Complete 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 of Kufa, Faculty of Education for Girls Department of Mathematics, Iraq Arkan Ajeal Atshan University of Kufa, Faculty of Computer science and Math Department of Mathematics, Iraq Copyright 2017 Habeeb Kareem Abdullah and Arkan Ajeal Atshan. This article is 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 In this paper, we defined and studied some types of ideals of B-algebra, which we called them, Complete Ideal, Closed Complete Ideal, n-ideal and Closed n-ideal respectively. In addition, we gave some propositions that explained some relationships between these ideals types. Keywords: B-algebra, subalgebra, normal, ideal, complete ideal, closed complete ideal, n-ideal, closed n-ideal and complete n-ideal of B-algebra Introduction In [2], [3] J. Neggers and H. Kim introduced the notation of B-algebra and they also studied some of its properties. A simple axiomatization of commutative B-algebras had been obtained by A. Walendziak [1]. T. Senapati introduced Fuzzy Closed Ideals and Fuzzy Subalgebras of B-algebras in [5]. The notation of ideals in subtraction algebras had been discussed by Y.B. Jun, H.S. Kim and E.H. Roh and they obtained significant results in [7]. Y.B. Jun, H.S. Kim and E.H. Roh gave a characterization of a prime ideal of a subtraction algebra in [8]. In this paper, firstly, we define the general concepts of some types of ideals of B-algebra and study their properties; secondly, we state some theorems that explained some relationships between these types of B-algebra.

1706 Habeeb Kareem Abdullah and Arkan Ajeal Atshan 1.1 Preliminaries of B-algebra In this section, we introduce general definitions of B-algebra, 0-commutative, subalgebra, normal and ideal. For more notations see [2], [3], [4] Definition (1.2) [2]: A B-algebra is a non-empty set X with a constant 0 and a binary operation satisfying the following axioms: (1) x x = 0, (2) x 0 = x, (3) (x y) z = x (z (0 y)), for all x, y, z X. Proposition:(1.3) [2]: If (X,,0) is B-algebra, then (1) ( x y) (0 y) = x, (2) x (y z) = (x (0 z)) y, (3) x y = 0 implies x = y, (4) 0 (0 x) = x, (5) ( x z) (y z) = x y, (6) 0 (x y) = y x, for all x, y, z X. Note that if we let (X,,0) be B-algebra, then (1) x y=0 iff y x= 0. (2) if 0 x=0, xϵx, then X contains only 0. Definition (1.4) [2]: A B-algebra (X,,0) is said to be 0- commutative if x (0 y) = y (0 x) for any x, y ϵx. Proposition (1.5) [2]: If (X,,0) is a 0-commutative B-algebra, then (1)(0 x) (0 y) = y x, (2)(z y) (z x) = x y, (3)( x y) z = (x z) y, (4)[x (x y)] y = 0, (5)(x z) (y t) = (t z) (y x), for all x, y, z, t X. Proposition (1.6): Let (X,,0) be B-algebra, then (X,,0) is 0-commutative B-algebra if and only if x (y z) = z (y x). Proof: )Let (X,,0) is 0-commutative and x, y, z X. x (y z) = (x 0) (y z) From proposition 1.5(5) we have(x 0) (y z) = (z 0) (y x) = z (y x) Thus x (y z) = z (y x). ) Let x (y z) = z (y x) and x, y, z X. Now,x (0 y) = y (0 x), then (X,,0) is 0-commutative.

Complete ideal and n-ideal of B-algebra 1707 Proposition (1.7): Let (X,,0) be B-algebra. If 0 x=x, then (X,,0) is a 0-commutative B-algebra. Proof: Let 0 x=x & x, y X. x (0 y) = x y = (0 x) y, from definition 1.2(3), we have: (0 x) y = 0 (y (0 x)), since, 0 x=x, then 0 (y (0 x)) = y (0 x), thus, X is a 0-commutative B-algebra. Remark (1.8) :-In general the converse of proposition 1.7 is not true, the following example illustrates this. Example (1.9): Let X = { 0, 1, 2} be a set with the following table: 0 1 2 0 0 2 1 1 1 0 2 2 2 1 0 clear (X;, 0) is B-algebra and 0-commutative but 0 2=1 2. Proposition (1.10): Let (X,,0) be a 0-commutative B-algebra. If x y = y x, then x=0 x. Proof: From proposition 1.3(1) we have, x = (x y) (0 y) and since x y = y x, then (x y) (0 y) = (0 y) (x y), from proposition1.5(5) we get: (0 y) (x y)=(y y) (x 0), thus x = 0 x and this complete the proof. Definition (1.11) [4]: A non-empty subset S of B-algebra X is called Subalgebra of X if x y S for any x, y S. Remark (1.12): If S is a subalgebra of X, then 0 S. Proof: Since S, then xϵs. Since S is subalgebra, then x x=0 S. Definition (1.13) [3]: A non-empty subset N of B-algebra X is called Normal if any x y, a b N implies (x a) (y b) N. Proposition (1.14): Every normal of B-algebra X is subalgebra. Proof: Let S be a normal subset of X and x, y S, then x 0, y 0 S, so x y= (x y) (0 0) = (x y) 0 S, thus S is subalgebra. Remark (1.15):-In general the converse of proposition 1.14 is not true, the following example illustrates this.

1708 Habeeb Kareem Abdullah and Arkan Ajeal Atshan Example (1.16): Let X = {0, 1, 2, 3, 4, 5} be a set with the following table: 0 1 2 3 4 5 0 0 2 1 3 4 5 1 1 0 2 4 5 3 2 2 1 0 5 3 4 3 3 4 5 0 2 1 4 4 5 3 1 0 2 5 5 3 4 2 1 0 It is clear (X;, 0) is B-algebra and S= {0,4} is subalgebra, since 0 4=4 S,4 0=4 S,but is not normal, since 3 1=4 S, 2 5 = 4 S, but (3 2) (1 5) =5 3=2 S. Proposition (1.17): If N subalgebra of 0-commutative B-algebra then N is normal. Proof:Let x y, a b N, from proposition1.5(5) we have: ( x a) (y b) = (b a) (y x), from definition 1.2(3) we have: (b a) (y x) = b [(y x) (0 a)], from proposition1.5(5) we have: b [(y x) (0 a)] = b [(a x) (0 y)], from definition 1.2(3) we have: b [(a x) (0 y)] = b [a ((0 y) (0 x))], by using proposition1.5(1) we have: b [a ((0 y) (0 x))] = b [a (x y)], from proposition1.3(2) we get: b [a (x y)] = [b (0 (x y))] a, from proposition1.3(6) we get: [b (0 (x y))] a = [b (y x)] a, by using proposition1.3(2) we have [b (y x)] a = [(b (0 x)) y] a, by using definition 1.2(3) we get [(b (0 x)) y] a = (b (0 x)) (a (0 y)), from proposition1.5(5) we have b (0 x)) (a (0 y)) = [ (0 y) (0 x)] (a b)), from proposition1.5(1) we get: [ (0 y) (0 x)] (a b)) = (x y) (a b), and since N subalgebra, (x y) (a b) N, then( x a) (y b) N, thus N is normal and this complete the proof.

Complete ideal and n-ideal of B-algebra 1709 Definition (1.18) [6]: A non- empty subset I of B-algebra X is called an Ideal of X if it satisfies: (1)0 I, (2) x y I & y I x I. Proposition (1.19): Let I be an ideal of B-algebra X. If x, y X and y I such that x y=0 then x I. Proof: Let yϵi. Since x y = 0 I & I is an ideal, then, x I. Proposition (1.20): If I is ideal of B-algebra X and x, y I, then (1) x (0 x) I. (2) x (0 y) I. Proof: (1)Let I is ideal & x I, from definition 1.2(3), we have, (x (0 x)) x = x (x x), since, x (x x) = x 0 = x I, (x (0 x)) x I, since, I is ideal& x I, then x (0 x) I this complete the proof. (2) Let I is ideal & y I, from definition 1.2(3), we have(x (0 y)) y = x (y y), since, x (y y) = x 0 = x I, then(x (0 y)) y I, since, I is ideal& y I, then x (0 y) I this complete the proof. Proposition (1.21): Every subalgebra of B-algebra X is ideal. Proof:Let I be a subalgebra of X, from remark 1.12, then 0 I. Now, let x y I and y I, since I is a subalgebra, then0 y I, hence(x y) (0 y) I, from proposition 1.3(1), (x y) (0 y) = x, hence x I, thus I is ideal this complete the proof. Corollary (1.22): Every normal of B-algebra is ideal. Proof: It is directly from proposition 1.14 and proposition 1.21. Remark (1.23): - The convers of corollary 1.22 is not true in general from example 1.16, S= {0,4} is ideal but not normal. Proposition(1.24): If I is ideal of B-algebra X and 0 x=x for all x X then I is subalgebra. Proof: Let I is ideal & x, y I. Now, (x y) y = (x y) (0 y), from proposition1.3(1), (x y) (0 y) = x, then x I,hence (x y) y I, since I is ideal & y I, then x y I, thus I is subalgebra. This complete the proof.

1710 Habeeb Kareem Abdullah and Arkan Ajeal Atshan Proposition(1.25): Let I be ideal of B-algebra X and x y = y x for all x X then I is subalgebra. Proof: Let I is ideal, x, y I & x y = y x, (x y) y = (x y) (y 0) = (x y) (0 y), from proposition1.3(1), (x y) (0 y) = x, then x I,hence (x y) y I, since I is ideal & y I, then x y I, thus I is subalgebra. This complete the proof. Remark(1.26): (1) The intersection of two ideals is an ideal. (2) Not necessary the union of two ideals is an ideal, the following example illustrates this. Example (1.27): Let X = {0,1,2,3} be a set with the following 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 It is clear that X is B-algebra, I={0, 1}and J={0,2} are ideals, however, since,3 1=2 I J,but 3 I J, then I J={0, 1, 2}is not ideal. 2.1. Complete Ideal In this section, we introduce definition of complete ideal of B-algebra. Also, we study its relationship with ideal of B-algebra. Definition (2.2): A subset I of B-algebra X is said to be Complete Ideal of B- algebra briefly (c-ideal), if (1) 0 I, (2) x y I, y I such that y 0 implies x I. The following example explains the definition above. Example (2.3): Let X = {0,1,2,3} be a set with the following table: Complete Ideal and n-ideal of B-algebra * 0 1 2 3 0 0 3 2 1 1 1 0 3 2 2 2 1 0 3 3 3 2 1 0 It is clear that (X,,0) is B-algebra and I= {0,2} is c-ideal. Proposition (2.4): In B-algebra X every ideal is c-ideal. Proof: Let I be an ideal and let x y I, y I such that y 0 1- If I= {0}, then I is c-ideal.

Complete ideal and n-ideal of B-algebra 1711 2- If I {0}, then y I, such that y 0 & x y I. Since I is ideal, then x I. Thus I is c-ideal. Remark(2.5):- The converse of proposition 2.4 is not true in general, I={0,2,3,4} in example 1.16 is c-ideal but not ideal, since 1 2=2 I but 1 I. Corollary (2.6): Every subalgebra of B-algebra is c-ideal. proof: It is directly from proposition 1.21 and proposition 2.4. Remark(2.7):-In general, the converse of corollary 2.6 is not true from example 1.16,I= {0,2,3,4} is c-ideal but not subalgebra, since, 2 3=5 I. Corollary (2.8): Every normal of B-algebra is c-ideal. proof: It is directly from proposition 1.14 and corollary 2.6. Definition (2.9): An c-ideal I of B-algebra X is said to be Closed c-ideal if it is also subalgebra. Example (2.10): In example 2.3, I={0,2} is closed c-ideal. Remark(2.11): (1) The intersection of two complete ideals is complete ideal. (2) Not necessary the union of two complete ideals is complete ideal, from example 1.27, I={0, 1}and J={0,2} are c-ideals, but I J={0, 1, 2}is not c-ideal, since,3*1=2 I J&3*2=1 I J but, 3 I J. 3.1 n-ideal of B-algebra In this section, we introduce definition of n-ideal of B-algebra. Moreover, we study its relationship with ideal and c-ideal of B-algebra. Definition (3.2):A subset I of B-algebra X is said to be n-ideal of B-algebra, if (1) 0 I, (2) x y I, & y I n Z +, x n 0,such that x n I, where x n =((x x) x) x x. The following example explains the definition above. Example( 3.3): In example 1.8,I= {0,2} is n-ideal, since,1 2=2 I, 1 3 = (1 1) 1 = 0 1 = 2 I. Proposition (3.4): In B-algebra X every ideal is n-ideal. Proof: Let I be ideal, x y I, & y I, since I is ideal, then x I. This complete the proof. Remark(3.5): The converse of proposition 3.4 is not true in general by example 1.9, I= {0,2} is n-ideal, (since,1 2=2 I, 1 3 = (1 1) 1 = 0 = I),but not ideal, since 1 2=2 I, 1 I. Corollary (3.6): Every subalgebra S of B-algebra X is n-ideal. Proof: It is directly from proposition 1.21 and proposition 3.4.

1712 Habeeb Kareem Abdullah and Arkan Ajeal Atshan Remark(3.7): The converse of corollary 3.6 is not true in general by. example 1.9, I= {0,2} is n-ideal but not subalgebra, since 0 2=1 I. Corollary (3.8): Every normal of B-algebra is n-ideal. proof: It is directly from proposition 1.14 and corollary 3.6. Remark (3.9): The converse of corollary 3.8 is not true in general by example 1.16, I={0, 4} is n-ideal, but not normal. Proposition (3.10):If I is n-ideal of B-algebra X and 0 x=x for all x X, then I is ideal. Proof: Let I is ideal & 0 x = x. Now, x = 0 x = ((x x) x) x x =x n, since I is n-ideal, then x =x n I,thus I is ideal. This complete the proof. Remark(3.11):- (1) It is clear that not every n-ideal is c-ideal and the converse I= {0,2} in example 1.9 is n-ideal but not c-ideal. Whereas I= {0,1,3}in example 1.16 is c-ideal but not n-ideal, since,4 3 = 1 I but 4 n I, n Z + s. t 4 n 0. (2) In general, not union of two n-ideals is n-ideal, from example 1.16, I= {0,1} and J= {0, 3} are n-ideals, but, I J= {0,1,3} is not n-ideal, since, 5 1=3 I J, but 5 n I J, n Z + s. t. 5 n 0). (3) In general, not necessary the intersection of two n-ideals is n-ideal, from example 1.16, I={0,1,3,4,5}and J={0, 2,3,4,5}are n-ideal,but, I J= {0,3,4,5} is not n-ideal, since,2 3=5 I J, but 2 n I J, n Z + s. t. 2 n 0. is c-ideal, Definition (3.12): A n-ideal I of B-algebra X is said to be Closed n-ideal if it is also subalgebra. The following example explains the definition above. Example (3.13): In example 2.3, I= {0,2} is closed n-ideal Remark(3.14):- In general not every n-ideal is closed n-ideal from example 1.9, I= {0,2}is n-ideal but not closed n-ideal, since 0 2=1 I. Definition (3.15): A subset I of B-algebra X is said to be Complete n-ideal briefly (c-n-ideal), if (1) 0 I, (2) x y I, y 0 I x n 0 I for some n Z +. Example (3.16): In example 2.3, I= {0,2} is c-n-ideal. Proposition (3.17): Every complete ideal of B-algebra X is complete n-ideal. Proof: Let I be c-ideal. Then 0 I. Now, let x y I, y 0 I, since I is c-ideal, then x I.This complete the proof.

Complete ideal and n-ideal of B-algebra 1713 Remark(3.18):- (1) The convers of the proposition 3.17 is not true in general by example 2.3, I= {0,1,3}is c-n-ideal, since 2 1=3 I, 2 3 = (2 2) 2 = 0 2 = 1 I,2 3=1 I, but not c-ideal, since 2 1=1 I, 2 3=3 I,but 2 I. (2) In general, not every c-n-ideal is subalgebra I= {0,1,3} in example 2.3 is c-n-ideal but not subalgebra, since 1 3=2 I.Also not every c-n-ideal is ideal, I= {0,1,3} in example 2.3 is c-n-ideal but not ideal, since,2 1=3 I,but 2 I. References [1] A. Walendziak, Some axiomization of B-algebras, Mathematica Slovaca, 56 (2006), no. 3, 301-306. [2] J. Neggers and H.S. Kim, On B-algebras, Mathematical Bechnk, 54 (2002), 21-29. [3] J. Neggers and H.S. Kim, A fundamental theorem of B-homomophism for B- algebras, International Mathematical Journal, 2 (2002), 215-219. [4] S.S. Ahn and K. Bang, On fuzzy subalgebras in B-algebras, Commun. Korean Math. Soc, 18 (2003), no. 3, 429-437. https://doi.org/10.4134/ckms.2003.18.3.429 [5] T. Senapati, M. Bhowmik and M. Pal, Fuzzy closed ideals of B-algebras, International Journal of Computer Science, Engineering and Technology, 1 (2011), no.10, 669-673. [6] T. Senapati, C.S. Kim, M. Bhowmik and M. Pal, Cubic subalgebras and cubic closed ideals of B-algebra, Fuzzy Information and Engineering, 7 (2015), 129-149. https://doi.org/10.1016/j.fiae.2015.05.001 [7] Y.B. Jun, H.S. Kim and E.H. Roh, Ideal theory of subtraction algebras, Scientiae Mathematicae Japonicae, 61 (2005), no. 3, 459-464. [8] Y.B. Jun and K.H. Kim, Prime and irreducible ideals in subtraction algebras, International Mathematical Forum, 3 (2008), no. 10, 457-462. Received: May 15, 2017; Published: June 23, 2017