Advanced
ON MEDIAL B-ALGEBRAS
ON MEDIAL B-ALGEBRAS
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 849-856
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 13, 2014
  • Accepted : March 17, 2014
  • Published : September 30, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
YOUNG HEE KIM

Abstract
In this paper we introduce the notion of medial B -algebras, and we obtain a fundamental theorem of B -homomorphism for B -algebras. AMS Mathematics Subject Classification : 06F35.
Keywords
1. Introduction
Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK -algebras and BCI -algebras [4 , 5] . It is known that the class of BCK -algebras is a proper subclass of the class of BCI -algebras. In [2 , 3] Q. P. Hu and X. Li introduced a wide class of abstract algebras: BCH -algebras. They have shown that the class of BCI -algebras is a proper subclass of the class of BCH -algebras. J. Neggers and H. S. Kim [8] introduced the notion of d -algebras, i.e., (I) x x = 0;(V) 0 ∗ x = 0; (VI) x y = 0 and y x = 0 implay x = y , which is another useful generalization of BCK -algebras, and then they investigated several relations between d -algebras and oriented digraphs. Recently, Y. B. Jun, E. H. Roh and H. S. Kim [6] introduced a new notion, called an BH -algebra, i.e., (I),(II) x ∗ 0 = 0 and (IV), which is a generalization of BCH / BCI / BCK -algebras. They also defined the notions of ideals and boundedness in BH -algebras, and showed that there is a maximal ideal in bounded BH -algebras. J. Neggers and H. S. Kim [9] introduced and investigated a class of algebras, i.e., the class of B -algebras, which is related to several classed of algebras of interest such as BCH / BCI / BCK -algebras and which seems to have rather nice properties without being excessively complicated otherwise. Furthermore, a digraph on algebras defined below demonstrates a rather interesting connection between B -algebras and groups. J. R. Cho and H. S. Kim [1] discussed further relations between B -algebras and other classed of algebras, such as quasigroups. J. Neggers and H. S. Kim [10] introduced the notion of normality in B -algebras and obtained a fundamental theorem of B -homomorphism for B -algebras.
In this paper we introduce the notion of medial B -algebras, and we obtain a fundamental theorem of B -homomorphism for B -algebras.
2. Preliminaries
In this section, we introduce some notions and results which have also been discussed in [1 , 9] . A B -algebra is a non-empty set X with a constant 0 and a binary operation "∗" satisfying the following axioms:
  • (I)x∗x= 0,
  • (II)x∗ 0 =x,
  • (III) (x∗y) ∗z=x∗ (z∗ (0 ∗y))
for all x , y , z in X .
Example 2.1. Let X := {0, 1, 2} be a set with the following table:
PPT Slide
Lager Image
Then ( X ; ∗, 0) is a B -algebra.
Example 2.2 ( [9] ). Let X be the set of all real numbers except for a negative integer − n . Define a binary operation ∗ on X by
PPT Slide
Lager Image
Then ( X ; ∗, 0) is a B -algebra.
Example 2.3. Let X := {0, 1, 2, 3, 4, 5} be a set with the following table:
PPT Slide
Lager Image
Then ( X ; ∗, 0) is a B -algebra (see [10] ).
Example 2.4 ( [9] ). Let F < x , y , z > be the free group on three elements. Define u v := vuv −2 . Thus u u = e and u e = u . Also e u = u −1 . Now, given a , b , c , ∈ F < x , y , z >, let
PPT Slide
Lager Image
Let N (∗) be the normal subgroup of F < x , y , z > generated by the elements w ( a , b , c ). Let G = F < x , y , z > / N (∗). On G define the operation "·" as usual and define
PPT Slide
Lager Image
It follows that ( uN (∗)) ∗ ( uN (∗)) = eN (∗), ( uN (∗)) ∗ ( eN (∗)) = uN (∗) and
PPT Slide
Lager Image
Hence ( G ; ∗, eN (∗)) is a B -algebra.
Lemma 2.5 ( [9] ). If ( X ; ∗, 0) is a B - algebra , then y z = y ∗ (0 ∗ (0 ∗ z )) for any y , z X .
Proposition 2.6 ( [9] ). If ( X ; ∗; 0) is a B - algebra , then
PPT Slide
Lager Image
for any x , y , z X .
Lemma 2.7 ( [1] ). Let ( X ; ∗, 0) be a B - algebra . Then we have the following statements .
  • (i)if x∗y= 0then x=y for any x,y∈X;
  • (ii)if0 ∗x= 0 ∗y then x=y for any x,y∈X;
  • (iii) 0 ∗ (0 ∗x) =x for any x∈X.
Let ( X ; ∗, 0 X ) and ( Y ; •, 0 Y ) be B -algebras. A mapping φ : X Y is called a B - homomorphism [10] if φ ( x y ) = φ ( x ) • φ ( y ) for any x , y X .
Example 2.8 ( [10] ). Let X := {0, 1, 2, 3} be a set with the following table:
PPT Slide
Lager Image
Then ( X ; ∗, 0) is a B -algebra [1] . If we define φ (0) = 0, φ (1) = 3, φ (2) = 3 and φ (3) = 0, then φ : X Y is a B -homomorphism.
A B -homomorphism φ : X Y is called a B - isomorphism [10] if φ is a bijection, and denote it by X Y . Note that if φ : X Y is a B -isomorphism then φ −1 : Y X is also a B -isomorphism. If we define φ (0) = 0, φ (1) = 2, φ (2) = 1 and φ (3) = 3 in Example 2.8, then φ : X Y is a B -isomorphism. Let φ : X Y be a B -homomorphism. Then the subset { x X | φ ( x ) = 0 Y } of X is called the kernel of the B -homomorphism φ , and denote it by Kerφ
Definition 2.9 ( [10] ). Let ( X ; ∗, 0) be a B -algebra. A non-empty subset N of X is called a subalgebra of X if x y N , for any x , y N .
In Example 2.8, N 1 := {0, 3} is a subalgebra of X , while N 2 := {0, 1} is not a subalgebra of X , since 0 ∗ 1 = 2 ∉ N 2 . Note that any subalgebra of a B -algebra is also a B -algebra.
Theorem 2.10 ( [10] ). Let ( X ; ∗, 0) be a B - algebra and
PPT Slide
Lager Image
Then the following are equivalent :
  • (a)N is a subalgebra of X.
  • (b)x∗ (0 ∗y), 0 ∗y∈N,for any x,y∈N.
Note that any kernel of a B -homomorphism is a subalgebra of X .
3. Medial B-algebras
Let ( X ; ∗, 0) be a B -algebra and let N be a subalgebra of X . The set X (resp., N ) is said to be medial if ( x n 1 ) ∗ ( y n 2 ) = ( x y ) ∗ ( n 1 n 2 ) for any x , y , n 1 , n 2 X (resp., for any x , y , n 1 , n 2 N ).
Example 3.1. The B -algebra in Example 2.8, is medial. The B -algebra in Example 2.3, is not medial, since (5∗2)∗(4∗3) = 4∗1 = 5 ≠ 3 = 1∗5 = (5∗4)∗(2∗3).
J. Neggers and H. S. Kim [10] introduced the notion of a normal subalgebra in B -algebras. A nonempty subset N of X is said to be normal (or normal subalgebra ) of X if ( x a ) ∗ ( y b ) ∈ N for any x a , y b N .
Example 3.2. The subalgebra N 1 = {0, 3} is both a normal and a medial subalgebra of X in Example 2.8, while the subalgebra N 2 = {0, 3} in Example 2.3 is medial, but not normal.
Example 3.3. Let X := {0, 1, 2, 3} be a set with the following table:
PPT Slide
Lager Image
Then ( X ; ∗, 0) is a B -algebra and the subalgebra N 3 = {0, 2} is a medial subal-gebra of X .
Let ( X ; ∗, 0) be a B -algebra and let N be a subalgebra of X . Define a relation ∼ N on X by x N y if and only if x N = y N , where x , y X . Then it is easy to show that ∼ N is an equivalence relation on X . Assume X is medial (or N is a medial subalgebra of X ). If x N y and a N b , where x , y , a , b N , then x N = y N and a N = b N and hence x = y n 1 , a = b n 2 for some n 1 , n 2 N . Hence x a = ( y n 1 ) ∗ ( b n 2 ) = ( y b ) ∗ ( n 1 n 2 ) ∈ ( y b ) ∗ N , since X (resp., N ) is medial. For any ( x a ) ∗ n 3 ∈ ( x a ) ∗ N , we have
PPT Slide
Lager Image
Hence ( x a ) ∗ N ⊆ ( y b ) ∗ N . Similarly, we obtain ( y b ) ∗ N ⊆ ( x a ) ∗ N . This means that x a N y b , i.e., ∼ N is a congruence relation on X . Denote the equivalence class containing x by [ x ] N , i.e., [ x ] N = { y X | x N y } and let X / N := {[ x ] N | x X }. We show that X / N is a B -algebra.
Theorem 3.4. Let X be a medial B - algebra and let N be a subalgebra of X . Then X / N is a medial B - algebra with N = [0] N .
Proof . If we define [ x ] N ∗ [ y ] N := [ x y ] N then the operation "∗" is well-defined, since ∼ N is a congruence relation on X . We claim that [0] N = N . If x ∈ [0] N , then x N = 0 ∗ N , and hence by (II) x = x ∗ 0 ∈ x N = 0 ∗ N , i.e., x = 0 ∗ n for some n N . Since N is a subalgebra and 0 ∈ N , x = 0 ∗ n N . Hence [0] N N .
For any x N , since N is subalgebra of X , 0 ∗ x N , say n 1 = 0 ∗ x . By applying Lemma 2.7-(iii), x = 0 ∗ (0 ∗ x ) ∈ 0 ∗ N . We show that x N = 0 ∗ N . For any x n x N ,
PPT Slide
Lager Image
Hence x N ⊆ 0 ∗ N . If y ∈ 0 ∗ N , then y = 0 ∗ n 2 for some n 2 N . Hence y = 0 ∗ n 2 = ( x x ) ∗ n 2 = x ∗ ( n 2 ∗ (0 ∗ x )). Since x N , by Theorem 2.10, n 2 ∗ (0 ∗ x ) ∈ N . Hence y x N , i.e., 0 ∗ N ⊆ x ∗ N . Thus x N = 0 ∗ N , i.e., x ∼ N 0. Hence x ∈ [0] N , proving N ⊆ [0] N . Checking three axioms and mediality is trivial and we omit the proof.
Theorem 3.4 can be replaced by the following statement:
Theorem 3.4′. Let X be a B - algebra and N be a medial subalgebra of X . Then X / N is a medial B - algebra with N = [0] N .
The B -algebra X / N discussed in Theorems 3.4 and 3.4′ is called the quotient B -algebra of X by N .
Proposition 3.5. Let N be a medial subalgebra of the B - algebra ( X ; ∗, 0). Then the mapping γ : X X / N , given by γ ( x ) := [ x ] N , is a surjective B - homomorphism , and Kerγ = N .
Proof . The mapping γ is obviously surjective. For all x , y X , γ ( x y ) = [ x y ] N = [ x ] N ∗ [ y ] N = γ ( x ) ∗ γ ( y ). Hence γ is a B -homomorphism. We claim that { x X | [ x ] N = [0] N } = N . For any n N , we show that n N = 0 ∗ N . If n 1 N , by Lemma 2.7-(iii), n n 1 = (0 ∗ (0 ∗ n )) ∗ n 1 = 0 ∗ ( n 1 ∗ (0 ∗ (0 ∗ n ))) = 0 ∗ ( n 1 n ) ∈ 0 ∗ N , i.e., n N ⊆ 0 ∗ N . For any 0 ∗ n 2 ∈ 0 ∗ N , 0 ∗ n 2 = ( n n ) ∗ n 2 = n ∗ ( n 2 ∗ (0 ∗ n )) ∈ n N , i.e., 0 ∗ N n N . This proves 0 ∗ N = n N , i.e., [ n ] N = [0] N . If [ x ] N = [0] N , then x N = 0 ∗ N , i.e., x = 0 ∗ n 1 for some n 1 N . Since N is a subalgebra of X , x = 0 ∗ n 1 N . Hence
PPT Slide
Lager Image
proving the proposition.
The mapping γ discussed in Proposition 3.5 is called the natural (or canonical ) B -homomorphism of X onto X / N .
Proposition 3.6. Let X be a medial B - algebra . If φ : X Y is a B - homomorphism , then the kernel Kerφ is a medial subalgebra of X .
Proof . Straightforward.
By Theorem 3.4 and Proposition 3.6, if φ : X Y is a B -homomorphism, then X / Kerφ is a B -algebra.
A B -algebra ( X ; ∗, 0) is said to be commutative [9] if a ∗ (0 ∗ b ) = b ∗ (0 ∗ a ) for any a , b X . The B -algebra in Example 2.1 is commutative, while the B -algebra in Example 2.3 is not commutative, since 3 ∗ (0 ∗ 4) = 2 ≠ 1 = 4 ∗ (0 ∗ 3).
Theorem 3.7. Let X be a commutative medial B - algebra and let φ : X Y be a B - homomorphism . Then X / Kerφ Imφ . In particular , if φ is surjective , then X / Kerφ Y .
Proof . Let K := Kerφ . If we define Ψ : X / K Imφ by Ψ([ x ] K ) := φ ( x ), then Ψ is well-defined. In fact, suppose that [ x ] K = [ y ] K . Then x K y and x K = y K , i.e., x = y k 1 , y = x k 2 for some k 1 , k 2 K . Hence φ ( x ) = φ ( y k 1 ) = φ ( y ) ∗ φ ( k 1 ) = φ ( y ) ∗ 0 = φ ( y ), i.e., Ψ([ x ] K ) = Ψ([ y ] K ). Suppose that Ψ([ x ] K ) = Ψ([ y ] K ), where [ x ] K , [ y ] K X / K . Then φ ( x ) = φ ( y ). If α ∈ [ x ] K , then α K x and α K = x K . This means that α = x k 1 , x = α k 2 for some k 1 , k 2 K . Hence φ ( α ) = φ ( x k 1 ) = φ ( x ) ∗ φ ( k 1 ) = φ ( x ) = φ ( y ), which implies φ ( α y ) = φ ( α ) ∗ φ ( y ) = 0. Hence α y Kerφ = K , i.e., α y = k 3 for some k 3 K . Similarly, φ ( y ) ∗ φ ( α ) = 0 implies y α = k 4 for some k 4 K . Sice X is commutative,
PPT Slide
Lager Image
For any α k 4 α K , α k = ( y k 4 ) ∗ k = y ∗ ( k ∗ (0 ∗ k 4 )) ∈ y K . Hence α K y K . Conversely, we have
PPT Slide
Lager Image
proving y K α K . Hence α K = y K , i.e., α K y . This proves α ∈ [ y ] K . Similarly, [ y ] K ⊆ [ x ] K . Thus [ x ] K = [ y ] K , proving that Ψ is injective. Obviously Ψ is surjective. Since Ψ([ x ] K ∗ [ y ] K ) = Ψ([ x y ] K ) = φ ( x y ) = φ ( x ) ∗ φ ( y ) = Ψ([ x ] K ) ∗ Ψ([ y ] K ), Ψ is a B -homomorphism. Hence X / Kerφ Imφ .
Example 3.8. In Example 2.8, since K = Kerφ = {0, 3}, we have [0] K = {0, 3} and [1] K = { x X | x ∗ 1 ∈ K } = {1, 2}. Hence X / Kerφ = {[0] K , [1] K } and X / Kerφ Imφ by defining Ψ([0] K ) = φ (0) and Ψ([1] K ) = φ (1).
BIO
Young Hee Kim is working as a professor in Department of Mathematics and is interested in BE-algebras.
Department of Mathematics, Chungbuk National University, Cheongju 361-763, Korea.
e-mail:yhkim@chungbuk.ac.kr
References
Kim J.R. , Cho H.S. (2001) On B-algebras and quasigroups Quasigroup and Related Systems 8 1 - 6
Hu Q.P. , Li X. (1983) On BCH-algebras Math. Seminar Notes 11 313 - 320
Hu Q.P. , Li X. (1985) On proper BCH-algebras Math. Japonica 30 659 - 661
Iséki K. 1980 On BCI-algebras Math. Seminar Notes 8 125 - 130
Iséki K. , Tanaka S. (1978) An introduction to the theory of BCK-algebras Math. Japonica 23 1 - 26
Jun Y.B. , Roh E.H. , Kim H.S. (1998) On BH-algebras Sci. Math. Japo. 1 347 - 354
Meng J. , Jun Y.B. 1994 BCK-algebras Kyung Moon Sa Co. Seoul
Neggers J. , Kim H.S. (1999) On d-algebras Math. Slovaca 49 19 - 26
Neggers J. , Kim H.S. (2002) On B-algebras Matematichki Vesnik 54 21 - 29
Neggers J. , Kim H.S. (2002) A fundamental theorem of B-homomorphism for B-algebras Int. Math. J. 2 207 - 214
So K.S. , Kim Y.H. (2013) Mirror d-Algebras J. Appl. Math. & Informatics 31 559 - 564    DOI : 10.14317/jami.2013.559