J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae19.
FUZZY TRANSLATIONS OF A FUZZY SET IN
UP-ALGEBRAS
Tanintorn Guntasow1 .
Supanat Sajak2 Apirat Jomkham3 , and Aiyared Iampan4
1,2,3,4
Department of Mathematics.
School of Science.
University of Phayao.
Phayao 56000.
Thailand ia@up.
Abstract.
In this paper, we apply the concept of fuzzy translations of a fuzzy set to UP-algebras.
For any fuzzy set AA in a UP-algebra, the concepts of fuzzy translations of AA of type I and of fuzzy -translations of AA of type II are introduced, their basic properties are investigated and some useful examples are discussed.
The concepts of prime fuzzy sets and of weakly prime fuzzy sets in UP-algebras are also Moreover, we discuss the concepts of extensions and of intensions of a fuzzy set in UP-algebras.
Key words and Phrases: UP-algebra, fuzzy set, fuzzy translation, extension, intension.
Abstrak.
Di dalam paper ini, diaplikasikan konsep translasi fuzzy dari suatu himpunan fuzzy pada aljabar-UP.
Untuk sebarang himpunan fuzzy AA di suatu aljabarUP, konsep -translasi fuzzy dari AA dengan tipe I dan konsep -translasi dari AA dengan tipe II diperkenalkan, kemudian diinvestigasi beberapa sifat dasarnya, dan didiskusikan beberapa contoh penting.
Konsep himpunan fuzzy prima dan himpunan fuzzy prima lemah di aljabar-UP juga dikaji di dalam paper ini.
Lebih jauh, didiskusikan juga konsep ekstensi dan intensi dari suatu himpunan fuzzy di aljabarUP.
Kata kunci: aljabar-UP, himpunan fuzzy, translasi fuzzy, ekstensi, intensi.
2000 Mathematics Subject Classification: 03G25.
Received: 26 Nov 2016, revised: 15 March 2017, accepted: 16 March 2017.
Guntasow.
Introduction and Preliminaries Among many algebraic structures, algebras of logic form important class of algebras.
Examples of these are BCK-algebras .
BCI-algebras .
BCHalgebras .
K-algebras .
KU-algebras .
SU-algebras .
and others.
They are strongly connected with logic.
For example.
BCI-algebras introduced by IseAki .
in 1966 have connections with BCI-logic being the BCI-system in combinatory logic which has application in the language of functional programming.
BCK and BCI-algebras are two classes of logical algebras.
They were introduced by Imai and IseAki .
, .
in 1966 and have been extensively investigated by many researchers.
It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras.
A fuzzy subset f of a set S is a function from S to a closed interval .
, .
The concept of a fuzzy subset of a set was first considered by Zadeh .
The fuzzy set theories developed by Zadeh and others have found many applications in the domain of mathematics and elsewhere.
After the introduction of the concept of fuzzy sets by Zadeh .
, several researches were conducted on the generalizations of the notion of fuzzy set and application to many logical algebras such as: In 2001.
Lele.
Wu.
Weke.
Mamadou and Njock .
studied fuzzy ideals and weak ideals in BCK-algebras.
Jun .
introduced the notion of Q-fuzzy subalgebras of BCK/BCIalgebras.
In 2002.
Jun.
Roh and Kim .
studied fuzzy B-algebras in B-algebras.
Jun .
introduced the concept of (, )-fuzzy ideals of BCK/BCI-algebras.
In 2005.
Akram and Dar .
introduced the notions of T -fuzzy subalgebras and of T -fuzzy H-ideals in BCI-algebras.
In 2007.
Akram and Dar .
introduced the notion of fuzzy ideals in K-algebras.
In 2008.
Akram .
introduced the notion of bifuzzy ideals of K-algebras.
In 2012.
Sitharselvam.
Priya and Ramachandran .
introduced the notion of anti Q-fuzzy KU-ideals of KU-algebras.
In 2016.
Somjanta.
Thuekaew.
Kumpeangkeaw and Iampan .
introduced and studied fuzzy UP-subalgebras and fuzzy UP-ideals of UP-algebras.
Moreover, fuzzy sets were extended to fuzzy translations in many logical algebras such as: In 2009.
Lee.
Jun and Doh .
investigated relations among fuzzy translations, .
ormalized, maxima.
fuzzy extensions and fuzzy multiplications of fuzzy subalgebras in BCK/BCI-algebras.
In 2011.
Jun .
discussed fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy ideals in BCK/BCIalgebras.
In 2013.
Chandramouleeswaran.
Muralikrishna and Srinivasan .
introduced the notions of fuzzy translations and fuzzy multiplications of fuzzy sets in BF-algebras.
In 2014.
Ansari and Chandramouleeswaran .
introduced the notions of fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy -ideals of -algebras.
In 2015.
Senapati.
Bhowmik.
Pal and Davvaz .
discussed the fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy H-ideals in BCK/BCI-algebras.
Bhowmik.
Senapati .
introduced the notions of fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy subalgebras in BGalgebras.
In 2016.
Hashemi .
introduced the notions of fuzzy translations of fuzzy associative ideals in BCK/BCI-algebras.
Fuzzy translations of a fuzzy set in UP-algebras In this paper, we apply the concept of fuzzy translations of a fuzzy set to UPalgebras.
For any fuzzy set AA in a UP-algebra, the concepts of fuzzy -translations of AA of type I and of fuzzy -translations of AA of type II are introduced, their basic properties are investigated and some useful examples are discussed.
The concepts of prime fuzzy sets and of weakly prime fuzzy sets in UP-algebras are also studied.
Moreover, we discuss the concepts of extensions and of intensions of a fuzzy set in UP-algebras.
Before we begin our study, we will introduce the definition of a UP-algebra.
Definition 1.
An algebra A = (A.
A, .
of type .
, .
is called a UP-algebra if it satisfies the following axioms: for any x, y, z OO A, (UP-.
: .
A .
A (.
A .
A .
A .
) = 0, (UP-.
: 0 A x = x, (UP-.
: x A 0 = 0, and (UP-.
: x A y = y A x = 0 implies x = y.
From .
, we know that the notion of UP-algebras is a generalization of KU-algebras .
Example 1.
Let X be a universal set.
Define a binary operation A on the power set of X by putting A A B = B O A0 = A0 O B = B Oe A for all A.
B OO P(X).
Then (P(X).
OI) is a UP-algebra and we shall call it the power UP-algebra of type Example 1.
Let X be a universal set.
Define a binary operation O on the power set of X by putting A O B = B O A0 = A0 O B for all A.
B OO P(X).
Then (P(X).
X) is a UP-algebra and we shall call it the power UP-algebra of type 2.
Example 1.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 0 0
2 0 1 0 3
3 0 1 2 0
Then (A.
A, .
is a UP-algebra.
In what follows, let A and B denote UP-algebras unless otherwise specified.
The following proposition is very important for the study of UP-algebras.
Proposition 1.
In a UP-algebra A, the following properties hold: for any x, y, z OO A, .
x A x = 0, .
x A y = 0 and y A z = 0 imply x A z = 0, .
x A y = 0 implies .
A .
A .
A .
= 0, .
x A y = 0 implies .
A .
A .
A .
= 0, .
x A .
A .
= 0.
Guntasow.
A .
A x = 0 if and only if x = y A x, and .
x A .
A .
= 0.
On a UP-algebra A = (A.
A, .
, we define a binary relation O on A .
as follows: for all x, y OO A, x O y if and only if x A y = 0.
Definition 1.
A subset S of A is called a UP-subalgebra of A if the constant 0 of A is in S, and (S.
A, .
itself forms a UP-algebra.
Proposition 1.
A nonempty subset S of a UP-algebra A = (A.
A, .
is a UP-subalgebra of A if and only if S is closed under the A multiplication on A.
Definition 1.
A subset F of A is called a UP-filter of A if it satisfies the following properties:
the constant 0 of A is in F , and .
for any x, y OO A, x A y OO F and x OO F imply y OO F .
Definition 1.
A subset B of A is called a UP-ideal of A if it satisfies the following properties:
the constant 0 of A is in B, and .
for any x, y, z OO A, x A .
A .
OO B and y OO B imply x A z OO B.
Definition 1.
A subset C of A is called a strongly UP-ideal of A if it satisfies the following properties:
the constant 0 of A is in C, and .
for any x, y, z OO A, .
A .
A .
A .
OO C and y OO C imply x OO C.
Definition 1.
A fuzzy set in a nonempty set X .
r a fuzzy subset of X) is an arbitrary function f : X Ie .
, .
, .
is the unit segment of the real line.
Definition 1.
A fuzzy set f in A is called a fuzzy UP-subalgebra of A if it satisfies the following property: for any x, y OO A, f .
A .
Ou min.
, f .
By Proposition 1.
5 1, we have f .
= f .
A .
Ou min.
, f .
} = f .
for all x OO A.
Definition 1.
A fuzzy set f in A is called a fuzzy UP-filter of A if it satisfies the following properties: for any x, y OO A, .
Ou f .
, and .
Ou min.
A .
, f .
Definition 1.
A fuzzy set f in A is called a fuzzy UP-ideal of A if it satisfies the following properties: for any x, y, z OO A, .
Ou f .
, and .
A .
Ou min.
A .
A .
), f .
Fuzzy translations of a fuzzy set in UP-algebras Definition 1.
A fuzzy set f in A is called a fuzzy strongly UP-ideal of A if it satisfies the following properties: for any x, y, z OO A, .
Ou f .
, and .
Ou min.
A .
A .
A .
), f .
For example, we see that AA7 and AA8 are fuzzy strongly UP-ideal of A in Example 3.
Fuzzy -Translation of a Fuzzy Set In this section, we study the basic properties of UP-subalgebras .
UPfilters.
UP-ideals, strongly UP-ideal.
and fuzzy UP-subalgebras .
fuzzy UPfilters, fuzzy UP-ideals, fuzzy strongly UP-ideal.
of a UP-algebra, and study the concept of prime and weakly prime of subsets and of fuzzy sets of a UP-algebra.
The proof of Theorems 2.
1, 2.
2, 2.
4, and 2.
6 can be verified easily.
Theorem 2.
A subset C of A is a strongly UP-ideal of A if and only if C = A.
Theorem 2.
Every UP-filter of A is a UP-subalgebra.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
Hence, (A.
A, .
is a UP-algebra.
Let S = .
, .
Then S is a UP-subalgebra of A.
Since 2 A 3 = 2 OO S, but 3 OO / S, we have S is not a UP-filter of A.
Theorem 2.
Every UP-ideal of A is a UP-filter.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
Hence, (A.
A, .
is a UP-algebra.
Let F = .
, .
Then F is a UP-filter of A.
Since 2 A .
A .
= 0 OO F , 1 OO F but 2 A 3 = 2 OO / F , we have F is not a UP-ideal of A.
Theorem 2.
Every strongly UP-ideal of A is a UP-ideal.
Guntasow.
Example 2.
Let A = .
, 1, 2, 3, .
be a set with a binary operation A defined by the following Cayley table:
Then (A.
A, .
is a UP-algebra.
Let B = .
, 1, .
Then B is UP-ideal of A.
Since .
A .
A .
A .
= 0 OO B, 0 OO B but 2 OO / B, we have B is not a strongly UP-ideal of A.
By Theorems 2.
2, 2.
4, and 2.
6 and Examples 2.
3, 2.
5, and 2.
7, we have that the notion of UP-subalgebras is a generalization of UP-filters, the notion of UPfilters is a generalization of UP-ideals, and the notion of UP-ideals is a generalization of strongly UP-ideals.
Definition 2.
A nonempty subset B of A is called a prime subset of A if it satisfies the following property: for any x, y OO A, x A y OO B implies x OO B or y OO B.
Definition 2.
A UP-subalgebra .
UP-filter.
UP-ideal, strongly UPidea.
B of A is called a prime UP-subalgebra .
prime UP-filter, prime UPideal, prime strongly UP-idea.
of A if it is a prime subset of A.
Definition 2.
A nonempty subset B of A is called a weakly prime subset of A if it satisfies the following property: for any x, y OO A and x 6= y, x A y OO B implies x OO B or y OO B.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 2 3
2 0 1 0 3
3 0 1 2 0
Hence, (A.
A, .
is a UP-algebra.
Let B = .
, .
Then B is a weakly prime subset of A.
Since 2 A 2 = 0 OO B, but 2 OO / B, we have B is not prime subset of A.
Definition 2.
A UP-subalgebra .
UP-filter.
UP-ideal, strongly UP-idea.
B of A is called a weakly prime UP-subalgebra .
weakly prime UP-filter, weakly prime UP-ideal, weakly prime strongly UP-idea.
of A if it is a weakly prime subset of A.
The proof of Theorems 2.
13, 2.
14, 2.
15, 2.
17, and 2.
19 and Lemma 2.
21 can be verified easily.
Theorem 2.
Let S be a subset of A.
Then the following statements are equivalent:
Fuzzy translations of a fuzzy set in UP-algebras .
S is a prime UP-subalgebra .
prime UP-filter, prime UP-ideal, prime strongly UP-idea.
of A, .
S = A, and .
S is a strongly UP-ideal of A.
Theorem 2.
A fuzzy set f in A is constant if and only if it is a fuzzy strongly UP-ideal of A.
Theorem 2.
Every fuzzy UP-filter of A is a fuzzy UP-subalgebra.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 1 2
2 0 0 0 1
3 0 0 0 0
Then (A.
A, .
is a UP-algebra.
We define a mapping f : A Ie .
, .
as follows:
= 1, f .
= 0.
6, f .
= 0.
4, and f .
= 0.
Then f is a fuzzy UP-subalgebra of A.
Since f .
= 0.
4 < 0.
6 = min.
, f .
}, we have f is not a fuzzy UP-filter of A.
Theorem 2.
Every fuzzy UP-ideal of A is a fuzzy UP-filter.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 2 2
2 0 1 0 2
3 0 1 0 0
Then (A.
A, .
is a UP-algebra.
We define a mapping f : A Ie .
, .
as follows:
= 1, f .
= 0.
2, f .
= 0.
1, and f .
= 0.
Then f is a fuzzy UP-filter of A.
Since f .
= 0.
1 < 0.
2 = min.
), f .
}, we have f is not a fuzzy UP-ideal of A.
Theorem 2.
Every fuzzy strongly UP-ideal of A is a fuzzy UP-ideal.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 2 3
2 0 1 0 3
3 0 1 2 0
Then (A.
A, .
is a UP-algebra.
We define a mapping f : A Ie .
, .
as follows:
Guntasow.
= 0.
6, f .
= 0.
4, f .
= 0.
3, and f .
= 0.
Then f is a fuzzy UP-ideal of A.
Since f .
= 0.
4 < 0.
6 = min.
A .
A .
A .
), f .
}, we have f is not a fuzzy strongly UP-ideal of A.
By Theorems 2.
15, 2.
17, and 2.
19 and Examples 2.
16, 2.
18, and 2.
20, we have that the notion of fuzzy UP-subalgebras is a generalization of fuzzy UP-filters, the notion of fuzzy UP-filters is a generalization of fuzzy UP-ideals, and the notion of fuzzy UP-ideals is a generalization of fuzzy strongly UP-ideals.
Lemma 2.
If f is a fuzzy UP-filter of A, then it is order preserving.
Definition 2.
A fuzzy set f in A is called a prime fuzzy set in A if it satisfies the following property: for any x, y OO A, f .
A .
O max.
, f .
Definition 2.
A fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
f of A is called a prime fuzzy UP-subalgebra .
fuzzy UP-filter, prime fuzzy UP-ideal, prime fuzzy strongly UP-idea.
of A if it is a prime fuzzy set in A.
Definition 2.
A fuzzy set f in A is called a weakly prime fuzzy set in A if it satisfies the following property: for any x, y OO A and x 6= y, f .
A .
O max.
, f .
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 2 3
2 0 1 0 3
3 0 1 2 0
Then (A.
A, .
is a UP-algebra.
We define a mapping f : A Ie .
, .
as follows:
= 0.
4, f .
= 0.
3, f .
= 0.
2, and f .
= 0.
Then f is a weakly prime fuzzy set of A.
Since f .
= f .
= 0.
4 > max.
), f .
} = 3, we have f is not a prime fuzzy set of A.
Definition 2.
A fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
f of A is called a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A if it is a weakly prime fuzzy set in A.
The proof of Theorems 2.
27, and 2.
29 can be verified easily.
Theorem 2.
Let f be a fuzzy set in A.
Then the following statements are .
f is a prime fuzzy UP-subalgebra .
prime fuzzy UP-filter, prime fuzzy UP-ideal, prime fuzzy strongly UP-idea.
of A.
Fuzzy translations of a fuzzy set in UP-algebras .
f is a constant fuzzy set in A, and .
f is a fuzzy strongly UP-ideal of A.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
Then (A.
A, .
is a UP-algebra.
We define a fuzzy set AA1 : A Ie .
, .
in A as follows:
AA1 .
= 0.
AA1 .
= 0.
AA1 .
= 0.
5, and AA1 .
= 0.
Then AA1 is weakly prime fuzzy UP-subalgebras of A.
Since AA1 .
A .
= 0.
6 > 0.
max{AA1 .
AA1 .
}, we have AA1 is not a prime fuzzy UP-subalgebras of A.
We define a fuzzy set AA2 : A Ie .
, .
in A as follows:
AA2 .
= 0.
AA2 .
= 0.
AA2 .
= 0.
5, and AA2 .
= 0.
Then AA2 is weakly prime fuzzy UP-filter of A.
Since AA2 .
A .
= 0.
6 > 0.
max{AA2 .
AA2 .
}, we have AA2 is not a prime fuzzy UP-filter of A.
We define a fuzzy set AA3 : A Ie .
, .
in A as follows:
AA3 .
= 0.
AA3 .
= 0.
AA3 .
= 0.
6, and AA3 .
= 0.
Then AA3 is weakly prime fuzzy UP-ideal of A.
Since AA3 .
A .
= 0.
6 > 0.
max{AA3 .
AA3 .
}, we have AA3 is not a prime fuzzy UP-ideal of A.
Theorem 2.
For UP-algebras, the notions of weakly prime fuzzy strongly UPideals and of prime fuzzy strongly UP-ideals coincide.
Definition 2.
The inclusion AuOIAy is defined by setting, for any fuzzy sets AA1 and AA2 in A.
AA1 OI AA2 Ni AA1 .
O AA2 .
for all x OO A.
We say that AA2 is a fuzzy extension of AA1 .
, and AA1 is a fuzzy intension of AA2 .
For any fuzzy set AA in A, we let A = 1 Oe sup{AA.
| x OO A}.
Definition 2.
Let AA be a fuzzy set in A and let OO .
A].
A mapping a : A Ie .
, .
defined by a .
= AA.
for all x OO A is said to be a fuzzy -translation of AA of type I or, in short, a fuzzy -translation of AA.
Guntasow.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 1 2
2 0 0 0 1
3 0 0 0 0
Then (A.
A, .
is a UP-algebra.
We define a mapping AA : A Ie .
, .
as follows:
AA.
= 0.
AA.
= 0.
AA.
= 0.
3, and AA.
= 0.
Then A = 1 Oe sup.
7, 0.
5, 0.
3, 0.
= 1 Oe 0.
7 = 0.
Let = 0.
2 OO .
, 0.
Then a mapping a : A Ie .
, .
defined by a .
= 0.
9, a .
= 0.
7, a .
= 0.
5, and a .
= 0.
The proof of Theorems 2.
33, 2.
34, 2.
35, 2.
36, 2.
37, 2.
38, 2.
39, and 2.
40 can be verified easily.
Theorem 2.
If AA is a fuzzy UP-subalgebra of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra of A for all OO .
A].
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-subalgebra of A, then AA is a fuzzy UP-subalgebra of A.
Theorem 2.
If AA is a fuzzy UP-filter of A, then the fuzzy -translation a of AA is a fuzzy UP-filter of A for all OO .
A].
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-filter of A, then AA is a fuzzy UP-filter of A.
Theorem 2.
If AA is a fuzzy UP-ideal of A, then the fuzzy -translation a of AA is a fuzzy UP-ideal of A for all OO .
A].
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-ideal of A, then AA is a fuzzy UP-ideal of A.
Theorem 2.
If AA is a fuzzy strongly UP-ideal of A, then the fuzzy -translation a of AA is a fuzzy strongly UP-ideal of A for all OO .
A].
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy strongly UP-ideal of A, then AA is a fuzzy strongly UP-ideal of A.
Theorem 2.
If AA is a fuzzy UP-filter .
fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is order preserving for all OO .
A].
Proof.
It follows from Theorem 2.
Theorems 2.
37 and 2.
Theorems 2.
and Lemma 2.
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-filter .
fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then AA is order preserving.
Fuzzy translations of a fuzzy set in UP-algebras Proof.
It follows from Theorem 2.
Theorems 2.
38 and 2.
Theorems 2.
and Lemma 2.
Example 2.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 2 3
2 0 0 0 3
3 0 0 0 0
Then (A.
A, .
is a UP-algebra.
We define a mapping f : A Ie .
, .
as follows:
= 1, f .
= 0.
2, f .
= 0.
3, and f .
= 0.
Then f is a fuzzy UP-subalgebra of A.
Since f .
= 0.
2 < min.
), f .
} = 0.
we have f is not a fuzzy UP-filter of A.
Since 3 O 2 but f .
= 0.
4 > 0.
3 = f .
, we have f is not order preserving.
The proof of Theorems 2.
44, 2.
45, 2.
46, and 2.
47 can be verified easily.
Theorem 2.
If AA is a prime fuzzy set in A, then the fuzzy -translation a of AA is a prime fuzzy set in A for all OO .
A].
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a prime fuzzy set in A, then AA is a prime fuzzy set in A.
Theorem 2.
If AA is a weakly prime fuzzy set in A, then the fuzzy -translation a of AA is a weakly prime fuzzy set in A for all OO .
A].
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a weakly prime fuzzy set in A, then AA is a weakly prime fuzzy set in A.
Theorem 2.
If AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A for all OO .
A].
Proof.
It follows from Theorems 2.
Theorem 2.
Theorem 2.
Theorem .
Theorem 2.
If there exists OO .
A] such that the fuzzy -translation a of AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A, then AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A.
Proof.
It follows from Theorems 2.
Theorem 2.
Theorem 2.
Theorem .
Guntasow.
Note 2.
If AA is a fuzzy set in A and OO .
A], then a .
= AA.
Ou AA.
for all x OO A.
Hence, the fuzzy -translation a of AA is a fuzzy extension of AA for all OO .
A].
Lemma 2.
Let AA and be fuzzy sets in A.
If ON a for OO .
A], there exists OO .
A] with Ou such that ON a ON a .
Proof.
Assume that ON a for OO .
A].
Then .
Ou a .
for all x OO A.
Putting = inf xOOA {.
Oe a .
Then inf {.
Oe a .
} = inf {.
Oe (AA.
)} xOOA xOOA O inf .
Oe (AA.
)} xOOA = 1 inf {OeAA.
Oe } xOOA = 1 inf {OeAA.
} Oe xOOA = 1 Oe sup {AA.
} Oe xOOA = A Oe , so = inf xOOA {.
Oe a .
} O A Oe = A.
a ON a .
Now, for all x OO A, we have Thus OO .
A] and Ou , so a .
= AA.
= AA.
inf {.
Oe a .
} xOOA = AA .
inf {.
Oe a .
} xOOA O a .
Oe a .
= .
, so ON a .
Hence.
ON a ON a for some OO .
A] with Ou .
Definition 2.
Let AA1 and AA2 be two fuzzy sets in A and AA1 OI AA2 .
If AA1 is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then AA2 is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, and we say that AA2 is a fuzzy UP-subalgebra .
UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
extension of AA1 .
Theorem 2.
If AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
extension of AA for all OO .
A].
Proof.
It follows from Theorem 2.
Theorem 2.
Theorem 2.
Theorem .
and Note 2.
Fuzzy translations of a fuzzy set in UP-algebras Theorem 2.
If AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
extension of the fuzzy -translation a of AA for all .
OO .
A] with Ou .
Proof.
It follows from Theorem 2.
Theorem 2.
Theorem 2.
Theorem Theorem 2.
Let AA be a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-idea.
of A and OO .
A].
For every fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
extension of the fuzzy -translation a of AA, there exists OO .
A] with Ou such that is the fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
extension of the fuzzy -translation a of AA.
Proof.
It follows from Theorem 2.
Theorem 2.
Theorem 2.
Theorem .
and Lemma 2.
Fuzzy -Translation of a Fuzzy Set For any fuzzy set AA in A, we let A = inf{AA.
| x OO A}.
Definition 3.
Let AA be a fuzzy set in A and let OO .
A].
A mapping a : A Ie .
, .
defined by a .
= AA.
Oe for all x OO A is said to be a fuzzy -translation of AA of type II or, in short, a fuzzy -translation of AA.
Example 3.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
A 0 1 2 3
0 0 1 2 3
1 0 0 1 2
2 0 0 0 1
3 0 0 0 0
Then (A.
A, .
is a UP-algebra.
We define a mapping AA : A Ie .
, .
as follows:
AA.
= 0.
AA.
= 0.
AA.
= 0.
3, and AA.
= 0.
Then A = inf.
7, 0.
5, 0.
3, 0.
= 0.
Let = 0.
05 OO .
, 0.
Then a mapping a : A Ie .
, .
defined by a .
= 0.
65, a .
= 0.
45, a .
= 0.
25, and a .
= 0.
The proof of Theorems 3.
3, 3.
5, 3.
6, 3.
7, 3.
8, 3.
9, 3.
10, and 3.
11 can be verified Theorem 3.
If AA is a fuzzy UP-subalgebra of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra of A for all OO .
A].
Guntasow.
Example 3.
In Example 2.
16, a mapping f : A Ie .
, .
defined by f .
= 1, f .
= 0.
6, f .
= 0.
4, and f .
= 0.
is a fuzzy UP-subalgebra of A.
Then A = inf.
, 0.
6, 0.
4, 0.
= 0.
For all OO .
, 0.
, we can show that a mapping a : A Ie .
, .
is a fuzzy UP-subalgebra of A.
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-subalgebra of A, then AA is a fuzzy UP-subalgebra of A.
Theorem 3.
If AA is a fuzzy UP-filter of A, then the fuzzy -translation a of AA is a fuzzy UP-filter of A for all OO .
A].
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-filter of A, then AA is a fuzzy UP-filter of A.
Theorem 3.
If AA is a fuzzy UP-ideal of A, then the fuzzy -translation a of AA is a fuzzy UP-ideal of A for all OO .
A].
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-ideal of A, then AA is a fuzzy UP-ideal of A.
Theorem 3.
If AA is a fuzzy strongly UP-ideal of A, then the fuzzy -translation a of AA is a fuzzy strongly UP-ideal of A for all OO .
A].
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy strongly UP-ideal of A, then AA is a fuzzy strongly UP-ideal of A.
Theorem 3.
If AA is a fuzzy UP-filter .
fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is order preserving for all OO .
A].
Proof.
It follows from Theorem 3.
Theorems 3.
8 and 2.
Theorems 3.
and Lemma 2.
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-filter .
fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then AA is order preserving.
Proof.
It follows from Theorem 3.
Theorems 3.
9 and 2.
Theorems 3.
and Lemma 2.
The proof of Theorems 3.
14, 3.
15, 3.
16, and 3.
17 can be verified easily.
Theorem 3.
If AA is a prime fuzzy set in A, then the fuzzy -translation a of AA is a prime fuzzy set in A for all OO .
A].
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a prime fuzzy set in A, then AA is a prime fuzzy set in A.
Theorem 3.
If AA is a weakly prime fuzzy set in A, then the fuzzy -translation a of AA is a weakly prime fuzzy set in A for all OO .
A].
Fuzzy translations of a fuzzy set in UP-algebras Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a weakly prime fuzzy set in A, then AA is a weakly prime fuzzy set in A.
Theorem 3.
If AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A for all OO .
A].
Proof.
It follows from Theorems 3.
Theorem 3.
Theorem 3.
Theorem .
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A, then AA is a weakly prime fuzzy UP-subalgebra .
weakly prime fuzzy UP-filter, weakly prime fuzzy UP-ideal, weakly prime fuzzy strongly UP-idea.
of A.
Proof.
It follows from Theorems 3.
Theorem 3.
Theorem 3.
Theorem .
Note 3.
If AA is a fuzzy set in A and OO .
A], then a .
= AA.
Oe O AA.
for all x OO A.
Hence, the fuzzy -translation a of AA is a fuzzy intension of AA for all OO .
A].
Lemma 3.
Let AA and be fuzzy sets in A.
If OI a for OO .
A], there exists OO .
A] with Ou such that OI a OI a .
Proof.
Assume that OI a for OO .
A].
Then .
O a .
for all x OO A.
Putting = inf xOOA .
Oe .
Then inf .
Oe .
} O inf .
} xOOA xOOA = inf {(AA.
Oe )} xOOA = inf {AA.
} Oe xOOA = A Oe .
Guntasow.
so = inf xOOA .
Oe .
} O A Oe = A.
Thus OO .
A] and Ou , so a OI a .
Now, for all x OO A, we have a .
= AA.
= AA.
Oe ( inf .
Oe .
}) xOOA = AA.
Oe Oe inf .
Oe .
} xOOA = AA .
sup {.
Oe a .
} xOOA Ou AA .
Oe a .
= .
, so OI a .
Hence.
OI a OI a for some OO .
A] with Ou .
Definition 3.
Let AA1 and AA2 be two fuzzy sets in A and AA1 OI AA2 .
If AA2 is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then AA1 is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, and we say that AA1 is a fuzzy UP-subalgebra .
UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
intension of AA2 .
Example 3.
Let A = .
, 1, 2, .
be a set with a binary operation A defined by the following Cayley table:
Then (A.
A, .
is a UP-algebra.
We define two fuzzy sets AA1 : A Ie .
, .
and AA2 : A Ie .
, .
in A as follows:
AA1 .
= 0.
AA1 .
= 0.
AA1 .
= 0.
3, and AA1 .
= 0.
AA2 .
= 0.
AA2 .
= 0.
AA2 .
= 0.
6, and AA2 .
= 0.
Then AA1 OI AA2 , and AA1 and AA2 are fuzzy UP-subalgebras of A.
Hence.
AA1 is a fuzzy UP-subalgebra intension of AA2 , and AA2 is a fuzzy UP-subalgebra extension of AA1 .
We define two fuzzy sets AA3 : A Ie .
, .
and AA4 : A Ie .
, .
in A as follows:
AA3 .
= 0.
AA3 .
= 0.
AA3 .
= 0.
1, and AA3 .
= 0.
AA4 .
= 0.
AA4 .
= 0.
AA4 .
= 0.
4, and AA4 .
= 0.
Then AA3 OI AA4 , and AA3 and AA4 are fuzzy UP-filter of A.
Hence.
AA3 is a fuzzy UPfilter intension of AA4 , and AA4 is a fuzzy UP-filter extension of AA3 .
We define two fuzzy sets AA5 : A Ie .
, .
and AA6 : A Ie .
, .
in A as follows:
AA5 .
= 0.
AA5 .
= 0.
AA5 .
= 0.
3, and AA5 .
= 0.
AA6 .
= 0.
AA6 .
= 0.
AA6 .
= 0.
6, and AA6 .
= 0.
Fuzzy translations of a fuzzy set in UP-algebras Then AA5 OI AA6 , and AA5 and AA6 are fuzzy UP-ideal of A.
Hence.
AA5 is a fuzzy UPideal intension of AA6 , and AA6 is a fuzzy UP-ideal extension of AA5 .
We define two fuzzy sets AA7 : A Ie .
, .
and AA8 : A Ie .
, .
in A as follows:
AA7 .
= 0.
5 and AA8 .
= 0.
for all x OO A.
Then AA7 OI AA8 , and AA7 and AA8 are fuzzy strongly UP-ideal of A.
Hence.
AA7 is a fuzzy strongly UP-ideal intension of AA8 , and AA8 is a fuzzy strongly UP-ideal extension of AA7 .
Theorem 3.
If AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-idea.
of A, than the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
intension of AA for all OO .
A].
Proof.
It follows from Theorem 3.
Theorem 3.
Theorem 3.
Theorem .
and Note 3.
Theorem 3.
If AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
intension of the fuzzy -translation a of AA for all .
OO .
A] with Ou .
Proof.
It follows from Theorem 3.
Theorem 3.
Theorem 3.
Theorem Theorem 3.
Let AA be a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UPideal, fuzzy strongly UP-idea.
of A and OO .
A].
For every fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
intension of the fuzzy -translation a of AA, there exists OO .
A] with Ou such that is the fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
intension of the fuzzy -translation a of AA.
Proof.
It follows from Theorem 3.
Theorem 3.
Theorem 3.
Theorem .
and Lemma 3.
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A for all OO .
A].
Proof.
It follows from Theorems 2.
34 and 3.
Theorems 2.
36 and 3.
Theorems 2.
38 and 3.
Theorems 2.
40 and 3.
Theorem 3.
If there exists OO .
A] such that the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A, then the fuzzy -translation a of AA is a fuzzy UP-subalgebra .
fuzzy UP-filter, fuzzy UP-ideal, fuzzy strongly UP-idea.
of A for all OO .
A].
Guntasow.
Proof.
It follows from Theorems 3.
5 and 2.
Theorems 3.
7 and 2.
Theorems 3.
9 and 2.
Theorems 3.
11 and 2.
Conclusions In the present paper, we have introduced the concepts of fuzzy -translations of a fuzzy set of type I and of fuzzy -translations of a fuzzy set of type II in UP-algebras and investigated some of its essential properties.
Finally, we have important relationships between two types of fuzzy translations of a fuzzy set.
think this work would enhance the scope for further study in this field of fuzzy sets.
It is our hope that this work would serve as a foundation for the further study in this field of fuzzy sets in UP-algebras.
Acknowledgement.
The authors wish to express their sincere thanks to the referees for the valuable suggestions which lead to an improvement of this paper.
References