International Journal of Algebra, Vol. 7, 2013, no. 6, 251-258 HIKARI Ltd, www.m-hikari.com Generalized Derivation on TM Algebras T. Ganeshkumar Department of Mathematics M.S.S. Wakf Board College Madurai-625020, India ganeshkumar.wbc@gmail.com M. Chandramouleeswaran Department of Mathematics Saiva Bhanu Kshatriya College Aruppukottai - 626101, India moulee59@gmail.com Copyright c 2013 T. Ganeshkumar and M. Chandramouleeswaran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Recently an algebra based on propositional calculi was introdued by Tamilarasi and Megalai in the year 2010 known as TM algebras [6]. In our paper [1], we introduced the notion of derivation on TM algebras. In this paper, we introduce the group structure of a TM-algebra, the notion of generalized derivation on TM-algebra. We study the properties of generalized derivations on TM algebra. Also the concept of a torsion free TM-algebra is introduced and the properties of generalized derivation on a torsion free TM-algebra are studied. Mathematics Subject Classification: 03G25, 06F35 Keywords: derivations BCK/BCI algebras, TM-algebras, derivations, generalized 1 Introduction Imai and Iseki [2, 3] introduced two classes of algebras of logic:bck and BCIalgebras. and have been extensively investigated by many researchers. It is
252 T. Ganeshkumar and M. Chandramouleeswaran known that the class of BCK-algebras is a proper sub class of the class of BCIalgebras. Recently another algebra based on propositional calculi was introdued by Tamilarasi and Megalai [6] in the year 2010 known as TM algebras. Motivated by the notion of derivations on rings and near-rings Jun and Xin [4]studied the notion of derivation on BCI-algebras. In [5], the authors introduced the notion of generalized derivation of BCI algebras. In our paper [1], we introduced the notion of derivation on TM algebras. In this paper, we discuss the relationship between an abelian group and a TMalgebra. Also, we introduce the notion of generalized derivation and some of its properties on TM algebras. Finally, we introduce the notion of torsion-free TM-algebra and study how the generalized derivation act on it. 2 Preliminaries Definition 2.1 A TM algebra (X,, 0) is a non-empty set X with a constant 0 and a binary operation satisfying the following axioms: 1. x 0=x 2. (x y) (x z) =z y x, y, z X. Definition 2.2 For any TM algebra (X,, 0). We define the set G(X) = {x X 0 x = x}. Definition 2.3 Let (X,, 0) be a TM algebra. A self map d : X X is said to be a (l, r) derivation on X, if d(x y) =(d(x) y) (x d(y)). dis said to be a (r, l) derivation on X, if d(x y) =(x d(y)) (d(x) y). The map d is said to be a derivation on X if d is both a (l, r) derivation and a (r, l) derivation on X. 3 Group structure and Generalized derivations on a TM algebra In this section, we first introduce the notion of obtaining an abelian group from a TM-algebra by defining suitable operation on X and also an abelian group determines a TM-algebra by defining the binary composition suitably. Then we introduce the notion of generalized derivation on a TM-algebra corresponding to a derivation on it. Definition 3.1 Let (X,, 0) be a TM algebra. In X we define a binary composition + as follows: x + y = x (0 y), x, y X.
Generalized derivations on TM-algebras 253 Theorem 3.2 In any TM-algebra (X,, 0) if we define + as x + y = x (0 y), x, y X, then the following hold: 1. x +0=x =0+x. 2. Addition is associative. 3. If x + y =0, then x =0 y. 4. Addition is commutative. 5. Additive inverse of x X is 0 x. Proof: 1. x +0=x (0 0) = x 0=x =0 (0 x) = (0 + x). 2. Repeatedly applying the definition of + and simplifying, we get the result. 3. Applying left cancellation law on x + y =0 x (0 y) =0=x x, we get 0 y = x. 4. Addtion is commutative. For, x + y = 0+(x + y) =(y y)+(x (0 y)) = (y y) (0 (x (0 y))) = (y y) ((0 x) (0 (0 y))) (Since 0 (x y) =(0 x) (0 y)) = (y y) ((0 x) y) (Since 0 (0 y) =y) = y (0 x) =y + x 5. The additive inverse of x is written as x =0 x. For, x +(0 x) =x (0 (0 x)) = x x =0. Definition 3.3 Let X be a TM algebra. If we define an addtion + as x+y = x (0 y) for all x, y X, then (X, +) is an abelian group with identity 0 and the additive inverse denoted by x =0 x x X. Remark 3.4 If we have a TM algebra (X,, 0) it follows from the above definition that (X, +) is an abelian group with y =0 y y X. Then we have x y = x y x, y X. On the other hand if we choose an abelian group (X, +) with an identity 0 and define x y = x y, we get a TM algebra (X,, 0) where x + y = x (0 y) x, y X.
254 T. Ganeshkumar and M. Chandramouleeswaran Example 3.5 Let (X,, 0) be a TM algebra with the Cayley table. Define x + y = x (0 y). Then (X, +) is an abelian group. 0 1 2 3 4 0 0 4 3 2 1 1 1 0 4 3 2 2 2 1 0 4 3 3 3 2 1 0 4 4 4 3 2 1 0 + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 Definition 3.6 Let X be a TM algebra. A mapping D : X X is called a generalized (l, r) derivation of X if there exist an (l, r) derivation d : X X such that D(x y) =(D(x) y) (x d(y)) for all x, y X. Example 3.7 For the TM algebra defined in example 3.5, if we define a map d : X X by d(0) = 3, d(1) = 4, d(2) = 0, d(3) = 1, d(4) = 2. Then d is a (l, r) derivation of X. But d is not a (r, l) derivation of X. Since d(x y) (x d(y)) (d(x) y) for x, y X. The map D : X X defined below is a generalized (l, r) derivation of X. D(0) = 1, D(1) = 2, D(2) = 3, D(3) = 4, D(4) = 0 Remark 3.8 In a TM algebra, x y = y (y x) =x for all x, y X. We can observe that by using the above property if we take D as a generalized (l, r) derivation of X, then D(x y) =D(x) y, for all x, y X. Hence for every (l, r) derivation d of X and any self map D : X X we have D(x y) =(D(x) y) (x d(y)) for all x, y X. Thus D is a generalized (l, r) derivation of X. Analoguosly we define a (r, l) derivation on a TM algebra (X,, 0) as follows. Definition 3.9 Let X be a TM algebra. A mapping D : X X is called a generalized (r, l) derivation of X if there exist an (r, l) derivation d : X X such that D(x y) =(x D(y)) (d(x) y) for all x, y X.
Generalized derivations on TM-algebras 255 Remark 3.10 In a TM algebra, it is observed that for every (r, l) derivation d of X and any self map D : X X we have D(x y) =(x D(y)) (d(x) y) for all x, y X, proving that any self map D is a generalized (r, l) derivation on X. Definition 3.11 Let X be a TM algebra. A mapping D : X X is called a generalized derivation of X, if there exist a derivation d : X X such that D is both a (l,r) generalized derivation and a (r, l) generalized derivation. Example 3.12 Let (X,, 0) be a TM algebra with the following Cayley table. 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 d : X X be defined by d(0) = 3, d(1) = 2, d(2) = 1, d(3) = 0 is a derivation on X. The map D : X X defined below is a generalized derivation on X : D(0) = 2, D(1) = 3, D(2) = 0, D(3) = 1. Lemma 3.13 Let D be a self map of a TM algebra X. Then 1. If D is a generalized (l, r) derivation of X, then D(x) =D(x) x for all x X. 2. If D is a generalized (r, l) derivation of X, then D(0) = 0 if and only if D(x) =x d(x) for all x X and for some (r, l) derivation d of X. Proof: 1. If D be a generalized (l, r) derivation on X, then there exist a (l, r) derivation d on X such that D(x y) =(D(x) y) (x d(y)) for all x, y X. D(x) = D(x 0) = (D(x) 0) (x d(0)) = D(x) (x d(0)) = (x d(0) ((x d(0)) D(x)) = (x d(0)) ((x D(x)) d(0)) (Since (x y) z =(x z) y) = x (x D(x)) (Since(x z) (y z) =x y) = D(x) x for all x X 2. Let D be a generalized (r, l) derivation on X such that D(0) = 0. Then, D(x y) =(x D(y)) (d(x) y) for all (r, l) derivation d (1).
256 T. Ganeshkumar and M. Chandramouleeswaran Putting y = 0 in (1),we get D(x 0) = (x D(0)) (d(x) 0) That is, D(x) =(x 0) d(x) =x d(x) x X. Conversely, if D(x) = x d(x), then D(0) = 0 d(0) = d(0) (d(0) 0) = d(0) d(0) = 0 Lemma 3.14 Let D be a generalized (l, r) derivation of a TM algebra X. Then 1. D(a) =D(0) + a a X. 2. D(a + x) =D(a)+x, a, x X. 3. D(a + b) =D(a)+b = a + D(b) a, b X. Proof: 1. D(a) =D(0 (0 a)) = (D(0) (0 a)) (0 d(0 a)) = D(0) (0 a). That is, D(a) =D(0) + a a X. 2. By using remark 3.8, we get D(a + x) =D(a (0 x)) = D(a) (0 x) =D(a)+x 3. Since (X, +) is an abelian group, the result follows from D(a)+b = D(a + b) =D(b + a) =D(b)+a Theorem 3.15 Let D be a generalized (r, l) derivation of a TM algebra X. Then 1. D(a) G(X) a G(X). 2. D(a) =a D(0) = a + D(0) a X. 3. D(a + b) =D(a)+D(b) D(0) a, b X. 4. D is the identity map on X if and only if D(0) = 0. Proof: 1. For a G(X), D(a) G(X). For, D(a) =D(0 a) =(0 D(a)) (d(0) a) =0 D(a).
Generalized derivations on TM-algebras 257 2. Now, D(a) =D(a 0) = (a D(0)) (d(a) 0). Therefore, D(a) =a D(0) = a D(0 0) = a (0 D(0)) = a + D(0). 3. By result (2) above, we have D(a + b) =(a + b)+d(0). Since (X, +) is an abelian group, on simplifying the right hand side, we get D(a + b) =D(a)+D(b) D(0). 4. If D(0) = 0, Dis the identity map. For, D(a) =D(a 0) = a D(0) = a 0=a a X, Conversely, if D is the identity map on X, then D(a) =a, a X. In particular D(0) = 0. 4 Torsion free TM algebras Definition 4.1 A TM algebra X is said to be torsion free if it satisfies x + x =0 x =0, x X. If there exist a non-zero element x X such that x + x =0, then X is not torsion free. Example 4.2 The TM-algebra X given by the Cayley table in example 3.5 is a Torsion-free TM algebra. For, 0 + 0 = 0, 1+1=1 (0 1) = 1 4=2 2+2=2 (0 2) = 2 3=4 3+3=3 (0 3) = 3 2=1 4+4= 4 (0 4) = 4 1=3. The TM-algebra X given by the Cayley table in example 3.12 is not a Torsion-free TM algebra. For, 1 + 1 = 1 (0 1) = 1 1=0. Theorem 4.3 Let X be a Torsion free TM algebra and let D 1 and D 2 be two generalized derivations. If D 1 D 2 =0on X, then D 2 =0on X. Proof: Let x X, then x + x X. 0 = (D 1 D 2 )(x + x) =D 1 (D 2 (x + x)) = D 1 (0) + D 2 (x + x) (Since D(a) =D(0) + a) = D 1 (0) + D 2 (x)+d 2 (x) D 2 (0) ( By proposition 3.15(3) ) = D 1 (0) D 2 (0) + D 2 (x)+d 2 (x) = (D 1 (0) D 2 (0)) + D 2 (x)+d 2 (x) = (D 1 (0) (0 D 2 (0)) + D 2 (x)+d 2 (x) = D 1 (0) + D 2 (0) + D 2 (x)+d 2 (x) = D 1 (D 2 (0)) + D 2 (x)+d 2 (x) = (D 1 D 2 )(0) + D 2 (x)+d 2 (x) = 0+D 2 (x)+d 2 (x) =D 2 (x)+d 2 (x)
258 T. Ganeshkumar and M. Chandramouleeswaran Since X is Torsion-free, D 2 (x) =0, x X, proving that D 2 =0onX. In the above theorem, if we replace both the generalized derivations D 1 and D 2 by a generalized derivation D itself, we get the following Corollary 4.4 Let X be a Torsion free TM algebra and D be a generalized derivation. If D 2 =0on X, then D =0on X. Proof: Let D 2 =0onX. That is D 2 (x) =0, x X. Now for any x X, 0 = D 2 (x + x) = D(D(x + x)) = D(0) + D(x + x) (Since D(a) =D(0) + a) = D(0) + D(x) + D(x) D(0) (Since By theorem 3.15(3) ) = D(x)+D(x) Since X is Torsion free, D(x) =0, x X, proving that D =0onX. References [1] Chandramouleeswaran M and Ganeshkumar T, Derivations On TM algerbas, International Journal of Mathematical Archive, 3(11), 2012, 3967-3974. [2] Iseki.K. and Tanaka.S, An introduction to theory of BCK-algebras, Math, Japan, 23 (1978), 1-26. [3] Iseki, K, On BCI-algebras, Math.Seminar Notes., 11 (1980), 313-320. [4] Jun.Y.B. and Xin.X.L, On derivations of BCI-algebras, Inform Sci., 159(2004),167-176. [5] Mehmet Ali Ozturk, Yilmaz Ceven and Young Bae Jun, Generalized Derivation of BCI algebras, Honam Mathematical J., No. 4, 31 (2009), 601-609. [6] Tamilarasi, A. and Megalai, K, TM algebra an introduction,casct., (2010). Received: January 10, 2013