)))min{t,r}∈µ ⇒ xt∈ ∨ qkµ or yr∈ ∨ qkµ).

(13)

An (∈,∈ ∨ qk)-fuzzy SUP-subalgebra with k = 0 is called an (∈,∈ ∨ q)-fuzzy SUP-
subalgebra.

Theorem 11. A fuzzy set µ in X is an (∈,∈∨ qk)-fuzzy SUP-subalgebra of X if and only
if the following inequality is valid:

(∀x, y ∈ X)(max{µ((x|(y|y))|(x|(y|y))), 1− k

2
} ≥ min{µ(x), µ(y)}). (14)

Proof. Let µ be an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X. Assume that (14) is not
valid. Then there exist a, b ∈ X such that max{µ((a|(b|b))|(a|(b|b))), 1−k

2 } < min{µ(a),
µ(b)} = t. Then 1−k

2 < t ≤ 1, at ∈ µ, bt ∈ µ and ((a|(b|b))|(a|(b|b)))t∈µ. It follows
from (13) that atqkµ or btqkµ. Hence, µ(a) ≥ t and µ(a) + t + k ≤ 1 or µ(b) ≥ t and
µ(b) + t + k ≤ 1. In either case, we have t ≤ 1−k

2 , which is a contradiction. Therefore,

max{µ((x|(y|y))|(x|(y|y))), 1−k
2 } ≥ min{µ(x), µ(y)} for all x, y ∈ X.

Conversely, suppose that (14) is valid. Let ((x|(y|y))|(x|(y|y)))min{t,r}∈µ for all x, y ∈
X and t, r ∈ (0, 1]. Then µ((x|(y|y))|(x|(y|y))) < min{t, r}. If max{µ((x|(y|y))|(x|(y|y))),
1−k
2 } = µ((x|(y|y))|(x|(y|y))), then min{t, r} > µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y)},

and so µ(x) < t or µ(y) < r. Thus, xt∈µ or yr∈µ, which implies that xt∈∨qkµ or yr∈∨qkµ.
If max{µ((x|(y|y))|(x|(y|y))), 1−k

2 } = 1−k
2 , then min{µ(x), µ(y)} ≤ 1−k

2 . Suppose xt ∈ µ

or yr ∈ µ. Then t ≤ µ(x) ≤ 1−k
2 or r ≤ µ(y) ≤ 1−k

2 , and so µ(x)+t+k ≤ 1−k
2 + 1−k

2 +k = 1

or µ(y) + r + k ≤ 1−k
2 + 1−k

2 + k = 1. Hence, xtqkµ or yrqkµ. Therefore, xt∈ ∨ qkµ or
yr∈ ∨ qkµ. This shows that µ is an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X.

Corollary 13 follows directly from Theorem 11 by setting k = 0. In this case, the
inequality given in Theorem 11 simplifies to

max{µ((x|(y|y))|(x|(y|y))), 0.5} ≥ min{µ(x), µ(y)},

which is exactly the condition stated in Corollary 13. Hence, the result is an immediate
specialization of the general case when k = 0.

Corollary 13. A fuzzy set µ in X is an (∈,∈∨ q)-fuzzy SUP-subalgebra of X if and only
if the following inequality is valid:

(∀x, y ∈ X)(max{µ((x|(y|y))|(x|(y|y))), 0.5} ≥ min{µ(x), µ(y)}). (15)

Theorem 12. A fuzzy set µ in X is an (∈,∈∨ qk)-fuzzy SUP-subalgebra of X if and only
if (X, ℓµU ) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1].



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 13 of 16

Proof. Assume that µ is an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X. Let x, y ∈
ℓµU (t) for t ∈ Ω = (1−k

2 , 1]. Then µ(x) ≥ t and µ(y) ≥ t. It follows from (14) that

max{µ((x|(y|y))|(x|(y|y))), 1−k
2 } ≥ min{µ(x), µ(y)} ≥ t. Since t > 1−k

2 , it follows that
µ((x|(y|y))|(x|(y|y))) ≥ t, and so that (x|(y|y))|(x|(y|y)) ∈ ℓµU (t). Thus, ℓµU (t) is an SUP-
subalgebra of X, and (X, ℓµU ) is a semidetached SUP-subalgebra over Ω = (1−k

2 , 1].

Conversely, suppose that (X, ℓµU ) is a semidetached SUP-subalgebra over Ω = (1−k
2 , 1].

If (14) is not valid, then there exist a, b ∈ X such that max{µ((a|(b|b))|(a|(b|b))), 1−k
2 } <

min{µ(a), µ(b)} = t. Then t ∈ (1−k
2 , 1], a, b ∈ ℓµU (t) and (a|(b|b))|(a|(b|b)) /∈ ℓµU (t). This is

a contradiction, and so (14) is valid. Using Theorem 11, we know that µ is an (∈,∈∨ qk)-
fuzzy SUP-subalgebra of X.

Corollary 14 is a direct consequence of Theorem 12 by setting k = 0. When k = 0,
the interval Ω =

(
1−k
2 , 1

]
becomes (0.5, 1], and the condition in Theorem 3.33 simplifies

accordingly. Thus, an (∈,∈ ∨q)-fuzzy SUP-subalgebra ofX is equivalent to a semidetached
SUP-subalgebra over (0.5, 1], establishing the result.

Corollary 14. A fuzzy set µ in X is an (∈,∈∨ q)-fuzzy SUP-subalgebra of X if and only
if µ is a semidetached SUP-subalgebra over Ω = (0.5, 1].

Theorem 13. A fuzzy set µ in X is an (∈,∈∨ qk)-fuzzy SUP-subalgebra of X if and only
if (X, ℓµQk

) is a semidetached SUP-subalgebra over Ω = (0, 1−k
2 ].

Proof. Assume that (X, ℓµU ) is a semidetached SUP-subalgebra over Ω = (0, 1−k
2 ].

If (14) is not valid, then there exist a, b ∈ X, t ∈ (0, 1] and k ∈ [0, 1) such that
max{µ((a|(b|b))|(a|(b|b))), 1−k

2 }+ t+k ≤ 1 < min{µ(a), µ(b)}+ t+k. It follows that atqkµ
and btqkµ, that is, a, b ∈ ℓµQk

(t), but ((a|(b|b))|(a|(b|b)))tqkµ, that is, (a|(b|b))|(a|(b|b)) /∈
ℓµQk

. This is a contradiction, and so (14) is valid. Using Theorem 11, we have µ is an
(∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X.

Conversely, suppose that µ is an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X. Let x, y ∈
ℓµQk

(t) for t ∈ Ω = (0, 1−k
2 ]. Then xtqkµ and ytqkµ, that is, µ(x) + t + k > 1 and µ(y) +

t+ k > 1. It follows from (14) that max{µ((x|(y|y))|(x|(y|y))), 1−k
2 } ≥ min{µ(x), µ(y)} >

1− t− k ≥ 1−k
2 and so that µ((x|(y|y))|(x|(y|y)))+ t+ k > 1, that is, (x|(y|y))|(x|(y|y)) ∈

µ. Therefore, ℓµQk
(t) is an SUP-subalgebra of X, and (X, ℓµQk

) is a semidetached SUP-

subalgebra over Ω = (0, 1−k
2 ].

Corollary 15 follows directly from Theorems 12 and 13. Theorem 12 states that (X, ℓUµ )

is a semidetached SUP-subalgebra over Ω =
(
1−k
2 , 1

]
if and only if µ is an (∈,∈ ∨qk)-

fuzzy SUP-subalgebra. Similarly, Theorem 13 establishes that (X, ℓQk
µ ) is a semidetached

SUP-subalgebra over Ω =
(
0, 1−k

2

]
under the same condition. Therefore, the equivalence

between the two semidetached structures directly follows.

Corollary 15. For a fuzzy set µ in X, the following are equivalent.

(1) (X, ℓµU ) is a semidetached SUP-subalgebra over Ω = (1−k
2 , 1],

(2) (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (0, 1−k

2 ].



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 14 of 16

Definition 11. A fuzzy set µ in X is called an (∈ ∨ qk,∈ ∨ qk)-fuzzy SUP-subalgebra of
X if it satisfies the following:

(∀x, y ∈ X)(∀t, r ∈ (0, 1])(((x|(y|y))|(x|(y|y)))min{t,r}∈ ∨ qkµ ⇒ xt∈ ∨ qkµ or yr∈ ∨ qkµ).

(16)

Theorem 14. Every (∈ ∨ qk,∈ ∨ qk)-fuzzy SUP-subalgebra is an (∈,∈ ∨ qk)-fuzzy SUP-
subalgebra.

Proof. Let x, y ∈ X and t, r ∈ (0, 1] be such that ((x|(y|y))|(x|(y|y)))min{t,r}∈µ. Then
((x|(y|y))|(x|(y|y)))min{t,r}∈ ∨ qkµ, and so xt∈ ∨ qkµ or yr∈ ∨ qkµ by (16). Therefore, µ is
an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X.

Corollary 16 is an immediate consequence of Theorem 14, which states that every
(∈ ∨qk,∈ ∨qk)-fuzzy SUP-subalgebra is also an (∈,∈ ∨qk)-fuzzy SUP-subalgebra. From
this, properties (2), (3), and (4) follow directly by invoking Theorems 11, 12, and 13,
respectively. Thus, the corollary summarizes the logical implications of Theorem 14 and
the previously established equivalences.

Corollary 16. If µ is an (∈ ∨ qk,∈ ∨ qk)-fuzzy SUP-subalgebra of X, then

(1) µ is an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra,

(2) µ satisfies the condition (14),

(3) (X, ℓµU ) is a semidetached SUP-subalgebra over Ω = (1−k
2 , 1],

(4) (X, ℓµQk
) is a semidetached SUP-subalgebra over Ω = (0, 1−k

2 ].

Definition 12. A fuzzy set µ in X is called a (qk,∈∨ qk)-fuzzy SUP-subalgebra of X if if
it satisfies the following:

(∀x, y ∈ X)(∀t, r ∈ (0, 1])(((x|(y|y))|(x|(y|y)))min{t,r}qkµ ⇒ xt∈ ∨ qkµ or yr∈ ∨ qkµ).

(17)

Theorem 15. Assume that min{t, r} ≤ 1−k
2 for any t, r ∈ (0, 1]. Then every (qk,∈∨ qk)-

fuzzy SUP-subalgebra is an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra.

Proof. Let µ be an (qk,∈ ∨ qk)-fuzzy SUP-subalgebra of X. Assume that

((x|(y|y))|(x|(y|y)))min{t,r}∈µ

for x, y ∈ X and t, r ∈ (0, 1] with min{t, r} ≤ 1−k
2 . Then µ((x|(y|y))|(x|(y|y))) <

min{t, r} ≤ 1−k
2 , and so µ((x|(y|y))|(x|(y|y))) + k + min{t, r} < 1−k

2 + 1−k
2 + k = 1,

that is, ((x|(y|y))|(x|(y|y)))min{t,r}qkµ. It follows from (17) that xt∈ ∨ qkµ or yr∈ ∨ qkµ.
Therefore, µ is an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X.

Corollary 17 directly follows from Theorem 15 by setting k = 0. In this case, the
condition min{t, r} ≤ 1−k

2 becomes min{t, r} ≤ 0.5. Theorem 15 ensures that under



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 15 of 16

this condition, every (qk,∈ ∨qk)-fuzzy SUP-subalgebra is also an (∈,∈ ∨qk)-fuzzy SUP-
subalgebra. Therefore, for k = 0, every (q,∈ ∨q)-fuzzy SUP-subalgebra becomes an
(∈,∈ ∨q)-fuzzy SUP-subalgebra.

Corollary 17. Assume that min{t, r} ≤ 0.5 for any t, r ∈ (0, 1]. Then every (q,∈ ∨ q)-
fuzzy SUP-subalgebra is an (∈,∈ ∨ q)-fuzzy SUP-subalgebra.

Theorem 16. Assume that min{t, r} > 1−k
2 for any t, r ∈ (0, 1]. Then every (∈,∈ ∨ qk)-

fuzzy SUP-subalgebra is a (qk,∈ ∨ qk)-fuzzy SUP-subalgebra.

Proof. Let µ be an (∈,∈ ∨ qk)-fuzzy SUP-subalgebra of X. Assume that

((x|(y|y))|(x|(y|y)))min{t,r}qkµ

for x, y ∈ X and t, r ∈ (0, 1] with min{t, r} > 1−k
2 . If ((x|(y|y))|(x|(y|y)))min{t,r} ∈ µ,

then µ((x|(y|y))|(x|(y|y))) ≥ min{t, r}, and so µ((x|(y|y))|(x|(y|y))) + k + min{t, r} >
1−k
2 + 1−k

2 + k = 1. Hence, ((x|(y|y))|(x|(y|y)))min{t,r}qkµ, a contradiction. Thus,

((x|(y|y))|(x|(y|y)))min{t,r}∈µ,

which implies from (13) that xt∈ ∨ qkµ or yr∈ ∨ qkµ. Therefore, µ is a (qk,∈ ∨ qk)-fuzzy
SUP-subalgebra of X.

Corollary 18. Assume that min{t, r} > 0.5 for any t, r ∈ (0, 1]. Then every (∈,∈ ∨ q)-
fuzzy SUP-subalgebra is a (q,∈ ∨ q)-fuzzy SUP-subalgebra.

5. Conclusion

In this paper, we have introduced the concept of semidetached SUP-algebras and
explored their fundamental properties. The investigation into these algebraic structures
has revealed several significant findings. The notion of semidetached SUP-subalgebras has
been clearly defined, providing a new perspective on the relationships within SUP-algebras.
This contributes to a deeper understanding of their algebraic properties and potential
applications in various fields of study, including logic and functional analysis. Additionally,
we have discussed various types of fuzzy SUP-subalgebras, including (∈,∈∨qk)-fuzzy SUP-
subalgebras and k-left (k-right) (qk,∈ ∨ qk)-fuzzy SUP-subalgebras. These classifications
are crucial for establishing the framework necessary for further research and applications
of these algebraic structures.

The paper has provided several conditions under which a semidetached structure can be
classified as a semidetached SUP-subalgebra. This is essential for validating the theoretical
framework and ensuring that the properties discussed are applicable in practical scenarios.
The findings of this study open avenues for future research, particularly in exploring the
applications of semidetached SUP-algebras in logical systems and operator theory. The
flexibility and completeness of the Sheffer stroke as a logical operator suggest that further
investigations could yield valuable insights into both algebraic and logical frameworks. In



T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5876 16 of 16

conclusion, the exploration of semidetached SUP-subalgebras in SUP-algebras not only
enhances our understanding of these structures but also sets the stage for future research
that could bridge the gap between algebra, logic, and analysis. The potential applications
of these findings are vast, and we encourage further exploration in this promising area of
study.

Acknowledgements

This research was supported by University of Phayao and Thailand Science Research
and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567).

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