On K-Homomorphisms of K-Algebras 1 Introduction - CiteSeerX

International Mathematical Forum, 2, 2007, no. 46, 2283 - 2293

On K-Homomorphisms of K-Algebras K. H. Dar1 Govt. College University Lahore Department of Mathematics Katchery Road, Lahore-54000, Pakistan prof [email protected] M. Akram University College of Information Technology University of the Punjab Old Campus, Lahore-54000, Pakistan [email protected] Abstract The authors have introduced a class of K-algebras in [1] and have further extended its scope of study in literature [2, 3, 4, 5]. In this paper, we introduce the notion of K-homomorphism of K-algebras, and investigate some of their properties and structure.

Mathematics Subject Classification: 06F35 Keywords: K-subalgebras; K-ideals; K-homomorphism; K-automorphism; Isomorphism

1

Introduction

The notion of a K-algebra (G, ·, , e) was first introduced by K. H. Dar and M. Akram in [1]. A K-algebra built on a group (G, ·) by adjoining induced binary operation defined by x y= x · y −1, for all x, y ∈ G, where e is the identity of the group. It is attached to an abstract K-algebra (G, , e), which is non-commutative and non-associative with right identity element e. It is characterized by using its left and right mappings in [2]. Recently, same authors have proved that a class of K-algebras as a generalization of a class of B-algebras [13] and that of a family of BCH/BCI/BCK-algebras [7, 8, 9, 10, 11, 12] in [4]. The notion of fuzzy structures of K(G)-algebras is also 1

The first author is supported by HEC, Islamabad.

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K. H. Dar and M. Akram

introduced in [5]. In this paper, we discuss some properties of K-subalgebras and K-ideals in K-algebras in Section 2. The notion of K-homomorphisms of K-algebras is introduced, and some of its related properties are investigated in Section 3.

2

K-subalgebras

In this section, we cite notion of K-algebras and extend study of K-subalgebra. Definition 2.1. [1] An algebra (G, , e) built on a group G with identity e is called a K-algebra on G(briefly, K(G)-algebra), if G is not elementary abelian 2-group and observes the following -axioms: (k1) (x y) (x z) = (x ((e z) (e y))) x, (k2) x (x y) = (x (e y)) x, (k3) x x = e, (k4) x e = x, (k5) e x = x−1 , for all x, y, z ∈ G. If the group G is abelian then the axioms (k1) and (k2) of K-algebra change to be (k1) and (k2) respectively where (k1) (x y) (x z) = z y, (k2) x (x y) = y, for all x, y, z ∈ G. Definition 2.2. A K(G)-algebra (G, , e) is abelian if, g(ex) = x(eg), for all x, g ∈ G. Proposition 2.3. A K(G)-algebra (G, , e) is abelian if and only if, ex = x, for all x ∈ G. Proof. Routine. Proposition 2.4. In a K(G)-algebra (G, , e), the following equations are valid if the group G is non-abelian: (k6) (x y) (u v) = (x (e v) (e y)) u. (k7) (x y) z = x (z (e y)).

On K-homomorphisms of K-algebras

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(k8) e (e x) = x. (k9) e (x y) = y x = (e x) (e y). (k10) x y = e if and only if x = y. Proof. The proof is straightforward and we omit it. Definition 2.5. [1] A non-empty subset H of a K-algebra (G, , e) is called K-subalgebra if (i) e ∈ H, (ii) h1 h2 ∈ H, for all h1 , h2 ∈ H. Example 2.6. [1] Consider the K(S3 )-algebra (S3 , , e) on the symmetric group S3 = {e, a, b, x, y, z} where e = (1), a = (123), b = (132), x = (12), y = (13), z = (23), and is given by the following Cayley table: e x y z a b

e e x y z a b

x x e b a z y

y y a e b x z

z z b a e y x

a b z x y e a

b a y z x b e

We see that K-subalgebra (A3 , , e) is a proper subalgebra having the following table: e a b

e e a b

a b e a

b a b e

If H = {e, x} = C2 then K(S3 )-subalgebra (C2 , , e) is an improper in K(S3 ) by the Cayley table. e x e e x x x e Remark. (1) For every subgroup H of group G there exists a proper Ksubalgebra (H, , e) if H is not an elementary abelian 2-subgroup of G.

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(2) If H is an elementary abelian 2-subgroup of G then K-subalgebra (H, , e) is improper. Proposition 2.7. In a K-algebra (G, , e), let g be a fixed element of G and H be a subgroup of a group G. Then Hg2 = {g (g x) : x ∈ G} forms a K-subalgebra. Proof. Since g (g e) = e ∈ Hg2 , where e ∈ H and (g (g x)) (g (g y)) = = = = =

(g ((e y) (e x))) g (g (e (y x))) g (g (x y)) g g (g (e (x y))) g (g (y x)) ∈ Hg2

for all x, y ∈ H and g (g x), g (g y) ∈ Hg2 . This complete the proof. Definition 2.8. A proper K-subalgebra (H, , e) of a K-algebra (G, , e) is called a K-ideal if for for all x ∈ G, g ∈ G, x g and g (g x) ∈ H implies x ∈ H. In example 2. 6, the K-subalgebra (A3 , , e) is a K-ideal of K-algebra (G, , e) . Proposition 2.9. Let H1 and H2 be two K-subalgebras(K-ideals) of a Kalgebra (G, , e) then (a) H1 ∩ H2 is a K-subalgebra(K-ideal) of (G, , e). (b) H1 H2 is a K-subalgebra(K-ideal) of (G, , e) if and only if H2 H1 = H1 H2 (either H1 or H2 is ideal). Proof. Straightforward. Proposition 2.10. For each proper normal subgroup H of G there corresponds a proper K- ideal (H, , e) of (G, , e). Proof. If H is a normal subgroup of G then gxg −1 ∈ H for all g ∈ G implies x ∈ H. Thus xg −1 = x g and g (g x) ∈ (H, , e) imply x ∈ H, which proves that (H, , e) is a K-ideal of K-algebra (G, , e). Conversely, if (H, , e) is a K-ideal of K-algebra then g (g x) and x g ∈ H. This implies that gxg −1 ∈ H for all g ∈ G and x ∈ H. This completes the proof Corollary 2.11. The improper K-subalgebras of a K-algebra correspond to the improper K-ideals of the K-algebra (G, , e).


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Corollary 2.12. If G is a simple group then the K-algebra is a simple algebra having no proper K-ideal. Theorem 2.13. Let K1 = (G, , e) be a K-algebra on G. If Z(G) is the centre of the group G then (Z(G), , e) = Z(G, , e), the centre of K1 . Proof. From definition 2.2, if h commutes with g in K1 then g (e h) = h (e g). If z1 , z2 ∈ Z(G) then z1 (e g) = g (e z1 ) and z2 (e g) = g (e z2 ) for all g ∈ G. Then z1 z2 ∈ Z(K1 ) since, for all g ∈ G, (z1 z2 ) (e g) = = = = = = = =

z1 ((e g) (e z2 )) z1 (e (g z2 )) z1 (z2 g) (z1 (e g) z2 (g (e z1 )) z2 g (z2 (e (e z1 ))) g (z2 z1 ) g (e (z1 z2 )) f or all g ∈ K1 .

Hence z1 z2 ∈ Z(K1 ), the centre of K1 and (Z(G), , e) ⊆ Z(K1 )

(a)

For the converse, if z1 , z2 ∈ Z(K1 ), then z1 (e g) = g (e z1 ) and z2 (e g) = g (e z2 ) Thus z1 z2 ∈ (Z(G), , e) since (z1 (e g)) (z2 (e g)) = z1 z2 = z1 z2−1 ∈ Z(G). Hence Z(K1 ) ⊆ (Z(G), , e)

(b)

Inequalities (a) and (b) imply the equality and the proof is completed.

3

K-homomorphisms of K-algebras

In a K-algebra K(G) = (G, , e) which is built on a group G, we see that set K(G)= set G and hence the induced operation on G is defined. We see easily that the mapping: endomorphisms and automorphisms of K(G)-algebra coincide set-wise [1] via actions on the respective group G. In this section we generalize the notion of a mapping from one K-algebra K1 = (G1 , , e1 ) into another K-algebra K2 = (G2 , , e2 ). Thus we define a K-homomorphism in a usual way.

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Definition 3.1. A mapping ψ from a K-algebra K1 into K2 is called a Khomomorphism if, for every x1 , y1 ∈ K1 , ψ(x1 y1 ) = ψ(x1 ) ψ(y1 ), where ψ(x1 ), ψ(y1) ∈ K2 . Example 3.2. If H = {g(gx) : x ∈ G} then mapping ψ : (G, ) → (H, ) is a K-homomorphism where, for a fixed g in G, ψ(x) = g (g x) is a homomorphism from (G, ) into (H, ). It is easy to see that ψ is a bijection. Remark. (1) A K-homomorphism ψ : K1 → K2 , fromK1 into K2 is a group-homomorphism, from G1 into G2 and the vice versa. (2) A K-homomorphism ψ is called as usual, a monomorphism, epimorphism and isomorphism if ψ is injective, surjective and bijective respectively. (3) If a K-homomorphism ψ from K1 into K2 is a K-isomorphism then the algebras K1 and K2 are called isomorphic and are written by K1 ∼ = K2 . (4) The relation of isomorphism (∼ =) defined on the set of all K-algebras is an equivalence relation which subdivides K-algebras into disjoint equivalence classes. Proposition 3.3. Let K1 = (G1 , , e1 ) and K1 = (G2 , , e2 ) be two Kalgebras and ψ ∈ Hom(K1 , K2 ). Then, for x1 , y1 ∈ K1 and ψ(x1 ), ψ(y1) ∈ K2 , we conclude that: (1) ψ(e1 ) = e2 . (2) ψ(x) = ψ(x−1 ). (3) ψ(e1 x1 ) = e2 ψ(x1 ). (4) ψ(x1 x2 ) = e2 , if and only if ψ(x1 ) = ψ(x2 ). (5) If H1 is a subalgebra of K1 then ψ(H1 ) is a subalgebra of K2 . (6) If H1 is an ideal of K1 then ψ(H1 ) is an ideal of K2 . Proof. The proofs of (1), (2), (3) and (4) follow easily from the definition 3.1. (5) If x1 , y1 ∈ H1 then x1 y1 ∈ H1 as H1 is a K-subalgebra of K1 then ψ(x1 ), ψ(y1 ) and ψ(x1 y1 ) ∈ ψ(H1 ), a subset of K2 . ψ(H1 ) is a Ksubalgebra of K2 since ψ(x1 y1 ) = ψ(x1 ) ψ(y1 ) ∈ ψ(H1 ). By (1) it follows that ψ(H1 ) is a K-subalgebra of K2 . (6) Suppose that H1 is an ideal of K1 and ψ(e1 ) = e2 ∈ ψ(H1 ). Then, for all g, x ∈ K1 , g (g x) ∈ H1 and x g ∈ H1 it implies that x ∈ H1 . Then ψ(g) (ψ(g) ψ(x)) ∈ ψ(H1 ) and ψ(x g) ∈ ψ(H1 ) → ψ(x) ∈ ψ(H1 ), which implies that ψ(H1 ) is an ideal of K2 .


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Corollary 3.4. ψ(x1 y1 ) = ψ(y1 x1 ), for all x1 , y1 ∈ K1 . Corollary 3.5. ψ(x1 ) = ψ(e1 x1 ), for all x1 ∈ K1 . Corollary 3.6. If ψ : K1 → K2 is an epimorphism then ψ −1 ∈ Hom (K2 , K1 ) and the Propositions (5) and (6) hold from K2 into K1 under ψ −1 ∈ Hom (K2 , K1 ). Definition 3.7. Let K1 = (G1 , , e1 ) and K2 = (G2 , , e2 ) be two K-algebras and ψ be a homomorphism from K1 into K2 . The subset Kerψ = {x ∈ K1 : ψ(x) = e2 } of K1 is called the kernel of ψ. We see that: (a) For each ψ ∈ Hom(K1 , K2 ) there exists Kerψ which contains at least the identity element e1 of K1 . (b) Kerψ= {e1 } if and only if ψ is a monomorphism. (c) For every x1 , y1 ∈ K1 , x1 y1 ∈ Kerψ if and only if y1 x1 ∈ Kerψ. (d) Kerψ is a subalgebra of K1 . (e) Kerψ is an ideal of K1 . Theorem 3.8. Any two groups G1 and G2 build their respective isomorphic K- algebras K1 = (G1 , , e1 ) and K2 = (G2 , , e2 ) if and only if the groups G1 and G2 are isomorphic. Proof. Let G1 ∼ = G2 and ψ be an isomorphism between G1 and G2 . Then ∼ ψ(G1 ) = G2 = G1 and ψ(G1 , ) = (G2 , ) ∼ = (G1 , ) → K1 ∼ = K2 . The commutative property ψ ◦ = ◦ ψ establishes the theorem since (ψ ◦ )(x, y) = = = =

ψ((x, y)) = ψ(x y) ψ(x) ψ(y) (ψ(x), ψ(y)) ( ◦ ψ)(x, y), f or every (x, y) ∈ G1 × G2 ,

and the following diagram commutes. ψ /G 2 JJ JJ JJ JJ ◦ψ=ψ◦ JJJ J J$ /K K

G1 dJ

1

ψ

2

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Theorem 3.9. Let K1 and K2 be two K-algebras. Let ψ ∈ Hom(K1 , K2 ). A relation ∼ defined on K1 by, x ∼ y, if and only if x1 y1 ∈ kerψ, for all x1 , y1 ∈ K1 , is an equivalence relation on K1 . Proof. The relation ∼ is reflexive on K1 by definition 2.1. The relation is symmetric by Corollary 3.5 of Proposition 3.4. For transitivity, take x ∼ y and y ∼ z. Then ψ(x y) ψ((x y) (y z)) ψ((x (y z) (e1 y))) ψ((x (y y) (e1 z))) ψ((x (e1 (e1 z))) ψ(x z) x ∼ z.

= = = = = =

e2 = ψ(y z) e2 e2 [by K7 ] e2 [by K7 ] e2 [by K3 ] e2 [by K8 ]

Thus it proves the assertion. For any homomorphism ψ ∈ Hom (K1 , K2 ) there corresponds an equivalence relation on K1 which subdivides K1 into a set of disjoint equivalence classes whose union is the whole of K1 . Since the subdivision of K1 is by kerψ, therefore, the set of all classes is called the quotient set of K1 by kerψ consisting of equivalence classes(cosets) denoted by K1 = {Cx : x ∈ K1 − kerψ} kerψ where Cx = kerψ x = x kerψ,

Cx Cy = Cxy , f or all x, y ∈ K1 and Ce1 = kerψ. K1 Theorem 3.10. The quotient set kerψ consisting of all equivalence classes of K1 K1 by kerψ forms a K-algebra ( kerψ , , kerψ), where Cx Cy = Cxy , for all K1 . x, y ∈ K1 and kerψ = e in the quotient algebra kerψ K1 and Cx Cy = Cxy , for all x, y ∈ K1 . The Proof. Let Cx , Cy ∈ kerψ properties (k1 ) to (k5 ) of definition 2.1 follow by the routine manipulation. K1 , , kerψ) verifies to be a K-algebra. Thus the quotient algebra ( kerψ


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Corollary 3.11. For each endomorphism ψ on K-algebra K1 , the quotient K1 K1 , , kerψ) is formed of order | kerψ |, if K1 is a finite K-algebra. algebra ( kerψ Corollary 3.12. For each ideal H of a K-algebra K1 = (G, , e) there exists a quotient algebra ( KH1 , , H) as a homomorphic image of K1 . That is, there exists a homomorphism θ : K1 → KH1 , defined by θ(x) = x H, where θ(x y) = (x y) H = (x H) (y H) = θ(x) θ(y), f or all x, y ∈ K1 having kerψ = H. The homomorphism θ is called natural homomorphism and denoted by nθ . If ψ in Hom(K1 , K2 ) is an K-epimorphism then: Theorem 3.13. (Fundamental homomorphism theorem) Let K1 and K2 be K1 ∼ any two K-algebras and ψ be an epimorphism from K1 into K2 . Then kerψ = K2 K1 Proof. Let there be a trio of K-algebras; K1 , K2 and kerψ for homomorphisms K1 ψ : K1 → K2 and nθ : K1 → kerψ , where ψ is an epimorphism. Let η be a K1 into K2 defined by η(Cx ) = ψ(x) for all x ∈ K1 . Then η is mapping from kerψ a homomorphism since

η(Cx Cy ) = η(Cxy ) = ψ(x y) = ψ(x) ψ(y) = η(Cx ) η(Cy ) ∀ Cx , Cy ∈

K1 . kerψ

In fact, η is an isomorphism because kerη=kerψ= Ce1 and ψ is an epimorphism onto K2 . Thus K1 ∼ = K2 . kerψ Consequently, the following diagram commutes. K1

K1 Kerψ

/K = 2 { {{ { {{ {{

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4


Structure of homomorphisms of K-algebras

Let there be three K-algebras K1 , K2 and K3 . If φ ∈ Hom(K1 , K2 ) and ψ ∈ Hom(K2 , K3 ) then their composition ψ ◦ φ ∈ Hom(K1 , K3 ) which observes the closure property if ψ, φ ∈Hom(K1 , K1 ). The set End(K1 ), consisting of all endomorphisms of K1 , forms a semigroup with the identity element under the operation of their composition (◦) which is defined by the rule (ψ ◦ φ)(x) = ψ(φ(x)), for all x ∈ X. It is known in [1] that Aut(K1 ) = Aut(G) is a group structure and K-algebra (Aut(K1 ), , Id) built on Aut(K1 ) is proper if the automorphisms of K1 are not all of order 2. Theorem 4.1. Let K1 = (G, , e) be a K(G)-algebra on a group G. Then Aut(K1 ) = (Aut(G), , Id) is K-algebra on the group (Aut(G), ◦), i.e., Aut(K1 ) = (Aut(G), , Id). Proof. If φ, ψ ∈ Aut(G) and φ ◦ ψ = φ ◦ ψ −1 then φ Id = φ, Id φ = φ−1 and ψ ◦ ψ = Id confirm to the properties K4, K5 and K3 respectively of Definition 2.1. The properties K1 and K2 follow in routine computation. Note that the homomorphic/ isomorphic K-algebraic models have similar structures as those of its group theoretic basis. Thus we state the following Theorems without proofs. Theorem 4.2. (The second isomorphism theorem) Let A be a K-subalgebra of K-algebra and let B be a K-ideal of K-algebra. Then (i) B is a K-ideal of A B, (ii) A ∩ B is a K-ideal of A, (iii)

AB B

∼ =

A . A∩B

Theorem 4.3. (The third isomorphism theorem) Let K1 be a K-algebra having K-ideals A and B with A B. Then (i) (ii)

B A

is K-ideal of

K1 B / A A

∼ =

K1 , A

K1 . B

Corollary 4.4. The K-algebra (G, , e) is simple if and only if the group G is simple. Lemma 4.5. (Zassenhaus lemma ) Let A and B be K-subalgebras of a Kalgebra (G, , e) and let A1 and B1 be K-ideals of A and B respectively. Then (a) A1 (A ∩ B1 ) is a K-ideal of A1 (A ∩ B), (b) B1 (A1 ∩ B) is a K-ideal of B1 (A ∩ B), (c)

A1 (A∩B) A1 (A∩B1 )

B1 (A∩B) . B1 (A1 ∩B)


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References [1] K. H. Dar and M. Akram, On a K-algebra built on a group, SEA Bull. Math. 29(1)(2005), 41-49. [2] K. H. Dar and M. Akram, Characterization of a K(G)-algebra by self maps, SEA Bull. Math. 28(4) (2004), 601-610. [3] K. H. Dar, M. Akram and A. Farooq, A note on left K(G)-algebras, SEA Bull. Math., 30 (2006). [4] K. H. Dar and M. Akram, On subclasses of K(G)-algebras, Annals of University of Craiova, Math. Comp. Sci. Ser., 33 (2006). [5] M. Akram, K. H. Dar, Y. B. Jun and E. H. Roh, Fuzzy structures of K(G)algebra, SEA Bull Math. (To appear). [6] Q. P. Hu and X. Li , On BCH-algebras, Math. Seminar Notes 11 (1983), 313-320. [7] Q. P. Hu and X. Li, On proper BCH-algebras, Math. Japonica 30 (1985), 659-661. [8] Y. Imai, and K. Iseki, On axiom System of propositional calculi XIV , Proc., Japonica Academy, 42(1966), 19-22. [9] K. Iseki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966), 26-29. [10] J. Meng, BCI-algebras and abelian groups, Math. Japo. 32 (1987), 693696. [11] C. S. Hoo, BCI-algebras with condition(S), Math. Japonica 32 (1987), 749-756. [12] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa, Co., Seoul, 1994. [13] J. Neggers and H. S. Kim, On B-algebras, Matematicki Vesnik , 54(2002), 21-29. Received: December 31, 2005