PURE MATHEMATICS RESEARCH ARTICLE Derivation, -derivation and generalized derivation o KUS-algebras Chiranjibe Jana 1 *, Tapan Senapati 2 and Madhumangal Pal 1 Received: 08 February 2015 Accepted: 10 June 2015 Published: 12 August 2015 *Corresponding autor: Chiranjibe Jana, Department o Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India E-mail: jana.chiranjibe7@gmail.com Reviewing editor: Xinguang Zhang, Curtin University, Australia Additional inormation is available at the end o the article Abstract: In this paper, the notion o let-right (respectively, right-let) derivation, -derivation, generalized derivation o KUS-algebras are introduced and their properties are established in details. Subjects: Advanced Mathematics; Algebra; Mathematics & Statistics; Pure Mathematics; Science Keywords: KUS-algebra; p-semisimple KUS-algebra; derivation; -derivation; generalized derivation AMS Mathematics Subject classiications (2010): 06F35; 03G25; 06A99 1. Introduction The study o BCK/BCI-algebras was initiated by Imai and Iseki (1966) and Iseki (1966) as a generalization o the concept o set-theoretic dierence and propositional calculus. Neggers and Kim (2002) introduced a new notion, called B-algebras which is related to several classes o algebras o interest such as BCK/BCI-algebras. Kim and Kim (2008) introduced the notion o BG-algebras, which is a generalization o B-algebras. Bhowmik, Senapati, and Pal (2014) and Senapati together with colleagues (Senapati, 2015; Senapati, Bhowmik, & Pal, 2012, 2013, 2014a, 2014b, 2015; Senapati, Bhowmik, Pal, & Davvaz, 2015; Senapati, Jana, Bhowmik, & Pal, 2015; Senapati, Kim, Bhowmik, & Pal, 2015) has done lot o works on B/BG/G-algebras. ABOUT THE AUTHORS Chiranjibe Jana has completed his MSc rom Vidyasagar University, India in 2009. Now, he is a research scholar in the Department o Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University since 2012. His scientiic interest concentrates on uzzy BCK/ BCI-algebras and related algebraic systems. Tapan Senapati obtained his PhD rom Vidyasagar University, India in 2013. He is working as an assistant teacher in Padima Janakalyan Banipith, India. His research interests include uzzy sets, uzzy algebras, triangular norms, BCK/BCIalgebras and related algebraic systems. Madhumangal Pal is a ull Proessor in the Department o Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University. He has published more than 180 articles in international journals. His specializations include Computational Graph Theory, Fuzzy Correlation and Regression, Fuzzy Game Theory, Fuzzy Matrices, Fuzzy Algebra. Pal is the author o the eight books publisherom India and UK. PUBLIC INTEREST STATEMENT In this paper, we introduce the notion o let-right (respectively, right-let) derivation, -derivation, generalized derivation o KUS-algebras. We characterize these derivations and prove that i X is a p-semisimple KUS-algebra, then (Der(X), ) reers to a semigroup. Finally, we determine the relationship between let-right (respectively, right-let) derivation and generalized derivation o KUS-algebras. This paper also considers regular derivations o KUSalgebras. This work is very useul in the ield o pure mathematics as well as applied mathematics. This research may be used to design complex system. 2015 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page 1 o 12
Prabpayak and Leerawat (2009a, 2009b) ormulated a new algebraic structure which is called KUalgebra and investigated some o its properties. In 2013, the concepts o KUS-algebras, KUSsubalgebras, KUS-ideals, homomorphism o KUS-algebras are introduced by Mostaa, Naby, Halim, and Hameed (2013). The relationship between some abelian groups and KUS-algebras, the G-part o KUS-algebras are studied and investigated some o its properties. The notion o derivation in rings and near-rings theory introduced by Posner (1957). Motivated by this, in this paper, the notion o derivation, -derivation and generalized derivation o KUS-algebras are introduced and a lot o their properties are investigated in detail. 2. Preliminaries In this section, some elementary aspects that are necessary or this paper are included. Deinition 2.1 (Prabpayak & Leerawat, 2009a, 2009b) A nonempty set X with the constant 0 and a binary operation is said to be KU-algebra i or all x, y, z X it satisies the ollowing axioms KU1. (x y) ((y z) (x z)) = 0, KU2. 0 x = x, KU3. x 0 = 0, KU4. x y = 0 and y x = 0 imply x = y. Lemma 2.2 (Prabpayak & Leerawat, 2009a, 2009b) Every KU-algebra X satisies the ollowing conditions, or any arbitrary x, y, z X, KU5. x (y z) =y (x z), KU6. x x = 0. Deinition 2.3 (Mostaa et al., 2013) A nonempty set X with the constant 0 and a binary operation is said to be KU-algebra i or all x, y, z X it satisies the ollowing axioms KUS1. (z y) (z x) =(y x), KUS2. 0 x = x, KUS3. x x = 0, KUS4. x (y z) =y (x z). We can deine a partial ordering by x y i and only i y x = 0. A KU-algebra (X,, 0) is called KUS-algebra i it satisies KU7. (z y) (z x) =(y x), or all x, y, z X. In any KUS-algebra X, the ollowing are true or all x, y, z X (a) x y = 0 and y x = 0 imply x = y, (b) x (y x) =y 0, (c) x y = 0 implies that x 0 = y 0, (d) (x y) 0 = y x, (e) x 0 = 0 implies that x = 0, () x = 0 (0 x), (g) 0 (x y) =(0 x) (0 y), (h) x z = y z implies that x 0 = y 0. Page 2 o 12
For a KUS-algebra X, we deine G(X) ={x X x 0 = x}. Then G(X) is the G-part o a KUS-algebra X. For any KUS-algebra X, the set B(X) ={x X x 0 = 0} is called p-radical o X. A KUS-algebra is called p-semisimple i B(X) ={0} and also G(X) B(X) ={0}. A mapping is said to be regular i (0) =0. A mapping o a KUS-algebra X into itsel is called an endomorphism (Mostaa e al., 2013) i (x y) = (x) (y). Note that (0) =0. In a KUS-algebra X, the ollowing properties hold, or any x, y, z X, (1) x (y x) =y, (2) y x = z imply x z = y, (3) x (0 y) =y (0 x), (4) x a = x b imply a = b, (5) a x = b x imply a = b. 3. Derivations o KUS-algebras From now and onwards, we assume that X is a KUS-algebra. For a KUS-algebra X, we denote x y = y (y x) or all x, y X. Deinition 3.1 Let X be a KUS-algebra. A sel map d: X X is said to be a (l, r)-derivation o X, i d(x y) =(d(x) y) (x d(y)), or all x, y X. d is said to be a (r, l)-derivation o X, i d(x y) =(x d(y)) (d(x) y), or all x, y X. The map d is said to be a derivation o X i d is both a (l, r)-derivation and a (r, l)-derivation o X. Example 3.2 Consider a KUS-algebra X ={0, a, b, c} with the ollowing Caley table: Deine a mapping d: X X by d(x) = c, i x = 0 b, i x = a a, i x = b 0, i x = c. * 0 a b c 0 0 a b c a a 0 c b b b c 0 a c c b a 0 It is now routine to veriy that d is a (l, r)-derivation o X. Remark 3.3 We can observe that (i) i the sel-map d:x X is a (l, r)-derivation o X, then d(x y) = d(x) y, (ii) i the sel-map d:x X is a (r, l)-derivation o X, then d(x y) = x d(y), (iii) i d is a derivation o X, then d(x y) = d(x) y = x d(y). Remark 3.4 In KUS-algebra, we can observe that x y = y (y x) =x or all x, y X. Proposition 3.5 A sel-map d: X X is regular (l, r)-derivation o X, i x d(x) or all x X. Proo Let d be a regular derivation o X. Then by Remark 3.3, we have d(0) =0 d(x x) =d(0) d(x x) =0 d(x) x = 0. Thereore, x d(x). Page 3 o 12
Proposition 3.6 Let d: X X be a derivation o KUS-algebra X. Then d is a regular derivation i d is either a (l, r)-derivation or a (r, l)-derivation. Proo I d is (l, r)-derivation then or all x X, d(x) x = 0. Now, by Remark 3.3, we have d(0) =d(x x) =d(x) x = 0. Thereore, d is regular. Again, i d is (r, l)-derivation then or all x X, x d(x) =0. Now, by Remark 3.3, we have d(0) =d(x x) =x d(x) =0. Thereore, d is regular. Proposition 3.7 Let d be a sel-map o a KUS-algebra X. Then (1) i d is regular (l, r)-derivation o X d(x) =x d(x), (2) i d is regular (r, l)-derivation o X d(x) =d(x) x. Proo (1) Given that d is regular (l, r)-derivation. Then d(0) =0. Now, by Deinition 3.1, x = 0 x d(x) =(d(0) x) (0 d(x)) = (0 x) d(x) =x d(x). (2) Given that d is regular (r, l)-derivation o X. Then, by Deinition 3.1, d(x) =d(0 x) =(0 d(x)) (d(0) x) =(0 d(x) (0 x) =d(x) x. Deinition 3.8 or all x X. Let d 1 be two sel-map o a KUS-algebra X. We deine d 1 as )(x) =d 1 (x)) Lemma 3.9 Let d 1 be two (l, r)-derivations o X. Then ) is a (l, r)-derivation o X. Proo Given that d 1 is a (l, r)-derivation o X. Hence, d 1 (x y) =d 1 (x) y or all x, y X. Similarly (x y) =d 2 (x) y or all x, y X. Now )(x y) =d 1 (x y)) = d 1 (x) y) = (x))) y = )(x) y. Thereore, ) is a (l, r)-derivation o X. Lemma 3.10 Let d 1 be two (r, l)-derivations o X. Then ) is a (r, l)- derivation o X. Proo Similar to the proo o Lemma 3.9. From Lemmas 3.9 and 3.10, we see that i d 1 be two derivations o X, then ) is a derivation o X. Theorem 3.11 Let d 1 be two derivations o X. Then )= d 1 ). Proo Since d 1 are two derivations o X, d 1 are both (l, r) and (r, l)-derivations o X. Now, by Remark 3.3, we have )(x y) =d 1 (x y)) = d 1 ( (x) y)) = d 2 (x) d 1 (y) and d 1 )(x y) =d 2 (x y)) = d 2 (x d 1 (y)) = d 2 (x) d 1 (y) = )(x y). This implies that d 1 = d 2 d 1. Deinition 3.12 Let d 1 be two sel-maps o X. We deine d 2 ):X X as d 2 )(x) =d 1 (x) d 2 (x) or all x X. Theorem 3.13 Let d 1 be two derivations o KUS-algebra X. Then d 1 d 2 = d 2 d 1. Proo Let x, y X. Now, by Remark 3.3, )(x y) =d 1 (x y)) = d 1 (x) y) =d 2 (x) d 1 (y) and )(x y) =d 1 (x y)) = d 1 (x d 2 (y)) = d 1 (x) d 2 (y). Thereore, we get d 2 (x) d 1 (y) =d 1 (x) d 2 (y). Substituting y = x, we get d 2 (x) d 1 (x) =d 1 (x) d 2 (x) d 1 )(x) = d 2 )(x), since this is true or all x X. Thereore d 1 = d 1 d 2. Page 4 o 12
Theorem 3.14 Let d be a (l, r)-derivation o KUS-algebra X. Then (1) d(0) =d(x) x, (2) d is one to one mapping, (3) I d regular then d is the identity mapping, (4) I there is an element x X such that d(y) x = 0, or x d(y) =0 or all y X, d(y) =x.that is d constant mapping. Proo (1 ) For a KUS-algebra X, x x = 0 and since d is a (l, r)-derivation which implies d(0) = d(x x) = d(x) x. (2) Let x, y X and d(x) =d(y). Now, d(0) =d(x x) =d(x) x. Again, d(0) =d(y y) =d(y) y. Thereore, d(x) x = d(y) y x = y by let cancellation law. Thereore, d is 1 1. (3) It is given that d is regular. Then d(0) =0. Now, d(0) =d(x) x 0 = d(x) x x x = d(x) x. Then by applying right cancellation law, we get x = d(x). This proves that d is an identity mapping. (4) We assume that there is an element x X such that d(y) x = 0. Then we get, d(y) x = x x d(y) =x, by right cancellation law. Again, we assume that x d(y) =0 x d(y) =x x d(y) =x, by let cancellation law. Hence, d(y) =x or all y X. Thereore, d is a constant mapping. Theorem 3.15 Let d be a (r, l)-derivation o a KUS-algebra X. Then (1) d(0) =x d(x). (2) d is one to one mapping. (3) I d is regular then d is the identity mapping. (4) I there is element x X, such that d(y) x = 0 or x d(y) =0, or all y X. Then d(y) =x. Thereore, d is a constant mapping. Proo Similar to the proo o Theorem 3.14. Theorem 3.16 Let d 1,, d n be regular derivations o X. Then it satisies the inequality d n (d (n 1) (d (n 2) (d (n 3) (d (n 4) (x)))) x. Proo By using Theorem 3.14 and 3.15, we get d n (d (n 1) (d (n 2) (d (n 3) (x))))))) = d n (d (n 1) (d (n 2) (d (n 3) (x))) ))) d (n 1) (d. (n 2) ( (x)))) ) d 1 (x) x Deinition 3.17 Let Der(X) be the set o all derivations o X and d 1 Der(X). We deine a binary operation o Der(X) as d2 )(x) =d 1 (x) d 2 (x) or all x X. Theorem 3.18 I d 1 are (l, r)-derivations o X, then d2 ) is a (l, r)-derivation o X. Proo Let d 1 be (l, r)-derivations o X and d 1 Der(X). Then or all x, y X, using Remarks 3.3 and 3.4, we get d2 )(x y) =d 1 (x y) d 2 (x y) = (x) y) (x) y) = (x) y) (( (x) y) (x) y))) = d 1 (x) y. Again, d2 )(x) y = (x) d 2 (x)) y = (x) (x) d 1 (x))) y = d 1 (x) y. This implies d2 )(x y) = d2 )(x) y. Thereore, d 1 d2 is a (l, r)-derivation o X. Page 5 o 12
Theorem 3.19 I d 1 are (r, l)-derivations o X, then d2 ) is a (r, l)-derivation o X. Proo Similar to the proo o Theorem 3.18. Theorem 3.20 The binary composition deined on Der(X) is associative. Proo Let d 1, d 3 Der(X). Then or all x, y X, we have ( d2 ) d 3 )(x y) = d2 )(x y) d 3 (x y) = (x y) d 2 (x y)) d 3 (x y) = (x) y d 2 (x) y) (d 3 (x) y) =d 1 (x) y d 3 (x) y =(d 3 (x) y) ((d 3 (x) y) (x) (y))) = d 1 (x) y and (d2 d3 ))(x y) =d 1 (x y) d3 )(x y) = (x) y) ( (x) y) (d 3 (x) y)) = (x) y) ((d 3 (x) y) (d 3 (x) y) (x) y)) = d 1 (x) y d 2 (x) y = d 2 (x) y( (x) y) (x) y)) = d 1 (x) y. Thereore, ( d2 ) d 3 )(x y) = (d2 d3 ))(x y). This proved that binary composition is associative. Combining Theorems 3.18, 3.19 and 3.20, we get the ollowing result. Theorem 3.21 In a KUS-algebra, (Der(X), ) orms a semigroup. 4. -Derivation o KUS-algebras In what ollows, let be an endomorphism o X unless otherwise speciied. Deinition 4.1 Let be an endomorphism o a KUS-algebra X. A sel map : X X is said to be a let-right - derivation (briely (l, r)--derivation) o X, i (x y) =( (x) (y)) ( (x) (y)), or all x, y X. is said to be a right-let -derivation (briely (r, l) derivation) o X, i (x y) =( (x) (y)) ( (x) (y)), or all x, y X. The map is said to be a -derivation o X i is both a (l, r) and (r, l)--derivation o X. Example 4.2 Let X ={0, a, b, c} be a KUS-algebra in Example 3.2 and a mapping : X X deined by (x) = c, i x = 0 b, i x = a a, i x = b 0, i x = c and an endomorphism o X deined by (x) = 0, i x = 0 a, i x = a b, i x = b c, i x = c. Then it is easy to veriy that is both derivation an-derivation o X. Example 4.3 Let X be a KUS-algebra in Example 3.2. Deine a mapping : X X by b, i x = 0 c, i x = a (x) = 0, i x = b a, i x = c. Page 6 o 12
Then it is easy to veriy that is a derivation o X. Deine an endomorphism o a unction o X by (x) =a, i x = 0, a, b, c. Now, (a b) = (c) =a and ( (a) (b)) ( (a) (b)) = ((c a) (a 0)) = (b a)=c. This implies (a b) ( (a) (b)) ( (a) (b)). Thereore is not a -derivation o X. Remark 4.4 We can observe that i (i) : X X is a (l, r)- -derivation o KUS-algebra X, then (x y) = (x) (y). (ii) : X X is (r, l)--derivation o KUS-algebra X, then (x y) = (x) (y). Theorem 4.5 Let be a (l, r)- -derivation o G-part o a KUS-algebra o X. Then (0) = (x) (x), or all x, y X. Proo For all x, y X, we have (0) = (x x) =( (x) (x)) ( (x) (x))=( (x) (x)) (( (x) (x)) ( (x) (x))) =( (x) (x)) (( (x) (x)) (( (x) (x)) 0))=( (x) (x)) (( (x) (x)) ( (x) (x))) =( (x) (x)) 0 = (x) (x). Theorem 4.6 Let be a (r, l)--derivation o G-part o a KUS-algebra X. Then (0) = (y) (y) or all x, y X. Proo For all x, y X, we have (0) = (y y) =( (y) (y)) ( (y) (y))=( (y) (y)) (( (y) (y)) ( (y) (y))) =( (y) (y)) (( (y) (y)) (( (y) (y)) 0)) = ( (y) (y)) (( (y) (y)) ( (y) (y))) =( (y) (y)) 0 = (y) (y). Theorem 4.7 Let be a sel-map o KUS-algebra X. I is a -derivation o X, then (x) (x) = (y) (y). Proo It is straightorward, ollowing rom Theorems 4.5 and 4.6. Corollary 4.8 Let be a (l, r)- -derivation (or, (r, l) -derivation) o KUS-algebra X. Then is injective i and only i is injective. Proo Let be a (l, r)- -derivation. Suppose is injective an (x) = (y) or all x, y X. Then (0) = (x) (x) and (0) = (y) (y). By applying Theorem 4.7, we get (x) (x) = (y) (y) (x) = (y) by right cancellation law. Thereore, x = y, since d is injective. Conversely, suppose is injective and (x) = (y) or all x, y X. Then (0) = (x) (x) and (0) = (y) (y). Thereore, (x) (x) = (y) (y) (x) = (y) by let cancellation law, x = y, since is injective. Deinition 4.9 Let be a -derivation o KUS-algebra X. Then is called regular -derivation o X i (0) =0. Theorem 4.10 I is a regular -derivation o a KUS-algebra X, then (x) = (x). Proo Let be a regular -derivation, thereore (0) =0. Since is a -derivation o X, thereore is a (l, r)- -derivation as well as (r, l)--derivation. When is a (l, r)- -derivation, 0 = (0) = (x x) =( (x) (x)) ( (x) (x)) = (x) (x) (x) (x). Page 7 o 12
Again, when is a (r, l)--derivation, 0 = (0) = (x x) =( (x) (x)) ( (x) (x)) = (x) (x) (x) (x). Thereore, we get (x) = (x). Deinition 4.11 Let, d be two sel mappings o KUS-algebra X. We deine d : X X such that ( d )(x) = (d (x)) or all x X. Theorem 4.12 I X is a p-semisimple KUS-algebra and, d are the (l, r)- -derivations o X. Then d is also a (l, r)- -derivation o X. Proo Let X be a p-semisimple KUS-algebra and, d are (l, r)- -derivations o X. Then by using Remarks 3.4 and 4.4, we get or all x, y X, ( d )(x y) = ((d (x) (y)) ( (y) d (y))) = (d (x) (y)) = (d (x)) (y) (d (x)) ( (y)) = (d (x)) (y) = (x) d (y)) ( (x) d (y)) ( (d (d (d (x)) (y))) =( d )(x) (y) (x) ( d )(y). This proves that d is a (l, r)- -derivation o X. Theorem 4.13 I, d are (r, l)--derivations o p-semisimple KUS-algebra X, then d is also a (r, l) --derivation o X. Proo Similar to the proo o Theorem 4.12. Theorem 4.14 Let X is a p-semisimple KUS-algebra and, d are -derivations o X. Then d is also -derivation o X. Proo It is straightorward, ollowing rom Theorems 4.12 and 4.13. Theorem 4.15 I X is a p-semisimple KUS-algebra and, d are -derivations o X such that =, d = d. Then d = d. Proo Let X be a p-semisimple KUS-algebra and, d are the -derivations o X. Since d is a (l, r)- -derivation o X, or all x, y X, d (x y) =d (x y)) = d (x) (y)) (d ((d ( (x) d (y))) = (d (x) (y)), by Remark 4.4. Again, is a (r, l)--derivation o X, so ( d )(x y) =d (x) (y))=( (x)) d ( (y))) (d (d ( (d (x)) (y)) = (x)) d ( (y)) = (x) (d d (y), thus we have, or all x, y X, ( d )(x y) = (x) d (y) (1) Also, since is a (r, l)--derivation X, then or all x, y X, (d d )(x y) ( (x) d (y) =d (x) (y)) = d ( (x) d (y)) by Remark 4.4. But d is a (l, r)- -derivation o X, so (d d )(x y) ( (x) d (y)) = ( (x)) (d (y)) =d d 2 (x) d (d (y)) = ( (x)) (d (y)) d = d (x) d (y) = (x) d (y). d Thus, we have or all x, y X, (d d )(x y) = (x) d (y) (2) From Equations 1 and 2, we get or all x, y X, Page 8 o 12
( d )(x y) =(d )(x y). (3) Putting y = 0 in Equation 3, we get ( d )(x) =(d )(x), or all x X. This implies that d = d. Deinition 4.16 Let, d be two sel maps o KUS-algebra X. We deine d d : X X such that ( d )(x) =d (x) d (x) or all x X. Theorem 4.17 Let X be a p-semisimple KUS-algebra, Epi(X), where Epi(X) is the set o identity mappings over X and, d are -derivations o X. Then ( d ) ( )=( ) ( d ). Proo Let X be a p-semisimple KUS-algebra and, d be -derivations o X. Since d is a (l, r)- -derivation o X, then or all x, y X, ( d )(x y) =d (x) (y)) (d ( (x) d (y)) = (d (x) (y)) by Remark 4.4. But is a (r, l)--derivation o X, thereore, (d (x) (y))=( (x)) d ( (y))) (d ( (d (x)) (y)) = (x)) d ( (y)) = (x) (d d (y). Hence, or all x, y X, ( d )(x y) =( )(x) ( d )(y). (4) Again, we have that d is a (r, l)--derivation o X, then or all x, y X, ( d )(x y) =d (( (x) (y)) d (d (x) (y))) = d ( (x) (y)). But d is a (l, r)- -derivation o X, so ( (x) d (y)) = d ( (x)) (y)) (d 2 (x) (d (y)) = d ( (x)) (y))=(d )(x) ( )(y). (d d Hence, or all x, y X, we get ( d )(x y) =( )(x) ( d )(y). (5) From Equations 4 and 5, we get, or all x, y X, d (x) (y) = (x) d (y). (6) By putting y = x in Equation 6, we get ( d d )(x) =(d d )(x), or all x X. This implies that ( d ) (d )=(d ) ( ). d Deinition 4.18 Let Der(X) be the set o all -derivations o X and, d Der(X). Deine a binary operation such that ( d )(x) = (x) d (x). Theorem 4.19 Let X be a p-semisimple KUS-algebra. Let, d be the (l, r)- -derivations o X. Then d is also a (l, r)- -derivation o X. Proo Let X be a p-semisimple KUS-algebra and, d be (l, r)- -derivations o X. Then, by using Remarks 3.4 and 4.4, we get ( d )(x y) =d (x y) d (x y) = {((d (x) (y)) ( (x) (y))} {(d (x) (y)) ( (x) d (y))}. Also, ( d )(x y) =(d (x) (y)) (d (x) (y)) = d (x) (y) (x) (x) =(d (d (x))) (y) =( (x) d (x)) (y) =(d d )(x) (y) =( (x) (d d )(y)) {( (x) (d d )(y)) ((d d )(x)) (y)} = (( d )(x) (y)) ( (x) ( d )(y)) = (d d )(x y), This proves that ( d ) is a (l, r)- -derivation o X. Theorem 4.20 Let X be a p-semisimple KUS-algebra and, d ( d ) is also a (r, l)--derivation o X. Proo Similarly, by Theorem 4.19, we can establish this proo. be (r, l)--derivations o X. Then Theorem 4.21 Let, d and d Der(X). Then associativity property (( d ) d )(x y) =(d (d d ))(x y) holds in Der(X). Page 9 o 12
Proo Since, d, d Der(X). Then by deinition, (( d ) d )(x y) =(d d )(x y) d (x y) =(d (x (x y) (d y)) (d d )(x y)) = ( d )(x y) =d (x y) d (x y) =((d (x) (y)) ( (x) (y))) ((d (x) (y)) ( (x) d (y))) = (x) (y) d (x) (y) = (x) (y). Also, ( (d d ))(x y) =d (x y) (d d )(x y) =d (x y) ((d (x y) d (x y)) = (x y) d (x y) =((d (x) (y)) ( (x) (y))) ((d (x) (y)) ( (x) d (y))) = d (x) (y) d (x) (y) = (x) (y). This shows that (d d )=(d d ) d ). From Theorems 4.13, 4.19 and 4.20, we conclude the ollowing. Theorem 4.22 I X is a p-semisimple KUS-algebra, then (Der(X), ) is a semigroup. 5. Generalized derivation o KUS-algebras Deinition 5.1 A mapping D: X X is called a generalized (l, r)-derivation o X i there exist a (l, r) -derivation d: X X such that D(x y) =(D(x) y) (x d(y)) or all x, y X. Deinition 5.2 A mapping D: X X is called generalized derivation o X i there exist a derivation d: X X such that D is both (l, r)-generalized derivation and a (r, l)-generalized derivation o X. Example 5.3 In Example 3.2, i we deine d:x X such that d(0) =c, d(a) =b, d(b) =a and d(c) =0 and the mapping D: X X such that D(0) =b, D(a) =c, D(b) =0 and D(c) =a, then D is a generalized (l, r)-derivation o X. Remark 5.4 In a KUS-algebra, x y = y (y x) =x or all x, y X. By using the above property, we can observe that i D is a generalized (l, r)-derivation o X then D(x y) =D(x) y, or all x, y X. Hence or every (l, r)-derivation d o X and any sel map D: X X, we have D(x y) =(D(x) y) (x d(y)) or all x, y X. Thus D is a generalized (l, r)-derivation o X. Deinition 5.5 A mapping D: X X is called a generalized (r, l)-derivation o X i there exist a (r, l) -derivation d:x X such that D(x y) =(x D(y)) (d(x) y) or all x, y X. Remark 5.6 It is observed that or every (r, l)-derivation d o X and any sel map D: X X, D(x y) =(x D(y)) (d(x) y) or all x, y X. This implies any sel map D is a generalized (r, l)-derivation o X. Example 5.7 In Example 3.2, we deine a mapping d: X X such that d(x) = a, i x = 0 0, i x = a c, i x = b b, i x = c. Then d is a derivation o X. Again, i we deine a mapping b, i x = 0 c, i x = a D(x) = 0, i x = b a, i x = c Page 10 o 12
then it is easy to veriy that D is satisying both the (l, r)-derivation and (r, l)-derivation o X. Thereore, D is a generalized derivation o X. Theorem 5.8 Let D be a generalized (r, l)-derivation on G-part o X. Then D(0) =0 i and only i D(x) =x d(x), or some (r, l)-derivation d o X. Proo Let D be a generalized (r, l)-derivation on G-part o X such that D(0) =0. Then D(x y) =(x D(y)) (d(x) y) or all (r, l)-derivation d. Putting y = 0, we get D(x 0) =(x D(0)) (d(x) 0) which implies D(x) =(x 0) d(x) =x d(x). Conversely, we assume that D(x) =x d(x). Putting x = 0, we get D(0) =0 d(0) =d(0) (d(0) 0) =d(0) d(0) =0. Theorem 5.9 Let D be a generalized (r, l)-derivation on G-part o X. Then D is the identity mapping o X i and only i D(0) =0. Proo I D(0) =0 then D(a) =D(a 0) =a D(0) =a 0 = a or all a G(X). Thereore, D is identity mapping. Conversely, i D is the identity map on G(X), then D(a) =a or all a G(X). In particular D(0) =0. Funding The authors received no direct unding or this research. Author details Chiranjibe Jana 1 E-mail: jana.chiranjibe7@gmail.com Tapan Senapati 2 E-mail: math.tapan@gmail.com ORCID ID: http://orcid.org/0000-0003-0399-7486 Madhumangal Pal 1 E-mail: mmpalvu@gmail.com 1 Department o Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India. 2 Department o Mathematics, Padima Janakalyan Banipith, 721517, Kukurakhupi, India. Citation inormation Cite this article as: Derivation, -derivation and generalized derivation o KUS-algebras, Chiranjibe Jana, Tapan Senapati & Madhumangal Pal, Cogent Mathematics (2015), 2: 1064602. Reerences Bhowmik, M., Senapati, T., & Pal, M. (2014). Intuitionistic L-uzzy ideals in BG-algebras. Journal o the Arican Mathematical Union, 25, 577 590. Imai, Y., & Iseki, K. (1966). On axiom system o propositional calculi. XIV. Proceedings o the Japan Academy, 42, 19 22. Iseki, K. (1966). An algebra related with a propositional calculus. Proceedings o the Japan Academy, 42, 26 29. Kim, C. B., & Kim, H. S. (2008). On BG-algebras. Demonstratio Mathematica, 41, 497 505. Mostaa, S. M., Naby, M. A., Halim, F. A., & Hameed, A. T. (2013). On KUS-algebra. International Journal o Algebra, 7, 131 144. Neggers, J., & Kim, H. S. (2002). On B-algebras. Matematicki Vesnik, 54, 21 29. Posner, E. (1957). Derivations in prime rings. Proceedings o the American Mathematical Society, 8, 1093 1100. Prabpayak, C., & Leerawat, U. (2009a). On isimorphism o KUalgebras. Scientia Magna, 5, 25 31. Prabpyak, C., & Leerawat, U. (2009b). On ideals and congruences in KU-algebras. Scientia Magna, 5, 54 57. Senapati, T. (2015). Bipolar uzzy structure o BG-subalgebras. Journal o Fuzzy Mathematics, 23, 209 220. Senapati, T., Bhowmik, M., & Pal, M. (2012). Interval-valued intuitionistic uzzy BG-subalgebras. Journal o Fuzzy Mathematics, 20, 707 720. Senapati, T., Bhowmik, M., & Pal, M. (2013). Atanassov s intuitionistic uzzy translations o intuitionistic uzzy H- ideals in BCK/BCI-algebras. Notes on Intuitionistic Fuzzy Sets, 19, 32 47. Senapati, T., Bhowmik, M., & Pal, M. (2014a). Fuzzy dot subalgebras anuzzy dot ideals o B-algebras. Journal o Uncertain Systems, 8, 22 30. Senapati, T., Bhowmik, M., & Pal, M. (2014b). Fuzzy dot structure o BG-algebras. Fuzzy Inormation and Engineering, 6, 315 329. Senapati, T., Bhowmik, M., & Pal, M. (2015). Triangular norm baseuzzy BG-algebras. Journal o the Arican Mathematical Union. doi:10.1007/s13370-015-0330-y Senapati, T., Bhowmik, M., Pal, M., & Davvaz, B. (2015). Fuzzy translations o uzzy H-ideals in BCK/BCI-algebras. Journal o the Indonesian Mathematical Society, 21, 45 58. Senapati, T., Jana, C., Bhowmik, M., & Pal, M. (2015). L-uzzy G-subalgebras o G-algebras. Journal o the Egyptian Mathematical Society, 23, 219 223. Senapati, T., Kim, C. S., Bhowmik, M., & Pal, M. (2015). Cubic subalgebras and cubic closed ideals o B-algebras. Fuzzy Inormation and Engineering, 7, 129 149. Page 11 o 12
2015 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. You are ree to: Share copy and redistribute the material in any medium or ormat Adapt remix, transorm, and build upon the material or any purpose, even commercially. The licensor cannot revoke these reedoms as long as you ollow the license terms. Under the ollowing terms: Attribution You must give appropriate credit, provide a link to the license, and indicate i changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others rom doing anything the license permits. Cogent Mathematics (ISSN: 2331-1835) is published by Cogent OA, part o Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics or your article Rapid online publication Input rom, and dialog with, expert editors and editorial boards Retention o ull copyright o your article Guaranteed legacy preservation o your article Discounts and waivers or authors in developing regions Submit your manuscript to a Cogent OA journal at www.cogentoa.com Page 12 o 12