Uploaded by IAEME PUBLICATION

IJARET 10 02 034

advertisement
International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 10, Issue 2, March - April 2019, pp. 350-361, Article ID: IJARET_10_02_034
Available online at http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=10&IType=02
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
© IAEME Publication
FUZZY IDEALS AND FUZZY DOT IDEALS ON
BH-ALGEBRAS
K. Anitha
Research Scholar,
PG and Research Department of Mathematics,
Saiva Bhanu Kshatriya College,
Aruppukottai - 626 101, Tamil Nadu, India
Dr. N. Kandaraj
Associate Professor,
PG and Research Department of Mathematics,
Saiva Bhanu Kshatriya College,
Aruppukottai - 626 101, Tamil Nadu, India
ABSTRACT
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
Keywords: BH-algebras, BH- Ideals, Fuzzy Dot BH-ideal.
Cite this Article: K. Anitha and Dr. N. Kandaraj, Fuzzy Ideals and Fuzzy Dot Ideals
on Bh-Algebras, International Journal of Advanced Research in Engineering and
Technology, 10(2), 2019, pp 350-361.
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=10&IType=2
Subject Classification: AMS (2000), 06F35, 03G25, 06D99, 03B47
1 INTRODUCTION
Y. Imai and K. Iseki [1, 2, and 3] introduced two classes of abstract algebras: BCK-algebras
and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class
of BCI-algebras. K. Iseki and S. Tanaka, [7] are introduced Ideal theory of BCK-algebras P.
Bhattacharya, N.P. Mukherjee and L.A. Zadeh [4] are introduced fuzzy relations and fuzzy
groups. The notion of BH-algebras is introduced by Y. B Jun, E. H. Roh and H. S. Kim[9]
Since then, several authors have studied BH-algebras. In particular, Q. Zhang, E. H. Roh and
Y. B. Jun [10] studied the fuzzy theory in BH-algebras. L.A. Zadeh [6] introduced notion of
fuzzy sets and A. Rosenfeld [8] introduced the notion of fuzzy group. O.G. Xi [5] introduced
the notion of fuzzy BCK-algebras. After that, Y.B. Jun and J. Meng [10] studied
Characterization of fuzzy sub algebras by their level sub algebras on BCK-algebras. J. Neggers
and H. S. Kim[11] introduced on d-algebras, M. Akram [12] introduced on fuzzy d-algebras In
this paper we classify the notion of Fuzzy Ideals on BH – algebras and the notion of Fuzzy dot
http://www.iaeme.com/IJARET/index.asp
350
[email protected]
K. Anitha and Dr. N. Kandaraj
Ideals on BH – algebras. And then we investigate several basic properties which are related to
fuzzy BH-ideals and fuzzy dot BH- ideals
2 PRELIMINARIES
In this section we cite the fundamental definitions that will be used in the sequel:
Definition 2.1 [1, 2, 3]
Let X be a nonempty set with a binary operation βˆ— and a constant 0. Then (X, βˆ—, 0) is called
a BCK-algebra if it satisfies the following conditions
1. ((π‘₯ βˆ— 𝑦) βˆ— (π‘₯ βˆ— 𝑧)) βˆ— (𝑧 βˆ— 𝑦) = 0
2.(π‘₯ βˆ— (π‘₯ βˆ— 𝑦)) βˆ— 𝑦 = 0
3.π‘₯ βˆ— π‘₯ = 0
4. π‘₯ βˆ— 𝑦 = 0, 𝑦 βˆ— π‘₯ = 0 β‡’ π‘₯ = 𝑦
5. 0 βˆ— π‘₯ = 0 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘₯, 𝑦, 𝑧 ∈ 𝑋
Definition 2.2 [1, 2, 3]
Let X be a BCK-algebra and I be a subset of X, then I is called an ideal of X if
(I1) 0∈ 𝐼
(I2) 𝑦 π‘Žπ‘›π‘‘ π‘₯ βˆ— 𝑦 ∈ 𝐼 β‡’ π‘₯ ∈ 𝐼 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘₯, 𝑦 ∈ 𝐼
Definition 2.3 [9, 10]
A nonempty set X with a constant 0 and a binary operation βˆ— is called a BH-algebra, if it
satisfies the following axioms
(BH1) π‘₯ βˆ— π‘₯ = 0
(BH2) π‘₯ βˆ— 0 = 0
(BH3) π‘₯ βˆ— 𝑦 = 0 π‘Žπ‘›π‘‘ 𝑦 βˆ— π‘₯ = 0 β‡’ π‘₯ = 𝑦 for all π‘₯, 𝑦 ∈ 𝑋
Example 2.4
Let X= {0, 1, 2} be a set with the following cayley table
*
0
1
2
0
0
1
2
1
1
0
1
2
1
1
0
Then (X, βˆ—, 0) is a BH-algebra
Definition 2.5[9, 10]
Let X be a BH-algebra and I be a subset of X, then I is called an ideal of X if
(BHI1) 0∈ 𝐼
(BHI2) 𝑦 π‘Žπ‘›π‘‘ π‘₯ βˆ— 𝑦 ∈ 𝐼 β‡’ π‘₯ ∈ 𝐼
(BHI3) π‘₯ ∈ 𝐼 and 𝑦 ∈ 𝑋 β‡’ π‘₯ βˆ— 𝑦 ∈ 𝐼 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘₯, 𝑦 ∈ 𝐼
A mapping 𝑓: 𝑋 β†’ π‘Œ of BH-algebras is called a homomorphism if 𝑓 (π‘₯ βˆ— 𝑦 ) = 𝑓(π‘₯) βˆ—
𝑓(𝑦) for all π‘₯, 𝑦 𝑋 ∈ 𝑋. Note that if 𝑓 ∢ 𝑋 β†’ π‘Œ is homomorphism of BH-algebras, Then
𝑓 (0) = 0’ . We now review some fuzzy logic concepts. A fuzzy subset of a set X is a function
πœ‡ ∢ 𝑋 β†’ [0, 1]. For a fuzzy subset ΞΌ of X and t ∈ [0, 1], define U (ΞΌ; t) to be the set U (ΞΌ; t) =
{x ∈ X | ΞΌ(x) β‰₯ t}. For any fuzzy subsets ΞΌ and Ξ½ of a set X, we define
(πœ‡ ∩ 𝜈)(π‘₯) = π‘šπ‘–π‘›{πœ‡(π‘₯), 𝜈(π‘₯)} π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘₯ ∈ 𝑋.
http://www.iaeme.com/IJARET/index.asp
351
[email protected]
Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
Let 𝑓 ∢ 𝑋 β†’ π‘Œ be a function from a set X to a set Y and let ΞΌ be a fuzzy subset of X. The
fuzzy subset v of Y defined by v(y) ο€½ {
sup  ( x) if
xοƒŽ f ο€­1 ( y )
f ( y ) ο‚Ή  , ο€’y οƒŽ Y
ο€­1
0 otherwise
is called the image of µ under𝑓, denoted by 𝑓(µ). If 𝜈 is a fuzzy subset o𝑓 π‘Œ, the fuzzy
subset ΞΌ of X given by πœ‡(π‘₯) = 𝜈(𝑓(π‘₯)) for all x ∈ X is called the Preimage of Ξ½ under f and is
denoted by 𝑓 βˆ’1 (𝑣) . A fuzzy subset ΞΌ in X has the sup property if for any 𝑇 βŠ† 𝑋 there exists
π‘₯0 ∈ T such that πœ‡(π‘₯0 ) = sup  ( z ) . A fuzzy relation ΞΌ on a set X is a fuzzy subset of 𝑋 × π‘‹,
xοƒŽ f
ο€­1
( y)
that is, a map πœ‡ ∢ 𝑋 × π‘‹ β†’ [0, 1].
Definition 2.6[4, 6, 8]
Let X be a nonempty set. A fuzzy (sub) set πœ‡ of the set X is a mapping πœ‡: 𝑋 β†’ [0,1
Definition 2.7[4, 6, 8]
Let πœ‡ be the fuzzy set of a set X. For a fixed s∈ [0,1], the set πœ‡π‘  = {π‘₯ ∈ 𝑋: πœ‡(π‘₯) β‰₯ 𝑠}is
called an upper level of ΞΌ or level subset of πœ‡
Definition 2.8[5, 7]
A fuzzy set πœ‡ in X is called fuzzy BCK-ideal of X if it satisfies the following inequalities
1.πœ‡(0) β‰₯ πœ‡(π‘₯)
2. πœ‡(π‘₯) β‰₯ min{πœ‡(π‘₯ βˆ— 𝑦), πœ‡(𝑦)}
Definition 2.9 [11]
Let X be a nonempty set with a binary operation βˆ— and a constant 0. Then (X, βˆ—, 0) is called
a d - algebra if it satisfies the following axioms.
1.π‘₯ βˆ— π‘₯ = 0
2. 0 βˆ— π‘₯ = 0
3. π‘₯ βˆ— 𝑦 = 0, 𝑦 βˆ— π‘₯ = 0 β‡’ π‘₯ = 𝑦 for all π‘₯, 𝑦 ∈ 𝑋
Definition 2.10 [12]
A fuzzy set πœ‡ in X is called fuzzy d-ideal of X if it satisfies the following inequalities
Fd1.πœ‡(0) β‰₯ πœ‡(π‘₯)
Fd2πœ‡(π‘₯) β‰₯ min{πœ‡(π‘₯ βˆ— 𝑦), πœ‡(𝑦)}
Fd3. πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑦)} For all π‘₯, 𝑦 ∈ 𝑋
Definition 2.11[12] A fuzzy subset ΞΌ of X is called a fuzzy dot d-ideal of X if it satisfies
The following conditions:
1. πœ‡(0) β‰₯ πœ‡(π‘₯)
2. πœ‡(π‘₯) β‰₯ πœ‡(π‘₯ βˆ— 𝑦) . πœ‡(𝑦)
3. πœ‡(π‘₯ βˆ— 𝑦) β‰₯ πœ‡(π‘₯). πœ‡(𝑦)for all π‘₯, 𝑦 ∈ 𝑋
3. FUZZY IDEALS ON BH-ALGEBRAS
Definition 3.1
A fuzzy set πœ‡ in X is called fuzzy BH-ideal of X if it satisfies the following inequalities
1.πœ‡(0) β‰₯ πœ‡(π‘₯)
2. πœ‡(π‘₯) β‰₯ min{πœ‡(π‘₯ βˆ— 𝑦), πœ‡(𝑦)}
3. πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑦)} for all π‘₯, 𝑦 ∈ 𝑋
http://www.iaeme.com/IJARET/index.asp
352
[email protected]
K. Anitha and Dr. N. Kandaraj
Example 3.2
Let X= {0, 1, 2, 3} be a set with the following cayley table
*
0
1
2
3
0
0
1
2
3
1
0
0
2
3
2
0
0
0
3
3
0
1
0
0
Then (X, βˆ— ,0) is not BCK-algebra. Since {(1βˆ— 3) βˆ— (1 βˆ— 2)} βˆ— (2 βˆ— 3) = 1 β‰  0
We define fuzzy set πœ‡ in X by πœ‡(0) = 0.8 and πœ‡(π‘₯) = 0.01 for all π‘₯ β‰  0 in X. then it is
easy to show that πœ‡ is a BH-ideal of X.
We can easily observe the following propositions
1. In a BH-algebra every fuzzy BH-ideal is a fuzzy BCK-ideal, and every fuzzy BCKideal is a fuzzy BH -Sub algebra
2. Every fuzzy BH-ideal of a BH- algebra is a fuzzy BH-subalgebra.
Example 3.3
Let X = {0, 1, 2} be a set given by the following cayley table
*
0
1
2
0
0
1
2
1
0
0
2
2
0
1
0
Then (X, βˆ— ,0) is a fuzzy BCK-algebra. We define fuzzy set πœ‡ in X by πœ‡(0) = 0.7, πœ‡(1) =
0.5, πœ‡(2) = 0.2
Then πœ‡ is a fuzzy BH-ideal of X.
Definition 3.4
Let Ξ» π‘Žπ‘›π‘‘ πœ‡ be the fuzzy sets in a set X. The Cartesian product Ξ» × πœ‡: 𝑋 × π‘‹ β†’ [0,1] is
defined by (Ξ» × πœ‡)(π‘₯, 𝑦) = min{πœ†(π‘₯), πœ‡(𝑦)} βˆ€π‘₯, 𝑦 ∈ 𝑋.
Theorem 3.5
If Ξ» π‘Žπ‘›π‘‘ πœ‡ be the fuzzy BH-ideals of a BH- algebra X, then Ξ» × πœ‡ is a fuzzy BH-ideals
of 𝑋 × π‘‹
Proof
For any (π‘₯, 𝑦) ∈ 𝑋 × π‘‹, wehave
(Ξ» × πœ‡)(0,0) = min{πœ†(0), πœ‡(0)} β‰₯ min{ πœ†(π‘₯), πœ‡(𝑦)}
= (Ξ» × πœ‡)(π‘₯, 𝑦)
That is (Ξ» × πœ‡)(0,0) = (Ξ» × πœ‡)(π‘₯, 𝑦)
Let (π‘₯1 , π‘₯2 ) π‘Žπ‘›π‘‘ (𝑦1 , 𝑦2) ∈ 𝑋 × π‘‹
Then, (Ξ» × πœ‡) (π‘₯1 , π‘₯2 ) = π‘šπ‘–π‘›{πœ†(π‘₯1 ), πœ‡(π‘₯2 )}
β‰₯ min{min{ πœ†(π‘₯1 βˆ— 𝑦1 ), πœ†(𝑦1 )} , min{πœ‡(π‘₯2 βˆ— 𝑦2 ), πœ‡(𝑦2 )}
= min{min πœ†(π‘₯1 βˆ— 𝑦1 ), πœ‡(π‘₯2 βˆ— 𝑦2 ), min{ πœ†(𝑦1 ), πœ‡(𝑦2 )}}
= min {(Ξ» × πœ‡) (π‘₯1 βˆ— 𝑦1 , π‘₯2 βˆ— 𝑦2 )) , ( Ξ» × πœ‡) (𝑦1 , 𝑦2 )}
= min {((Ξ» × πœ‡) (π‘₯1 , π‘₯2 ) βˆ— (𝑦1 , 𝑦2 )), ( Ξ» × πœ‡) (𝑦1 , 𝑦2 )}
That is ((Ξ» × πœ‡) (π‘₯1 , π‘₯2 )) = min {(Ξ» × πœ‡) (π‘₯1 , π‘₯2 ) βˆ— (𝑦1 , 𝑦2 )), (Ξ» × πœ‡)(𝑦1 , 𝑦2 )}
And (Ξ» × πœ‡)( (π‘₯1 , π‘₯2 ) βˆ— (𝑦1 , 𝑦2 ))
http://www.iaeme.com/IJARET/index.asp
353
[email protected]
Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
= (Ξ» × πœ‡) (π‘₯1 βˆ— 𝑦1 , π‘₯2 βˆ— 𝑦2 )
= min{ πœ†(π‘₯1 βˆ— 𝑦1 ), πœ‡(π‘₯2 βˆ— 𝑦2 )}
β‰₯ min {min {Ξ» (π‘₯1 ), Ξ» (𝑦1 )}, min {πœ‡(π‘₯2 ), πœ‡(𝑦2 )}}
= min {min (Ξ» ( π‘₯1 ), πœ‡(π‘₯2 ), min{( Ξ» (𝑦1 ), πœ‡(𝑦2 ))}
= min {(Ξ» × πœ‡)(( π‘₯1 , π‘₯2 ), ( Ξ» × πœ‡)(𝑦1 , 𝑦2 ))
That is (Ξ» × πœ‡)(( π‘₯1 , π‘₯2 ) βˆ— (𝑦1 , 𝑦2 ))
=min {(Ξ» × πœ‡)(( π‘₯1 , π‘₯2 ), ( Ξ» × πœ‡) (𝑦1 , 𝑦2 )}
Hence Ξ» × πœ‡ is a fuzzy BH-ideal of X× π‘‹
Theorem 3.6
Let Ξ» π‘Žπ‘›π‘‘ πœ‡ be fuzzy sets in a BH-algebra such that πœ† × πœ‡ is a fuzzy BH-ideal of 𝑋 ×
𝑋.Then
i)Either Ξ»(0)β‰₯ Ξ» (x)π‘œπ‘Ÿ πœ‡(0) β‰₯ πœ‡(π‘₯) βˆ€π‘₯ ∈ 𝑋.
ii) If Ξ» (0)β‰₯ Ξ» (x) βˆ€π‘₯ ∈ 𝑋, then either πœ‡(0) β‰₯ Ξ»(π‘₯) or πœ‡(0) β‰₯ πœ‡(π‘₯)
iii) If πœ‡(0) β‰₯ πœ‡(π‘₯) βˆ€π‘₯ ∈ 𝑋, then either Ξ» (0)β‰₯ Ξ» (x) or Ξ» (0)β‰₯ µ(x)
Proof
We use reduction to absurdity
i) Assume Ξ»(x)> Ξ» (0) and πœ‡(π‘₯) β‰₯ πœ‡(0) for some π‘₯, 𝑦 ∈ 𝑋.
Then (Ξ» × πœ‡)(π‘₯, 𝑦) = min{Ξ»(x), πœ‡(𝑦)}
>min {{Ξ»(0), πœ‡(0)}
= (Ξ» × πœ‡)(0,0)
(Ξ» × πœ‡)(π‘₯, 𝑦) > (Ξ» × πœ‡)(0,0) βˆ€π‘₯, 𝑦 ∈ 𝑋
Which is a contradiction to (Ξ» × πœ‡) is a fuzzy BH-ideal of X× π‘‹
Therefore either Ξ» (0)β‰₯ Ξ» (x)π‘œπ‘Ÿ πœ‡(0) β‰₯ πœ‡(π‘₯) βˆ€π‘₯ ∈ 𝑋.
ii) Assume πœ‡(0) < Ξ»(π‘₯) and πœ‡(0) < πœ‡(𝑦) for some π‘₯, 𝑦 ∈ 𝑋.
Then (Ξ» × πœ‡)(0,0) = min{Ξ»(0), πœ‡(0)}= πœ‡(0)
And (Ξ» × πœ‡)(π‘₯, 𝑦) = min{Ξ»(x), πœ‡(𝑦)}> πœ‡(0)
= (Ξ» × πœ‡)(0,0)
This implies (Ξ» × πœ‡)(π‘₯, 𝑦) > and ( Ξ» × πœ‡)(0,0)
Which is a contradiction to Ξ» × πœ‡ is a fuzzy BH-ideal of X× π‘‹
Hence if Ξ» (0)β‰₯ Ξ» (x) βˆ€π‘₯ ∈ 𝑋, π‘‘β„Žπ‘’π‘›
Either πœ‡(0) β‰₯ Ξ»(π‘₯) or πœ‡(0) β‰₯ πœ‡(π‘₯) βˆ€π‘₯ ∈ 𝑋
iii) Assume Ξ» (0)< Ξ» (x) or Ξ» (0)< µ(y) βˆ€π‘₯, 𝑦 ∈ 𝑋
Then ((Ξ» × πœ‡)(0, 0) = min{Ξ»(0), πœ‡(0)}= πœ†(0)
And (Ξ» × πœ‡)(π‘₯, 𝑦) = min{Ξ»(x), πœ‡(𝑦)}> πœ†(0)
= (Ξ» × πœ‡)(0,0)
This implies (Ξ» × πœ‡)(π‘₯, 𝑦) >(Ξ» × πœ‡)(0,0)
Which is a contradiction to (Ξ» × πœ‡) is a fuzzy BH-ideal of 𝑋 × π‘‹
Hence if πœ‡(0) β‰₯ πœ‡(π‘₯) βˆ€π‘₯ ∈ 𝑋 then either Ξ» (0)β‰₯ Ξ» (x) or Ξ» (0)β‰₯ µ(x)
This completes the proof
Theorem 3.7
http://www.iaeme.com/IJARET/index.asp
354
[email protected]
K. Anitha and Dr. N. Kandaraj
If Ξ» × πœ‡ is a fuzzy BH-idela of 𝑋 × π‘‹ then Ξ» π‘œπ‘Ÿ πœ‡ is a fuzzy BH-ideal of X.
Proof
First we prove that πœ‡ is a fuzzy BH-ideal of X.
Given Ξ» × πœ‡ is a fuzzy BH-ideal of 𝑋 × π‘‹, then by theorem 3.6(i), either Ξ» (0)β‰₯
(x)π‘œπ‘Ÿ
Ξ»
πœ‡(0) β‰₯ πœ‡(π‘₯) βˆ€π‘₯ ∈ 𝑋.
Let πœ‡(0) β‰₯ πœ‡(π‘₯)
By theorem 3.6(iii) then either Ξ» (0)β‰₯ Ξ» (x) or Ξ» (0)β‰₯ µ(x)
Now πœ‡(π‘₯) = min{Ξ»(0), πœ‡(π‘₯)}
= (Ξ» × πœ‡)(0, π‘₯)
β‰₯ min{ ( (Ξ» × πœ‡)(0, π‘₯) βˆ— (0, 𝑦)), (Ξ» × πœ‡)(0, 𝑦)}
= min{ (Ξ» × πœ‡)(0 βˆ— 0), π‘₯ βˆ— 𝑦),(Ξ» × πœ‡)(0, 𝑦)}
= min{ (Ξ» × πœ‡)(0, π‘₯ βˆ— 𝑦),(Ξ» × πœ‡)(0, 𝑦)}
= min{ (Ξ» × πœ‡)(0 βˆ— 0), π‘₯ βˆ— 𝑦),(Ξ» × πœ‡)(0, 𝑦)}
= min{ πœ‡(π‘₯ βˆ— 𝑦), ( πœ‡)(𝑦)}
That is πœ‡(π‘₯) β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦), πœ‡(𝑦)}
πœ‡(π‘₯ βˆ— 𝑦) = min{ Ξ» (0) , πœ‡(π‘₯ βˆ— 𝑦)}
= (Ξ» × πœ‡)(0, π‘₯ βˆ— 𝑦)
= (Ξ» × πœ‡)(0 βˆ— 0, π‘₯ βˆ— 𝑦)
= (Ξ» × πœ‡)(0, π‘₯) βˆ— (0, 𝑦)
πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{ (Ξ» × πœ‡)(0, π‘₯), (Ξ» × πœ‡)(0, 𝑦)}
=min{πœ‡(π‘₯), πœ‡(𝑦)}
That is, πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑦)}
This proves that πœ‡ is a fuzzy BH-ideal of X.
Secondly to prove that Ξ» is a Fuzzy BH-ideal of X.
Using theorem 4.6(i) and (ii) we get
This completes the proof.
Theorem 3.8
If πœ‡ is a fuzzy BH-idela of 𝑋, then πœ‡π‘‘ is a BH-idela of X for all 𝑑 ∈ [0,1]
Proof
Let πœ‡ be a fuzzy BH-ideal of X,
Then By the definition of BH-ideal
πœ‡(0) β‰₯ πœ‡(π‘₯)
πœ‡(π‘₯) β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦), (πœ‡(𝑦)}
πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{ πœ‡(π‘₯), (πœ‡(𝑦)} βˆ€π‘₯, 𝑦 ∈ 𝑋
To prove that πœ‡π‘‘ 𝑖𝑠 π‘Ž BH-ideal of x.
By the definition of level subset of πœ‡
πœ‡π‘‘ = {π‘₯/ πœ‡(π‘₯) β‰₯ 𝑑}
Let π‘₯, 𝑦 ∈ πœ‡π‘‘ and πœ‡ is a fuzzy BH-ideal of X.
Since πœ‡(0) β‰₯ πœ‡(π‘₯) β‰₯ 𝑑 implies 0∈ πœ‡π‘‘ , for all 𝑑 ∈ [0,1]
Let π‘₯, 𝑦 ∈ πœ‡π‘‘ and 𝑦 ∈ πœ‡π‘‘
Therefore πœ‡(π‘₯ βˆ— 𝑦) β‰₯ 𝑑 and πœ‡(𝑦) β‰₯ 𝑑
http://www.iaeme.com/IJARET/index.asp
355
[email protected]
Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
Now πœ‡(π‘₯) β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦), (πœ‡(𝑦)}
β‰₯ min{𝑑, 𝑑} β‰₯ 𝑑
Hence (πœ‡(π‘₯) β‰₯ 𝑑
That is π‘₯ ∈ πœ‡π‘‘ .
Let π‘₯ ∈ πœ‡π‘‘ , 𝑦 ∈ 𝑋
Choose y in X such that πœ‡(𝑦) β‰₯ 𝑑
Since π‘₯ ∈ πœ‡π‘‘ implies πœ‡(π‘₯) β‰₯ 𝑑
We know that πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{ πœ‡(π‘₯), (πœ‡(𝑦)}
β‰₯ min{𝑑, 𝑑}
β‰₯𝑑
That is
πœ‡(π‘₯ βˆ— 𝑦) β‰₯ 𝑑 implies π‘₯ βˆ— 𝑦 ∈ πœ‡π‘‘
Hence πœ‡π‘‘ is a BH-ideal of X.
Theorem 3.9
If X be a BH-algebra, βˆ€π‘‘ ∈ [0,1] and πœ‡π‘‘ 𝑖𝑠 π‘Ž 𝐡𝐻 βˆ’ideal of X, then πœ‡ is a fuzzy BH-ideal of
X.
Proof
Since πœ‡π‘‘ 𝑖𝑠 π‘Ž 𝐡𝐻 βˆ’ideal of X
0∈ πœ‡π‘‘
ii) π‘₯ βˆ— 𝑦 ∈ πœ‡π‘‘ And 𝑦 ∈ πœ‡π‘‘ implies π‘₯ ∈ πœ‡π‘‘
iii)π‘₯ ∈ πœ‡π‘‘ , y ∈ 𝑋 implies π‘₯ βˆ— 𝑦 ∈ πœ‡π‘‘
To prove that πœ‡ 𝑖𝑠 π‘Ž 𝑓𝑒𝑧𝑧𝑦 BH-ideal of X.
Let π‘₯, 𝑦 ∈ πœ‡π‘‘ then πœ‡(π‘₯) β‰₯ 𝑑 and πœ‡(𝑦) β‰₯ 𝑑
Let πœ‡(π‘₯) = 𝑑1 and πœ‡(𝑦) = 𝑑2
Without loss of generality let 𝑑1 ≀ 𝑑2
Then π‘₯ ∈ πœ‡π‘‘ 1
Now π‘₯ ∈ πœ‡π‘‘ 1 and y ∈ 𝑋 π‘–π‘šπ‘π‘™π‘–π‘’π‘  π‘₯ βˆ— 𝑦 ∈ πœ‡π‘‘ 1
That is πœ‡(π‘₯ βˆ— 𝑦) β‰₯ 𝑑1
= min {𝑑1 , 𝑑2 }
= min { πœ‡(π‘₯), πœ‡(𝑦)}
πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min { πœ‡(π‘₯), πœ‡(𝑦)}
𝑖𝑖) 𝐿𝑒𝑑 πœ‡(0) = πœ‡(π‘₯ βˆ— π‘₯) β‰₯ min { πœ‡(π‘₯), πœ‡(𝑦)} β‰₯ πœ‡(π‘₯) (by proof (i))
That is πœ‡(0) β‰₯ πœ‡(π‘₯) for all x ∈ 𝑋
iii) Let πœ‡(π‘₯) = πœ‡(π‘₯ βˆ— 𝑦) βˆ— (0 βˆ— 𝑦))
β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦), πœ‡(0 βˆ— 𝑦)) (By (i))
β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦), min{πœ‡(0), πœ‡(𝑦)}
β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦),{πœ‡(𝑦)) (By (ii))
πœ‡(π‘₯) β‰₯ min{ πœ‡(π‘₯ βˆ— 𝑦),{πœ‡(𝑦))
Hence πœ‡ is a fuzzy BH-ideal of X.
Definition 3.10
A fuzzy set πœ‡ in X is said to be fuzzy BH- πž† ideal if πœ‡(π‘₯ βˆ— 𝑒 βˆ— 𝑣 βˆ— 𝑦) β‰₯ min { πœ‡(π‘₯), πœ‡(𝑦)}
http://www.iaeme.com/IJARET/index.asp
356
[email protected]
K. Anitha and Dr. N. Kandaraj
Theorem 3.11
Every Fuzzy BH -ideal is a fuzzy BH- πž†- ideal
Proof
It is trivial
Remark
Converse of the above theorem is not true. That is every fuzzy BH-πž† -ideal is not true. That
is every fuzzy BH -ideal. Let us prove this by an example
Let X = {0, 1, 2} be a set given by the following cayley table
*
0
1
2
0
0
1
2
1
1
0
2
2
2
1
0
Then (X,βˆ— ,0) is a BH-algebra. We define fuzzy set: Xβ†’ [0,1] by πœ‡(0) = 0.8, πœ‡(π‘₯) =
0.2 βˆ€π‘₯ β‰  0 clearly πœ‡ is a fuzzy BH-ideal of X But πœ‡ is not a BH-πž†-ideal of X.
For Let π‘₯ = 0 𝑒 = 1 𝑣 = 1 𝑦 = 1
πœ‡(π‘₯ βˆ— 𝑒 βˆ— 𝑣 βˆ— 𝑦 = πœ‡(0 βˆ— 1 βˆ— 1 βˆ— 1)= πœ‡(1) = 0.2
min{ πœ‡(π‘₯), πœ‡(𝑦)} = min{ πœ‡(0), πœ‡(0)}
= πœ‡(0) = 0.8
πœ‡(π‘₯ βˆ— 𝑒 βˆ— 𝑣 βˆ— 𝑦) ≀min { πœ‡(π‘₯), πœ‡(𝑦)}
Hence πœ‡ is not a fuzzy BH-πž†-ideal of X.
πœ‡(π‘₯ βˆ— 𝑦)= πœ‡ (𝑓(π‘Ž) βˆ— 𝑓(𝑏))
β‰₯ πœ‡ (𝑓(π‘Ž βˆ— 𝑏))
= πœ‡ 𝑓 (π‘Ž βˆ— 𝑏)
β‰₯ min{ πœ‡ 𝑓 (π‘Ž), πœ‡ 𝑓 (𝑏)}
= min{πœ‡(𝑓(π‘Ž), πœ‡(𝑓(𝑏))}
= min{πœ‡(π‘₯), πœ‡(𝑦)}
Hence πœ‡(π‘₯ βˆ— 𝑦) β‰₯ min{πœ‡(π‘₯), πœ‡(𝑦)}
Hence πœ‡ is a fuzzy BH- πž† ideal of Y.
And
4. FUZZY DOT BH-IDEALS OF BH-ALGEBRAS
Definition 4.1. A fuzzy subset ΞΌ of X is called a fuzzy dot BH-ideal of X if it satisfies
The following conditions:
(FBH1). πœ‡(0) β‰₯ πœ‡(π‘₯)
(FBH2). πœ‡(π‘₯) β‰₯ πœ‡(π‘₯ βˆ— 𝑦) . πœ‡(𝑦)
(FBH3). πœ‡(π‘₯ βˆ— 𝑦) β‰₯ πœ‡(π‘₯). πœ‡(𝑦)for all π‘₯, 𝑦 ∈ 𝑋
Example 4.2. Let X = {0, 1, 2, 3} be a BH-algebra with Cayley table (Table 1) as follows:
(Table 1)
*
0
1
2
3
0
0
1
2
3
http://www.iaeme.com/IJARET/index.asp
1
0
0
2
3
357
2
0
0
0
3
3
0
2
0
0
[email protected]
Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
Define πœ‡ ∢ 𝑋 β†’ [0,1] by ΞΌ (0) =0.9, ΞΌ (a) = ΞΌ (b) =0.6, ΞΌ (c) =0.3 .It is easy to verify that ΞΌ
is a Fuzzy dot BH-ideal of X.
Proposition 4.3. Every fuzzy BH-ideal is a fuzzy dot BH-ideal of a BH-algebra.
Remark. The converse of Proposition 4.3 is not true as shown in the following Example
4.2. Let 𝑋 = {0 ,1 ,2 ,3} be a BH-algebra with Cayley table (Table 1) as follows:
Example 4.4. Let 𝑋 = {0,1, 2} be a BH-algebra with Cayley table (Table 2) as follows:
(Table 2)
*
0
1
2
0
0
1
2
1
0
0
1
2
0
2
0
Define ΞΌ: X β†’ [0, 1] by ΞΌ (0) =0.8, ΞΌ (1) = 0.5, ΞΌ (2) =0.4. It is easy to verify that ΞΌ is a
fuzzy Dot BH-ideal of X, but not a fuzzy d-ideal of X because
πœ‡(π‘₯) ≀ min{πœ‡(π‘₯ βˆ— 𝑦), πœ‡(𝑦)}
πœ‡(1) = min{πœ‡(1 βˆ— 2), πœ‡(2)}
= πœ‡(2)
Proposition 4.5. Every fuzzy dot BH-ideal of a BH-algebra X is a fuzzy dot subalgebra of
X.
Remark. The converse of Proposition 4.5 is not true as shown in the following
Example:
Example 4.6. Let X be the BH-algebra in Example 4.4 and define ΞΌ: X β†’ [0, 1] by
ΞΌ (0)= ΞΌ (1)=0.9, ΞΌ (2)= 0.7. It is easy to verify that ΞΌ is a fuzzy dot sub algebra of X, but
not a fuzzy dot BH-ideal of X because ΞΌ (2)=0.7 ≀ 0.81=ΞΌ (2 βˆ—1)β‹… ΞΌ (1) .
Proposition 4.7. If μ and ν are fuzzy dot BH-ideals of a BH-algebra X, then so is μ ∩ ν.
Proof. Let x, y ∈ X. Then
(πœ‡ ∩ 𝜈)(0) = π‘šπ‘–π‘› {πœ‡ (0), 𝜈 (0)}
β‰₯ π‘šπ‘–π‘› {πœ‡(π‘₯), 𝜈(π‘₯)}
= (πœ‡ ∩ 𝜈) (π‘₯).
Also, (πœ‡ ∩ 𝜈) (π‘₯) = π‘šπ‘–π‘› {πœ‡(π‘₯), 𝜈(π‘₯)}
β‰₯ π‘šπ‘–π‘› {πœ‡(π‘₯ βˆ— 𝑦) · πœ‡(𝑦), 𝜈(π‘₯ βˆ— 𝑦) · 𝜈(𝑦)}
β‰₯ (π‘šπ‘–π‘› {πœ‡(π‘₯ βˆ— 𝑦), 𝜈(π‘₯ βˆ— 𝑦)}) · (π‘šπ‘–π‘› {πœ‡(𝑦), 𝜈(𝑦)})
= ((πœ‡ ∩ 𝜈) (π‘₯ βˆ— 𝑦)) · ((πœ‡ ∩ 𝜈) (𝑦)).
𝐴𝑛𝑑, (πœ‡ ∩ 𝜈) (π‘₯ βˆ— 𝑦) = π‘šπ‘–π‘› {πœ‡(π‘₯ βˆ— 𝑦), 𝜈(π‘₯ βˆ— 𝑦)}
β‰₯ π‘šπ‘–π‘› {πœ‡(π‘₯) · πœ‡(𝑦), 𝜈(π‘₯) · 𝜈(𝑦)}
β‰₯ (π‘šπ‘–π‘› {πœ‡(π‘₯), 𝜈(π‘₯)}) · (π‘šπ‘–π‘› {πœ‡(𝑦), 𝜈(𝑦)})
= ((πœ‡ ∩ 𝜈) (π‘₯)) · ((πœ‡ ∩ 𝜈) (𝑦)).
Hence μ ∩ ν is a fuzzy dot BH-ideal of a d-algebra X.
Theorem 4.8. If each nonempty level subset U (ΞΌ; t) of ΞΌ is a fuzzy BH-ideal of X then
μ is a fuzzy dot BH-ideal of X, where t ∈[0, 1].
Definition 4.9
http://www.iaeme.com/IJARET/index.asp
358
[email protected]
K. Anitha and Dr. N. Kandaraj
Let Οƒ be a fuzzy subset of X. The strongest fuzzy Οƒ-relation on BH-algebra X is the fuzzy
subset µΟƒ of X ×X given by πœ‡πœŽ (x, y) = Οƒ(x) · Οƒ(y) for all x, y ∈ X. A fuzzy relation ΞΌ on BHalgebra X is called a Fuzzy Οƒ-product relation if µ(x, y) β‰₯ Οƒ(x) · Οƒ(y) for all x, y ∈ X. A
fuzzy relation ΞΌ on BH-algebra is called a left fuzzy relation on Οƒ if µ(x, y) = Οƒ(x) for all x, y
∈ X.
Note that a left fuzzy relation on Οƒ is a fuzzy Οƒ-product relation.
Remark. The converse of Theorem 4.8 is not true as shown in the following example:
Example 4.10. Let X be the BH-algebra in Example 4.4 and define ΞΌ: X β†’ [0, 1] by
ΞΌ (0)=0.6, ΞΌ (1)=0.7, ΞΌ (2)= 0.8. We know that ΞΌ is a fuzzy dot BH-ideal of X, but
U (ΞΌ; 0.8) = {x ∈X | ΞΌ(x) β‰₯ 0.8} = {2, 2} is not BH-ideal of X since 0βˆ‰U (ΞΌ; 0.8).
Theorem 4.11. Let 𝑓 ∢ 𝑋 β†’ 𝑋 β€² be an onto homomorphism of BH-algebras, Ξ½ be a fuzzy
Dot BH-ideal of Y. Then the Preimage 𝑓 βˆ’1 (𝑣) of Ξ½ under f is a fuzzy dot BH-ideal of X.
Proof. Let x∈ X,
𝑓 βˆ’1 (𝑣)(0) = 𝑣(𝑓(0)) = 𝑣(0β€² )
β‰₯ 𝑣(𝑓(π‘₯)) = 𝑓 βˆ’1 (𝑣)(π‘₯)
For any x, y∈ X, we have
𝑓 βˆ’1 (𝑣)(π‘₯) = Ξ½ (f (x)) β‰₯ Ξ½ (f (x)* f(y)) · Ξ½ (f (y))
= Ξ½ (f (x* y)) · Ξ½ (f (y)) = 𝑓 βˆ’1 (𝑣)(π‘₯ βˆ— 𝑦). 𝑓 βˆ’1 (𝑣)(𝑦)
Also,
𝑓 βˆ’1 (𝑣)(π‘₯ βˆ— 𝑦)= Ξ½ (f (x * y)) = Ξ½ (f (x) * f (y))
β‰₯ Ξ½ (f (x)) · Ξ½(f (y)) = 𝑓 βˆ’1 (𝑣)(π‘₯). 𝑓 βˆ’1 (𝑣)(𝑦)
Thus 𝑓 βˆ’1 (𝑣)is a fuzzy dot BH-ideal of X.
Theorem: 4.12 An onto homomorphic image of a fuzzy dot BH-ideal with the sup
Property is a fuzzy dot BH-ideal.
Theorem 4.13. If Ξ» and ΞΌ are fuzzy dot BH-ideal of a BH-algebra X, then Ξ» × ΞΌ is a fuzzy
Dot BH-ideal of X × X.
Proof.
Let π‘₯, 𝑦 ∈ 𝑋
Ξ» × ΞΌ (0,0)=Ξ»(0), ΞΌ(0)
β‰₯ Ξ»(x).ΞΌ(y)= (Ξ» × ΞΌ) (x, y)
For any π‘₯, π‘₯ β€² , 𝑦, 𝑦′ ∈ 𝑋 wehave
(Ξ» × µ) (x, y) = Ξ»(x). µ(y)
β‰₯ Ξ»(x βˆ— x β€² ). Ξ»(xβ€²))µ(y βˆ— y β€² ). µ(yβ€²)
= Ξ»(x βˆ— x β€² ). µ(y βˆ— y β€² )). Ξ»(x β€² ). µ(yβ€²)
= (Ξ»× µ)(x, y) βˆ— (x β€² , y β€² )). (Ξ»× µ)(x β€² , y β€² )
Also (Ξ»× µ)(x, y) βˆ— (x β€² , y β€² )) = (Ξ»× µ)(x βˆ— xβ€²) βˆ— (y βˆ— y β€² )).
= Ξ»(x βˆ— x β€² ). µ(y βˆ— y β€² )
β‰₯ Ξ»(x). Ξ»(x β€² )). ( µ(y). µ(yβ€²))
(Ξ»(x). µ(y)). (Ξ»(x β€² ). µ(y β€² ))
= (Ξ»× µ)(x, y). (Ξ» × µ)(xβ€², yβ€²)
Hence Ξ» × ΞΌ is a fuzzy dot BH-ideal of X × X.
http://www.iaeme.com/IJARET/index.asp
359
[email protected]
Fuzzy Ideals and Fuzzy Dot Ideals on Bh-Algebras
Theorem 4.14. Let Οƒ be a fuzzy subset of a BH-algebra X and πœ‡πœŽ be the strongest fuzzy
Οƒ -relation on BH-algebra X. Then Οƒ is a fuzzy dot BH-ideal of X if and only if πœ‡πœŽ is a
Fuzzy dot BH-ideal of X × X.
Proof. Assume that Οƒ is a fuzzy dot BH-ideal of X. For any x, y ∈ X we have
πœ‡πœŽ (0, 0)= Οƒ (0) · Οƒ (0) β‰₯ Οƒ(x) · Οƒ(y) = πœ‡πœŽ (x, y).
Let x, x β€², y, y β€²βˆˆ X. Then
πœ‡πœŽ ((π‘₯, π‘₯ β€²) βˆ— (𝑦, 𝑦 β€²)) · πœ‡πœŽ (𝑦, 𝑦 β€²)
= πœ‡πœŽ (π‘₯ βˆ— 𝑦, π‘₯ β€² βˆ— 𝑦 β€²) · πœ‡πœŽ (𝑦, 𝑦 β€²)
= (𝜎(π‘₯ βˆ— 𝑦) · 𝜎(π‘₯ β€² βˆ— 𝑦 β€²)) · (𝜎(𝑦) · 𝜎(𝑦 β€²))
= (𝜎(π‘₯ βˆ— 𝑦) · 𝜎(𝑦)) · (𝜎(π‘₯ β€² βˆ— 𝑦 β€²) · 𝜎(𝑦 β€²))
≀ 𝜎 (π‘₯) · 𝜎(π‘₯ β€² ) = πœ‡πœŽ (π‘₯, π‘₯ β€² ). 𝐴𝑛𝑑,
πœ‡πœŽ (x, x β€² )· πœ‡πœŽ (y, y β€²) = (Οƒ(x) · Οƒ(x β€²)) · (Οƒ(y) · Οƒ(y β€²))
= (Οƒ(x) · Οƒ(y)) · (Οƒ(x β€²) · Οƒ(y β€²))
≀ 𝜎 (π‘₯ βˆ— 𝑦) · 𝜎( π‘₯ β€² βˆ— 𝑦 β€² )
= πœ‡πœŽ (π‘₯ βˆ— 𝑦, π‘₯ β€² βˆ— 𝑦 β€²) = πœ‡πœŽ ((π‘₯, π‘₯ β€²) βˆ— (𝑦, 𝑦 β€²)).
Thus πœ‡πœŽ is a fuzzy dot BH-ideal of X × X.
Conversely suppose that πœ‡πœŽ is a fuzzy dot BH-ideal of X × X. From (FBH1) we get
(Οƒ(0) )2 = Οƒ (0) · Οƒ(0)= 𝑐 (0, 0)
And so Οƒ (0) β‰₯ Οƒ(x) for all x ∈ X. Also we have
( Οƒ(x) )2 = µΟƒ ((x, x )
β‰₯ µΟƒ ((x, x)*(y, y)). µΟƒ (y, y)
=µΟƒ ((xβˆ—y ), (x βˆ— y)). µΟƒ (y, y)
=((Οƒ(x βˆ— y) · Οƒ(y)) )2
Which implies that 𝜎(π‘₯) β‰₯ 𝜎(π‘₯ βˆ— y) · Οƒ(y) for all x, y ∈ X.
Also we have
(𝜎(π‘₯ βˆ— 𝑦)2=πœ‡πœŽ ((xβˆ—y ), π‘₯ βˆ— 𝑦)
=πœ‡πœŽ ((π‘₯, π‘₯) βˆ— (𝑦, 𝑦)) β‰₯ µΟƒ (π‘₯, π‘₯). µΟƒ (𝑦, 𝑦)
= (Οƒ(π‘₯) · Οƒ( 𝑦 ))2
So 𝜎(π‘₯ βˆ— 𝑦) β‰₯ 𝜎(π‘₯) · 𝜎(𝑦) for all x, y ∈X.
Therefore Οƒ is a fuzzy dot BH-ideal of X.
Proposition 4.15. Let ΞΌ be a left fuzzy relation on a fuzzy subset Οƒ of a BH-algebra X. If
ΞΌ is a fuzzy dot BH-ideal of X × X, then Οƒ is a fuzzy dot BH-ideal of a BH-algebra X.
Proof.
Suppose that a left fuzzy relation ΞΌ on Οƒ is a fuzzy dot BH-ideal of X × X.
Then
Οƒ (0) = ΞΌ (0, z ) βˆ€z ∈ X
By putting z=0
Οƒ (0) = ΞΌ (0,0) β‰₯ ΞΌ (x , y )=Οƒ (x ),
For all x∈ X.
For any x, x β€², y, y β€²βˆˆ X
http://www.iaeme.com/IJARET/index.asp
360
[email protected]
K. Anitha and Dr. N. Kandaraj
Οƒ(x) = µ(x, y) β‰₯ µ((x, y) βˆ— (x β€² , y β€² )). µ(xβ€², yβ€²)
= µ((x βˆ— xβ€²), (y βˆ— y β€² )). µ(y, y β€² ))
= Οƒ(x βˆ— xβ€²). Οƒ(x β€²)
Also
Οƒ(x βˆ— x β€² ).= µ(x βˆ— x β€² , y βˆ— y β€² ) = µ((x, y) βˆ— (x β€² , y β€² ))
β‰₯ µ(x, y). µ(x β€² , y β€² )
=Οƒ(x). Οƒ(π‘₯β€²)
Thus Οƒ is a fuzzy dot BH-ideal of a BH-algebra X.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Y. Imai and K. Iseki, on axiom systems of propositional calculi XIV, Proc. Japan Acad 42
(1966), 19-22.
K. Iseki, An algebra related with a propositional calculus, Proc. Japan, Acad 42 (1966), 26-29.
Iseki K and Tanaka S: β€œAn introduction to theory of BCK-algebras ”, Math. Japan. 23(1978),
1-26.
P. Bhattacharya and N. P. Mukherjee, Fuzzy relations and fuzzy groups, Inform. Sci. 36(1985),
267-282.
O. G. Xi, Fuzzy BCK-algebras, Math. Japan. 36(1991), 935-942.
L. A. Zadeh, Fuzzy sets, Information Control, 8(1965) 338-353.
K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japonica, 21(1976), 351-366.
A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512–517.
Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Scientiae Mathematician 1(1) (1998),
347–354.
Q. Zhang, E. H. Roh and Y. B. Jun, on fuzzy BH-algebras, J. Huanggang Normal Univ. 21(3)
(2001), 14–19.
J. Neggers and H. S. Kim, on d-algebras, Math. Slovaca, 49 (1996), 19-26.
M. Akram, on fuzzy d-algebras, Punjab University Journal of Math.37 (2005), 61-76.
http://www.iaeme.com/IJARET/index.asp
361
[email protected]
Download
Random flashcards
Arab people

15 Cards

Pastoralists

20 Cards

Radioactivity

30 Cards

Ethnology

14 Cards

Create flashcards