Derivation, f-derivation and generalized derivation of KUS-algebras
|
|
- Bertha Boyd
- 5 years ago
- Views:
Transcription
1 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 , India 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 Now, he is a research scholar in the Department o Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University since His scientiic interest concentrates on uzzy BCK/ BCI-algebras and related algebraic systems. Tapan Senapati obtained his PhD rom Vidyasagar University, India in 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 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page 1 o 12
2 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
3 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
4 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
5 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 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
6 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 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
7 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
8 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 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 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
9 ( 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
10 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
11 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 jana.chiranjibe7@gmail.com Tapan Senapati 2 math.tapan@gmail.com ORCID ID: Madhumangal Pal 1 mmpalvu@gmail.com 1 Department o Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore , India. 2 Department o Mathematics, Padima Janakalyan Banipith, , 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: Reerences Bhowmik, M., Senapati, T., & Pal, M. (2014). Intuitionistic L-uzzy ideals in BG-algebras. Journal o the Arican Mathematical Union, 25, Imai, Y., & Iseki, K. (1966). On axiom system o propositional calculi. XIV. Proceedings o the Japan Academy, 42, Iseki, K. (1966). An algebra related with a propositional calculus. Proceedings o the Japan Academy, 42, Kim, C. B., & Kim, H. S. (2008). On BG-algebras. Demonstratio Mathematica, 41, Mostaa, S. M., Naby, M. A., Halim, F. A., & Hameed, A. T. (2013). On KUS-algebra. International Journal o Algebra, 7, Neggers, J., & Kim, H. S. (2002). On B-algebras. Matematicki Vesnik, 54, Posner, E. (1957). Derivations in prime rings. Proceedings o the American Mathematical Society, 8, Prabpayak, C., & Leerawat, U. (2009a). On isimorphism o KUalgebras. Scientia Magna, 5, Prabpyak, C., & Leerawat, U. (2009b). On ideals and congruences in KU-algebras. Scientia Magna, 5, Senapati, T. (2015). Bipolar uzzy structure o BG-subalgebras. Journal o Fuzzy Mathematics, 23, Senapati, T., Bhowmik, M., & Pal, M. (2012). Interval-valued intuitionistic uzzy BG-subalgebras. Journal o Fuzzy Mathematics, 20, 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, Senapati, T., Bhowmik, M., & Pal, M. (2014a). Fuzzy dot subalgebras anuzzy dot ideals o B-algebras. Journal o Uncertain Systems, 8, Senapati, T., Bhowmik, M., & Pal, M. (2014b). Fuzzy dot structure o BG-algebras. Fuzzy Inormation and Engineering, 6, Senapati, T., Bhowmik, M., & Pal, M. (2015). Triangular norm baseuzzy BG-algebras. Journal o the Arican Mathematical Union. doi: /s 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, Senapati, T., Jana, C., Bhowmik, M., & Pal, M. (2015). L-uzzy G-subalgebras o G-algebras. Journal o the Egyptian Mathematical Society, 23, Senapati, T., Kim, C. S., Bhowmik, M., & Pal, M. (2015). Cubic subalgebras and cubic closed ideals o B-algebras. Fuzzy Inormation and Engineering, 7, Page 11 o 12
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: ) 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 Page 12 o 12
Fuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras
Journal of Uncertain Systems Vol.8, No.1, pp.22-30, 2014 Online at: www.jus.org.uk Fuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras Tapan Senapati a,, Monoranjan Bhowmik b, Madhumangal Pal c a
More informationSome aspects on hesitant fuzzy soft set
Borah & Hazarika Cogent Mathematics (2016 3: 1223951 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Some aspects on hesitant fuzzy soft set Manash Jyoti Borah 1 and Bipan Hazarika 2 * Received:
More informationOn some properties of p-ideals based on intuitionistic fuzzy sets
PURE MATHEMATICS RESEARCH ARTICLE On some properties of p-ideals based on intuitionistic fuzzy sets Muhammad Touqeer 1 * and Naim Çağman 2 Received: 13 June 2016 Accepted: 23 June 2016 First Published:
More informationA note on the unique solution of linear complementarity problem
COMPUTATIONAL SCIENCE SHORT COMMUNICATION A note on the unique solution of linear complementarity problem Cui-Xia Li 1 and Shi-Liang Wu 1 * Received: 13 June 2016 Accepted: 14 November 2016 First Published:
More informationThe plastic number and its generalized polynomial
PURE MATHEMATICS RESEARCH ARTICLE The plastic number and its generalized polynomial Vasileios Iliopoulos 1 * Received: 18 December 2014 Accepted: 19 February 201 Published: 20 March 201 *Corresponding
More informationOn a mixed interpolation with integral conditions at arbitrary nodes
PURE MATHEMATICS RESEARCH ARTICLE On a mixed interpolation with integral conditions at arbitrary nodes Srinivasarao Thota * and Shiv Datt Kumar Received: 9 October 5 Accepted: February 6 First Published:
More informationGraded fuzzy topological spaces
Ibedou, Cogent Mathematics (06), : 8574 http://dxdoiorg/0080/85068574 PURE MATHEMATICS RESEARCH ARTICLE Graded fuzzy topological spaces Ismail Ibedou, * Received: August 05 Accepted: 0 January 06 First
More informationMatrix l-algebras over l-fields
PURE MATHEMATICS RESEARCH ARTICLE Matrix l-algebras over l-fields Jingjing M * Received: 05 January 2015 Accepted: 11 May 2015 Published: 15 June 2015 *Corresponding author: Jingjing Ma, Department of
More informationInternational Journal of Algebra, Vol. 7, 2013, no. 3, HIKARI Ltd, On KUS-Algebras. and Areej T.
International Journal of Algebra, Vol. 7, 2013, no. 3, 131-144 HIKARI Ltd, www.m-hikari.com On KUS-Algebras Samy M. Mostafa a, Mokhtar A. Abdel Naby a, Fayza Abdel Halim b and Areej T. Hameed b a Department
More informationFinding the strong defining hyperplanes of production possibility set with constant returns to scale using the linear independent vectors
Rafati-Maleki et al., Cogent Mathematics & Statistics (28), 5: 447222 https://doi.org/.8/233835.28.447222 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Finding the strong defining hyperplanes
More informationGeneralized Derivation on TM Algebras
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,
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationPetrović s inequality on coordinates and related results
Rehman et al., Cogent Mathematics 016, 3: 1798 PURE MATHEMATICS RESEARCH ARTICLE Petrović s inequality on coordinates related results Atiq Ur Rehman 1 *, Muhammad Mudessir 1, Hafiza Tahira Fazal Ghulam
More informationHomomorphism on T Anti-Fuzzy Ideals of Ring
International Journal o Computational Science and Mathematics. ISSN 0974-3189 Volume 8, Number 1 (2016), pp. 35-48 International esearch Publication House http://www.irphouse.com Homomorphism on T nti-fuzzy
More informationResponse surface designs using the generalized variance inflation factors
STATISTICS RESEARCH ARTICLE Response surface designs using the generalized variance inflation factors Diarmuid O Driscoll and Donald E Ramirez 2 * Received: 22 December 204 Accepted: 5 May 205 Published:
More informationResearch Article Generalized Derivations of BCC-Algebras
International Mathematics and Mathematical Sciences Volume 2013, Article ID 451212, 4 pages http://dx.doi.org/10.1155/2013/451212 Research Article Generalized Derivations of BCC-Algebras S. M. Bawazeer,
More informationA detection for patent infringement suit via nanotopology induced by graph
PURE MATHEMATICS RESEARCH ARTICLE A detection for patent infringement suit via nanotopology induced by graph M. Lellis Thivagar 1 *, Paul Manuel 2 and. Sutha Devi 1 Received: 08 October 2015 Accepted:
More informationScientiae Mathematicae Japonicae Online, Vol.4 (2001), a&i IDEALS ON IS ALGEBRAS Eun Hwan Roh, Seon Yu Kim and Wook Hwan Shim Abstract. In th
Scientiae Mathematicae Japonicae Online, Vol.4 (2001), 21 25 21 a&i IDEALS ON IS ALGEBRAS Eun Hwan Roh, Seon Yu Kim and Wook Hwan Shim Abstract. In this paper, we introduce the concept of an a&i-ideal
More informationFuzzy Closed Ideals of B-algebras with Interval- Valued Membership Function
Intern. J. Fuzzy Mathematical Archive Vol. 1, 2013, 79-91 ISSN: 2320 3242 (P), 2320 3250 (online) Published on 3 March 2013 www.researchmathsci.org International Journal of Fuzzy Closed Ideals of B-algebras
More informationBG/BF 1 /B/BM-algebras are congruence permutable
Mathematica Aeterna, Vol. 5, 2015, no. 2, 351-35 BG/BF 1 /B/BM-algebras are congruence permutable Andrzej Walendziak Institute of Mathematics and Physics Siedlce University, 3 Maja 54, 08-110 Siedlce,
More informationCoupled -structures and its application in BCK/BCI-algebras
IJST (2013) 37A2: 133-140 Iranian Journal of Science & Technology http://www.shirazu.ac.ir/en Coupled -structures its application in BCK/BCI-algebras Young Bae Jun 1, Sun Shin Ahn 2 * D. R. Prince Williams
More informationBounds on Hankel determinant for starlike and convex functions with respect to symmetric points
PURE MATHEMATICS RESEARCH ARTICLE Bounds on Hankel determinant for starlike convex functions with respect to symmetric points Ambuj K. Mishra 1, Jugal K. Prajapat * Sudhana Maharana Received: 07 November
More informationBM-ALGEBRAS AND RELATED TOPICS. 1. Introduction
ao DOI: 10.2478/s12175-014-0259-x Math. Slovaca 64 (2014), No. 5, 1075 1082 BM-ALGEBRAS AND RELATED TOPICS Andrzej Walendziak (Communicated by Jiří Rachůnek ) ABSTRACT. Some connections between BM-algebras
More informationPure Mathematical Sciences, Vol. 1, 2012, no. 3, On CS-Algebras. Kyung Ho Kim
Pure Mathematical Sciences, Vol. 1, 2012, no. 3, 115-121 On CS-Algebras Kyung Ho Kim Department of Mathematics Korea National University of Transportation Chungju 380-702, Korea ghkim@ut.ac.kr Abstract
More informationStolarsky Type Inequality for Sugeno Integrals on Fuzzy Convex Functions
International Journal o Mathematical nalysis Vol., 27, no., 2-28 HIKRI Ltd, www.m-hikari.com https://doi.org/.2988/ijma.27.623 Stolarsky Type Inequality or Sugeno Integrals on Fuzzy Convex Functions Dug
More informationOn Fuzzy Dot Subalgebras of d-algebras
International Mathematical Forum, 4, 2009, no. 13, 645-651 On Fuzzy Dot Subalgebras of d-algebras Kyung Ho Kim Department of Mathematics Chungju National University Chungju 380-702, Korea ghkim@cjnu.ac.kr
More informationSome Results About Generalized BCH-Algebras
International Journal of Algebra, Vol. 11, 2017, no. 5, 231-246 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.712 Some Results About Generalized BCH-Algebras Muhammad Anwar Chaudhry 1
More information(, q)-fuzzy Ideals of BG-algebras with respect to t-norm
NTMSCI 3, No. 4, 196-10 (015) 196 New Trends in Mathematical Sciences http://www.ntmsci.com (, q)-fuzzy Ideals of BG-algebras with respect to t-norm Saidur R. Barbhuiya Department of mathematics, Srikishan
More informationComplete Ideal and n-ideal of B-algebra
Applied Mathematical Sciences, Vol. 11, 2017, no. 35, 1705-1713 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.75159 Complete Ideal and n-ideal of B-algebra Habeeb Kareem Abdullah University
More informationMarichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function
Mishra et al. Cogent Mathematics 2017 4: 1320830 PURE MATHEMATICS RESEARCH ARTICLE Marichev-Saigo-Maeda ractional calculus operators Srivastava polynomials and generalized Mittag-Leler unction Received:
More informationFuzzy ideals of K-algebras
Annals of University of Craiova, Math. Comp. Sci. Ser. Volume 34, 2007, Pages 11 20 ISSN: 1223-6934 Fuzzy ideals of K-algebras Muhammad Akram and Karamat H. Dar Abstract. The fuzzy setting of an ideal
More informationOn new structure of N-topology
PURE MATHEMATICS RESEARCH ARTICLE On new structure of N-topology M. Lellis Thivagar 1 *, V. Ramesh 1 and M. Arockia Dasan 1 Received: 17 February 2016 Accepted: 15 June 2016 First Published: 21 June 2016
More informationExistence and uniqueness of a stationary and ergodic solution to stochastic recurrence equations via Matkowski s FPT
Arvanitis, Cogent Mathematics 2017), 4: 1380392 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Existence and uniqueness of a stationary and ergodic solution to stochastic recurrence equations
More information(, q)-fuzzy Ideals of BG-Algebra
International Journal of Algebra, Vol. 5, 2011, no. 15, 703-708 (, q)-fuzzy Ideals of BG-Algebra D. K. Basnet Department of Mathematics, Assam University, Silchar Assam - 788011, India dkbasnet@rediffmail.com
More informationφ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers
Joshi et al., Cogent Mathematics 2016, 3: 1162484 PURE MATHEMATICS RESEARCH ARTICLE φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers Received: 25 October
More informationMappings of the Direct Product of B-algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 133-140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.615 Mappings of the Direct Product of B-algebras Jacel Angeline V. Lingcong
More informationSOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS
International Journal o Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 2017, 179-187 DOI: 10.28924/2291-8639-15-2017-179 SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM
More information(, q)-interval-valued Fuzzy Dot d-ideals of d-algebras
Advanced Trends in Mathematics Online: 015-06-01 ISSN: 394-53X, Vol. 3, pp 1-15 doi:10.1805/www.scipress.com/atmath.3.1 015 SciPress Ltd., Switzerland (, q)-interval-valued Fuzzy Dot d-ideals of d-algebras
More informationDouble domination in signed graphs
PURE MATHEMATICS RESEARCH ARTICLE Double domination in signed graphs P.K. Ashraf 1 * and K.A. Germina 2 Received: 06 March 2016 Accepted: 21 April 2016 Published: 25 July 2016 *Corresponding author: P.K.
More informationSubalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant fuzzy set theory
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 2, 2018, 417-430 ISSN 1307-5543 www.ejpam.com Published by New York Business Global Subalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant
More informationVague Set Theory Applied to BM-Algebras
International Journal of Algebra, Vol. 5, 2011, no. 5, 207-222 Vague Set Theory Applied to BM-Algebras A. Borumand Saeid 1 and A. Zarandi 2 1 Dept. of Math., Shahid Bahonar University of Kerman Kerman,
More informationON BP -ALGEBRAS. Sun Shin Ahn, Jeong Soon Han
Hacettepe Journal of Mathematics and Statistics Volume 42 (5) (2013), 551 557 ON BP -ALGEBRAS Sun Shin Ahn, Jeong Soon Han Received 06 : 05 : 2011 : Accepted 25 : 11 : 2012 Abstract In this paper, we introduce
More informationOn weak Γ-(semi)hypergroups
PURE MATHEMATICS RESEARCH ARTICLE On weak Γ-(semi)hypergroups T Zare 1, M Jafarpour 1 * and H Aghabozorgi 1 Received: 29 September 2016 Accepted: 28 December 2016 First Published: 14 February 2017 *Corresponding
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationDerivations of B-algebras
JKAU: Sci, Vol No 1, pp: 71-83 (010 AD / 1431 AH); DOI: 104197 / Sci -15 Derivations of B-algebras Department of Mathematics, Faculty of Education, Science Sections, King Abdulaziz University, Jeddah,
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationIntroduction to Neutrosophic BCI/BCK-Algebras
Introduction to Neutrosophic BCI/BCK-Algebras A.A.A. Agboola 1 and B. Davvaz 2 1 Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria aaaola2003@yahoo.com 2 Department of Mathematics,
More informationNOVEL CONCEPTS OF DOUBT BIPOLAR FUZZY H-IDEALS OF BCK/BCI-ALGEBRAS. Anas Al-Masarwah and Abd Ghafur Ahmad. Received February 2018; revised June 2018
International Journal of Innovative Computing, Information Control ICIC International c 2018 ISS 1349-4198 Volume 14, umber 6, December 2018 pp. 2025 2041 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS OF BCK/BCI-LGEBRS
More informationExtreme Values of Functions
Extreme Values o Functions When we are using mathematics to model the physical world in which we live, we oten express observed physical quantities in terms o variables. Then, unctions are used to describe
More informationNon-linear unit root testing with arctangent trend: Simulation and applications in finance
STATISTICS RESEARCH ARTICLE Non-linear unit root testing with arctangent trend: Simulation and applications in finance Deniz Ilalan 1 * and Özgür Özel 2 Received: 24 October 2017 Accepted: 18 March 2018
More informationA NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3185 3189 S 0002-9939(97)04112-9 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES BENJI FISHER (Communicated by
More informationA new iteration process for approximation of fixed points of mean nonexpansive mappings in CAT(0) spaces
PURE MATHEMATICS RESEARCH ARTICLE A new iteration process for approximation of fixed points of mean nonexpansive mappings in CAT(0) spaces Received: 3 July 07 Accepted: 6 October 07 First Published: 30
More informationCATEGORIES. 1.1 Introduction
1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the
More informationExample: When describing where a function is increasing, decreasing or constant we use the x- axis values.
Business Calculus Lecture Notes (also Calculus With Applications or Business Math II) Chapter 3 Applications o Derivatives 31 Increasing and Decreasing Functions Inormal Deinition: A unction is increasing
More informationResearch Article Introduction to Neutrosophic BCI/BCK-Algebras
International Mathematics and Mathematical Sciences Volume 2015, Article ID 370267, 6 pages http://dx.doi.org/10.1155/2015/370267 Research Article Introduction to Neutrosophic BCI/BCK-Algebras A. A. A.
More informationNON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS AND RETARDED EQUATIONS. M. Filali and M. Moussi
Electronic Journal: Southwest Journal o Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 83-464 Issue 2, December, 23, pp. 26 35. Submitted: December 24, 22. Published:
More informationNeutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCI-Algebras
mathematics Article Neutrosophic Permeable Values Energetic Subsets with Applications in BCK/BCI-Algebras Young Bae Jun 1, *, Florentin Smarache 2 ID, Seok-Zun Song 3 ID Hashem Bordbar 4 ID 1 Department
More informationNON-ARCHIMEDEAN BANACH SPACE. ( ( x + y
J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. ISSN(Print 6-0657 https://doi.org/0.7468/jksmeb.08.5.3.9 ISSN(Online 87-608 Volume 5, Number 3 (August 08, Pages 9 7 ADDITIVE ρ-functional EQUATIONS
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationON FILTERS IN BE-ALGEBRAS. Biao Long Meng. Received November 30, 2009
Scientiae Mathematicae Japonicae Online, e-2010, 105 111 105 ON FILTERS IN BE-ALGEBRAS Biao Long Meng Received November 30, 2009 Abstract. In this paper we first give a procedure by which we generate a
More informationClassification of effective GKM graphs with combinatorial type K 4
Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationON A PERIOD OF ELEMENTS OF PSEUDO-BCI-ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 35 (2015) 21 31 doi:10.7151/dmgaa.1227 ON A PERIOD OF ELEMENTS OF PSEUDO-BCI-ALGEBRAS Grzegorz Dymek Institute of Mathematics and Computer Science
More informationProblem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
More informationON STRUCTURE OF KS-SEMIGROUPS
International Mathematical Forum, 1, 2006, no. 2, 67-76 ON STRUCTURE OF KS-SEMIGROUPS Kyung Ho Kim Department of Mathematics Chungju National University Chungju 380-702, Korea ghkim@chungju.ac.kr Abstract
More informationInternational Journal of Scientific and Research Publications, Volume 6, Issue 10, October 2016 ISSN f -DERIVATIONS ON BP-ALGEBRAS
ISSN 2250-3153 8 f -DERIVATIONS ON BP-ALGEBRAS N.Kandaraj* and A.Arul Devi**, *Associate Professor in Mathematics **Assistant professor in Mathematics SAIVA BHANU KSHATRIYA COLLEGE ARUPPUKOTTAI - 626101.
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license
More informationON SUB-IMPLICATIVE (α, β)-fuzzy IDEALS OF BCH-ALGEBRAS
ON SUB-IMPLICATIVE (α, β)-fuzzy IDEALS OF BCH-ALGEBRAS MUHAMMAD ZULFIQAR Communicated by the former editorial board In this paper, we introduce the concept of sub-implicative (α, β)-fuzzy ideal of BCH-algebra
More informationResearch Article Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the Theory of Falling Shadows
International Mathematics and Mathematical Sciences Volume 2010, Article ID 819463, 11 pages doi:10.1155/2010/819463 Research Article Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the
More informationOn Homomorphism and Algebra of Functions on BE-algebras
On Homomorphism and Algebra of Functions on BE-algebras Kulajit Pathak 1, Biman Ch. Chetia 2 1. Assistant Professor, Department of Mathematics, B.H. College, Howly, Assam, India, 781316. 2. Principal,
More informationInternational Journal of Mathematical Archive-7(1), 2016, Available online through ISSN
International Journal of Mathematical Archive-7(1), 2016, 200-208 Available online through www.ijma.info ISSN 2229 5046 ON ANTI FUZZY IDEALS OF LATTICES DHANANI S. H.* Department of Mathematics, K. I.
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More informationSoft Matrices. Sanjib Mondal, Madhumangal Pal
Journal of Uncertain Systems Vol7, No4, pp254-264, 2013 Online at: wwwjusorguk Soft Matrices Sanjib Mondal, Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming Vidyasagar
More informationELECTRICAL & ELECTRONIC ENGINEERING RESEARCH ARTICLE
Received: 21 June 2018 Accepted: 04 December 2018 First Published: 18 December 2018 *Corresponding author: Shantanu Sarkar, Machinery, L&T Technology Services, Houston, TX, USA E-mail: Shantanu75@gmail.com
More informationON THE CLASS OF QS-ALGEBRAS
IJMMS 2004:49, 2629 2639 PII. S0161171204309154 http://ijmms.hindawi.com Hindawi Publishing Corp. ON THE CLASS OF QS-ALGEBRAS MICHIRO KONDO Received 19 September 2003 We consider some fundamental properties
More informationDOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, ON BI-ALGEBRAS
DOI: 10.1515/auom-2017-0014 An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, 177 194 ON BI-ALGEBRAS Arsham Borumand Saeid, Hee Sik Kim and Akbar Rezaei Abstract In this paper, we introduce a new algebra,
More informationUniversity of Cape Town
The copyright o this thesis rests with the. No quotation rom it or inormation derived rom it is to be published without ull acknowledgement o the source. The thesis is to be used or private study or non-commercial
More informationOn regularity of sup-preserving maps: generalizing Zareckiĭ s theorem
Semigroup Forum (2011) 83:313 319 DOI 10.1007/s00233-011-9311-0 RESEARCH ARTICLE On regularity o sup-preserving maps: generalizing Zareckiĭ s theorem Ulrich Höhle Tomasz Kubiak Received: 7 March 2011 /
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationAdditive functional inequalities in Banach spaces
Lu and Park Journal o Inequalities and Applications 01, 01:94 http://www.journaloinequalitiesandapplications.com/content/01/1/94 R E S E A R C H Open Access Additive unctional inequalities in Banach spaces
More informationON FUZZY TOPOLOGICAL BCC-ALGEBRAS 1
Discussiones Mathematicae General Algebra and Applications 20 (2000 ) 77 86 ON FUZZY TOPOLOGICAL BCC-ALGEBRAS 1 Wies law A. Dudek Institute of Mathematics Technical University Wybrzeże Wyspiańskiego 27,
More informationThe combined reproducing kernel method and Taylor series to solve nonlinear Abel s integral equations with weakly singular kernel
Alvandi & Paripour, Cogent Mathematics (6), 3: 575 http://dx.doi.org/.8/33835.6.575 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE The combined reproducing kernel method and Taylor series to
More informationFactors of words under an involution
Journal o Mathematics and Inormatics Vol 1, 013-14, 5-59 ISSN: 349-063 (P), 349-0640 (online) Published on 8 May 014 wwwresearchmathsciorg Journal o Factors o words under an involution C Annal Deva Priya
More informationHSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS
HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS MICHAEL BARR Abstract. Given a triple T on a complete category C and a actorization system E /M on the category o algebras, we show there is a 1-1 correspondence
More informationMathematica Bohemica
Mathematica Bohemica Young Hee Kim; Hee Sik Kim Subtraction algebras and BCK-algebras Mathematica Bohemica, Vol. 128 (2003), No. 1, 21 24 Persistent URL: http://dml.cz/dmlcz/133931 Terms of use: Institute
More informationQ-fuzzy sets in UP-algebras
Songklanakarin J. Sci. Technol. 40 (1), 9-29, Jan. - Feb. 2018 Original Article Q-fuzzy sets in UP-algebras Kanlaya Tanamoon, Sarinya Sripaeng, and Aiyared Iampan* Department of Mathematics, School of
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationSTATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION
Journal o athematical Analysis ISSN: 227-342, URL: http://www.ilirias.com Volume 4 Issue 2203), Pages 6-22. STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION AAR JYOTI DUTTA, AYHAN
More informationDynamics of the equation complex plane
APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Dynamics of the equation in the complex plane Sk. Sarif Hassan 1 and Esha Chatterjee * Received: 0 April 015 Accepted: 08 November 015 Published:
More informationOn KS-Semigroup Homomorphism
International Mathematical Forum, 4, 2009, no. 23, 1129-1138 On KS-Semigroup Homomorphism Jocelyn S. Paradero-Vilela and Mila Cawi Department of Mathematics, College of Science and Mathematics MSU-Iligan
More informationSyllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.
Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in
More informationVariations on a Casselman-Osborne theme
Variations on a Casselman-Osborne theme Dragan Miličić Introduction This paper is inspired by two classical results in homological algebra o modules over an enveloping algebra lemmas o Casselman-Osborne
More informationInclusion Relationship of Uncertain Sets
Yao Journal of Uncertainty Analysis Applications (2015) 3:13 DOI 10.1186/s40467-015-0037-5 RESEARCH Open Access Inclusion Relationship of Uncertain Sets Kai Yao Correspondence: yaokai@ucas.ac.cn School
More informationDescent on the étale site Wouter Zomervrucht, October 14, 2014
Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and
More informationProbabilistic Observations and Valuations (Extended Abstract) 1
Replace this ile with prentcsmacro.sty or your meeting, or with entcsmacro.sty or your meeting. Both can be ound at the ENTCS Macro Home Page. Probabilistic Observations and Valuations (Extended Abstract)
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationOn the complex k-fibonacci numbers
Falcon, Cogent Mathematics 06, 3: 0944 http://dxdoiorg/0080/33835060944 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE On the complex k-fibonacci numbers Sergio Falcon * ceived: 9 January 05
More information