Soft BCL-Algebras of the Power Sets

International Journal of lgebra, Vol. 11, 017, no. 7, 39-341 HIKRI Ltd, www.m-hikari.om https://doi.org/10.1988/ija.017.7735 Soft BCL-lgebras of the Power Sets Shuker Mahmood Khalil 1 and bu Firas Muhamad Jawad al Musawi 1 Department of Mathematis, College of Siene University of Basrah, Basrah, Iraq Corresponding author Department of Mathematis, College of Eduation University of Basrah, Basrah, Iraq Copyright 017 Shuker Mahmood Khalil and bu Firas Muhamad Jawad al Musawi. This artile is distributed under the Creative Commons ttribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. bstrat In this work, at the beginning we attempted to introdue the notion of BCL algebra of the power set and some other new onepts onneted to it are introdued and studied in this work like BCL subalgebra of the power set, soft BCL algebra of the power set and soft BCL subalgebra of the power set. Then some binary operations between two soft BCL algebras of the power set are studied. Moreover, we find relations between soft BCL algebra of the power set and soft BCK / d / d / algebra of the power set. Moreover, several examples are given to illustrate the onepts introdued in this paper. Mathematis Subjet Classifiation (010): 0G05, 0D06, 0B30, 0B35 Keywords: BCL / BCK / d / d / algebras, Soft BCK / d / d / algebras of the power sets 1 Introdution Y. Imai and K. Iseki introdued two lasses of abstrat algebras: BCK algebras and BCI-algebras ([5], [6]). It is known that the lass of BCK-algebras is a proper sublass of the lass of BCI-algebras. fter then the notion of d-algebras, whih is another useful generalization of BCK algebras are introdued (see [], [3], [19]). Next, the notion of algebra is introdued and disussed [9]. In [17] the basi notions of soft sets theory are introdued by Molodtsov to deal with unertainties

330 S. Mahmood Khalil and. Firas Muhamad Jawad al Musawi when solving problems in pratie as in engineering, environment, soial siene and eonomis. This notion is onvenient and easy to apply as it is free from the diffiulties that appear when using other mathematial tools as theory of theory of rough sets, theory of vague sets and fuzzy sets et. Next, many studies on soft sets theory and their appliations are disussed (see [10]-[15]). On the other hand, many authors applied the notion of soft set on several lasses of algebras like soft BCK / BCL algebras [7] and soft algebras [16]. In 017, for any finite set, the notion of d algebra of the power set, BCK algebra of the power set, d algebra of the power set, soft d algebra of the power set, soft BCK algebra of the power set, soft d algebra of the power set, soft edge d algebra of the power set, soft edge BCK algebra of the power set, soft edge d algebra of the power set are investigated. The aim of this work is to introdue new branh of the pure algebra it's alled BCI-algebra of the power set of (see [16]). Then some binary operations between two soft BCL algebras of the power set are studied. Moreover, we study the relations between soft BCL algebra of the power set and soft BCK / d / d / algebra of the power set. Moreover, several examples are given to illustrate the onepts introdued in this paper.. Preliminaries In this setion we reall the basi bakground needed in our present work. Definition.1: ([19]) d algebra is a non-empty set with a onstant 0 and a binary operation satisfying the following axioms: (i)- x x 0 (ii)- 0 x 0 (iii)- x y 0 and y x 0 imply that y x for all x, y in. Definition.: ([18]) d algebra (,, 0) is alled BCK algebra if satisfies the following additional axioms: (1). (( x y) ( x z)) ( z y) 0, (). ( x ( x y)) y 0, for all x, y. Definition.3 ([8]) algebra (,, f ) is a non-empty set with a onstant f and a binary operation satisfying the following axioms: (i)- x x f, (ii)- f x f, (iii)- x y f y x imply that y x, (iv)- For all y x {f } imply that x y y x f. Definition.4: ([0]) BCL algebra is a non-empty set with a onstant 0 and a binary operation satisfying the following axioms: (i)- x x 0,

Soft BCL-algebras of the power sets 331 (ii)- x y 0 and y x 0 imply that y x, (iii)- (( x y) z)(( x z) y)(( z y) x) 0, for all x, y,z in. Definition.5: ([17]) Let be an initial universe set and let E be a set of parameters. The power set of is denoted by P ( ). Let K be a subset of E. pair ( F, K) is alled a soft set over if F is a mapping of K into the set of all subsets of the set. Definition.6:([11]) Let ( F, and ( G, be two soft sets over, then their union is the soft set ( H, C), where C B and for all e C, H( e) F( e) if e B, G (e) if e B, F( e) G( e) if e B. We write ( F, ( G, ( H, C). Further, [4] for any two soft sets ( F, and ( G, over their intersetion is the soft set ( H, C) over, and we write ( H, C) ( F, ( G,, where C B, and H( e) F( e) G( e) for all e C. Definition.7: ([8]) Let ( F, K) be a soft set over. Then ( F, K) is alled a soft d algebra over if ( F( x),,0) is a d algebra for all x K. Definition.8: ([7]) Let ( F, K) be a soft set over. Then ( F, K) is alled a soft BCK algebra over if ( F( x),,0) is a BCK algebra for all x K. Definition.9: ([7]) Let ( F, K) be a soft set over. Then ( F, K) is alled a soft BCL algebra over if ( F( x),,0) is a BCL algebra for all x K. Definition.10: ([16]) Let ( F, K) be a soft set over. Then ( F, K) is alled a soft algebra over if ( F( x),,0) is a algebra for all x K. Definition.11: ([16]) Let be non-empty set and P ( ) be a power set of. Then ( P( ),, with a onstant and a binary operation is alled d algebra of the power set of if P( ) satisfying the following axioms: (i)- B B (ii)- B (iii)- B C and C B imply that B C for all B, C P( ). Definition.1: ([16]) Let ( P( ),, be a d algebra of the power set of. Then P ( ) is alled BCK algebra of the power set of if it satisfies the following additional axioms: (1). (( B C) ( B D)) ( DC), (). ( B ( B C)) C, for all B, C P( ).

33 S. Mahmood Khalil and. Firas Muhamad Jawad al Musawi Definition:.13: ([16]) Let ( P( ),, be a d algebra of the power set of. Then P ( ) is alled a ρ algebra of the power set of if it satisfies the identity ( B C) ( C, for all B C P( ). Definition.14: ([16]) Let ( P( ),, be a d algebra of the power set of. Then P ( ) is alled a d algebra of the power set of if it satisfies the identity ( B C) B, for all B, C P( ). Definition.15: ([16]) Let ( P( ),, be a d algebra (ρ algebra, BCK algebra, d algebra) of the power set of and let H { hi } ii P( ) be a olletion of some random subsets of. Then H is alled d subalgebra (resp. ρ subalgebra, BCK subalgebra, d subalgebra) of the power set of, if h h H, for any h, h H. m k m k Definition.16:([16]) Let ( P( ),, be a d algebra (resp. ρ algebra, BCK algebra, d algebra) of the power set of and let F : H P( ), be a set valued funtion, where H is a olletion of some random subsets of defined by F( h) { q P( ) h q} for all h H where is an arbitrary binary operation from H to P ( ). Then the pair ( F, H) is a soft set over. Further, ( F, H) is alled a soft d algebra (resp. soft ρ-algebra, soft BCK algebra, soft d algebra) of the power set of, if ( F( h),, is a d subalgebra (resp. ρ subalgebra, BCK subalgebra, d subalgebra) of the power set of for all h H. 3. Soft BCL-algebra of the power sets In this setion we introdue the notion of soft BCL-algebra of the power set and soft BCK subalgebra of the power set. We will illustrate the definitions with examples. Definition 3.1 Let be non-empty set and P ( ) be a power set of. Then ( P( ),, with a onstant and a binary operation ( ) is alled BCL algebra of the power set of if P( ) satisfying the following axioms: (i)- B B, (ii)- B C and C B imply that B C for all B, C P( ), (iii)- (( B C) D) (( B D) C) (( DC) 0, for all B, C, D P( ). Definition 3. Let ( P( ),, be a BCL algebra of the power set of and let H { h } P( ) be a olletion of some random subsets of. Then H is i i I

Soft BCL-algebras of the power sets 333 alled h, h H. m k BCL subalgebra of the power set of, if h h H, for any Example 3.3 Let {1,,3 } and let / : P( ) P( ) P( ) m k be a binary operation defined by /( B, C) B / C { x x B & x C} B C, for all B, C P( ). Then ( P ( ),/, ) is a BCL algebra of the power set of with the following table: / Table (1) On the other hand, ( P ( ),/, ) is a BCK / d / d algebra of the power set of. Further,,{} P ( ) { }, but /{} {}/{1 }. Then ( P ( ),/, ) is not ρ- algebra. lso, for example, q 1 { }, q {,{1 }}, q 3 {,{}}, q4 {,{3}}, q {,{1},{}}, q {,{1},{3}}, and q {,{},{3}} are BCL subalgebra 5 and 6 7 BCK / d / d subalgebra of the power set of. However, K {,{1,3},{,3}} is not BCL / BCK / d / d subalgebra of the power set of. Moreover, if is a non-empty set ontains exatly two elements. Then for any we have ( P ( ),/, ) is a BCK / d / d algebra of the power set of with the following table: / Table ()

334 S. Mahmood Khalil and. Firas Muhamad Jawad al Musawi Example 3.4: Let {1,,3 } and let : P( ) P( ) P( ) be a binary B, if B, operation defined by (, B, for all, Otherwise., B P( ). Then ( P ( ),, ) is a BCL algebra of the power set of with the following table: Table (3) Remarks 3.5: (1) It is not neessary every BCL algebra of the power set of is BCK / d / d / algebra of the power set. In example (3.4), let, B {}. Then ( P ( ),, ) is not d algebra of the power set of, sine ( {1,}. lso, let, B {}, C {3}. Then ( P ( ),, ) is not BCK algebra of the power set of, sine (( ( C)) ( C. lso, ( P ( ),, ) is a d / algebra of the power set of. On the other hand, q { }, q {,{1 }}, q {,{}} 1 3 1 { and q 4 {,{3}} are ρ subalgebras. However, K,{1},{}}, K {,{1},{3}}, and K 3 {,{},{3}} are not ρ subalgebras. Further, see example (3.3) ( P ( ),/, ) is BCL algebra of the power set of, but is not ρ- algebra. () Further, if ( P( ),, f ) is a BCL algebra of the power set of satisfying f h f for any h P( ). Then ( P( ),, f ) is a d-algebra of the power set of. Definition 3.6 Let ( P( ),, be a BCL algebra of the power set of and let F : H P( ), be a set valued funtion, where H is a olletion of some random subsets of defined by F( h) { q P( ) h q} for all h H where is an arbitrary binary operation from H to P ( ). Then the pair ( F, H) is a soft

Soft BCL-algebras of the power sets 335 set over. Further, ( F, H) is alled a soft BCL algebra of the power set of, if ( F( h),, is a BCL subalgebra of the power set of for all h H. Example: 3.7 Let ( P ( ),, ) be a BCL algebra of the power set of with the following table: Table (4) Let ( F, H) be a soft set over {1, }, where H {,{1},{}} and F : H P( ) is a set valued funtion defined by F( h) { k P( ) : h k t k h, t N} for all h H. Then F ( ) { }, F ({1}) {,{1 }}, F() {,{}} whih are soft BCL subalgebras of the power set of. Hene ( F, H) is a soft BCL algebra of the power set of. The next example shows that there exist set-valued funtions G : B P( ), where ( G, the soft set is not a soft BCL algebra of the power set of. Example 3.8: Consider the BCL algebra in example (3.7) with a set valued funtion defined by G( h) { k P( ) : h k h k {,{ }}} for all h B {, }. We have ( G, is not a soft BCL algebra of the power set of, sine there exists B, but G( ) {{1}, } is not soft BCL subalgebras of the power set of. Definition: 3.9 Let ( F, H) be a soft BCL algebra of the power set of, and let D H, where F : H P( ) is defined by F( h) { q P( ) h q} for all h H. Then F D : D P( ) is defined by F D ( h) { q P( ) h q} for all h D. Lemma 3.10: If ( F, H) is a soft BCL algebra of the power set of, then ( F D, H) is a soft BCL algebra of the power set of, for any D H. Proof: Let ( P( ),, be a BCL algebra of the power set of and let ( F, H) be a soft d algebra of the power set of, then ( F( h),, is a d subalgebra of the power set of, for all h H. Moreover, for all h D H we have F D ( h) F( h), but D D H (sine D H ). Hene ( F D ( h),, is a BCL

336 S. Mahmood Khalil and. Firas Muhamad Jawad al Musawi subalgebra of the power set of, for all algebra of the power set of. h D. Then ( F D, H) is a soft BCL Definition 3.11: Let ( F, H) be a soft BCL algebra of the power set of. Then ( F, H) is alled a null soft BCL algebra of the power set of if F( h) { } for all h H. lso, ( F, H) is alled an absolutely soft BCL algebra of the power set of if F( h) P( ) for all h H. Example 3.1 Let ( P ( ),, ) be the BCL algebra of the power set of in example (3.3) and let F1 : H1 P( ), F : H P( ), where H 1 {,{}}, H {,{1},{3}} are defined by F1 ( h) { k P( ) h k h k H1}, h H 1, and F ( h) { k P( ) h k k h }, h H. Thus, F1 ( ) F1 ({}) P ( ), and hene ( F 1, H1) is an absolutely soft BCL algebra of the power set. Moreover, F ({ }) F () F3 ({3}) { } and hene F, ) is a null soft BCL algebra of the power set. ( H Lemma: 3.13 If ( P( ),, L) is a BCK / d / d / algebra of the power set of. Then ( P( ),, L) is a BCL algebra of the power set of, if ( C ( C) B, for all, B, C P( ). Proof: Sine ( P( ),, L) is a BCK / d / d / algebra of the power set of. Then P( ) satisfying the following axioms: (i)- B B L, (ii)- L B L, (iii)- B C L and C B L imply that B C for all B, C P( ). Now, sine ( C ( C) B, for all, B, C P( ). Hene, from (i) we onsider that (( C) (( C) L. Further, from (ii) we have L (( C L and this implies that (( C) (( C) (( C L. Therefore, the following are hold: (1)- B B L, ()- B C L and C B L imply that B C, (3)- (( C) (( C) (( C L, for all, B, C P( ). The ( P( ),, L) is a BCL algebra of the power set of. Corollary 3.14 If ( P( ),, L) is a algebra of the power set of. Then ( P( ),, L) is a BCL algebra of the power set of, if C ( ( C) B, for all, B, C P( ).

Soft BCL-algebras of the power sets 337 Proof: Sine ( P( ),, L) is a algebra of the power set of. Then P( ) satisfying the following axioms: (i)- B B L, (ii)- L B L, (iii)- B C L and C B L imply that B C, for all B, C P( ). (iv)- ( B C) ( C L, for all B C P( ) L, Therefore, we onsider only the following are hold: 1)- B B L, )- B C L and C B L imply that B C, Then to prove that ( P( ),, L) is a BCL algebra of the power set of, we need also to prove that (( C) (( C) (( C L, for any, B, C P( ). Thus we have to show that (( C) L and hene we have (( C) (( C) (( C L(( C) (( C L (( C L. For any, B, C P ( ) we have the all ases that are over all probabilities as following: (1) If B, C P( ) (( C) L. () If L B, C P( ) (( C) L. (3) If L B, C P( ) B ( C) L, but B ( C) ( C ( C L. (4) If B C, B L P( ). Then, B ( C) ( C B L ( C ( B ( ( C ( B LC L ( B B LC L ( L (( C) L. (5) If B C, B L P( ). Thus, sine C P( ) L and ( P( ),, L) is a algebra of the power set of. Then from (iv) we have ( C) L and this implies that B ( C) ( C) B and hene ( C) B ( C. Then by lemma (3.13) we have ( P( ),, L) is a BCL algebra of the power set of. Theorem 3.15: Let ( F, and ( G, be two soft BCL algebras over. If B, then the union ( H, C) ( F, ( G, is a soft BCL algebra of the power set of. Proof: Sine B and by definition (.6), we have for all k C, F( k), if k \ B, H ( k) G( k), if k B \. If k \ B then H( k) F( k) is a BCL subalgebra of the power set of. Similarly, if k B \, then H( k) G( k) is a BCL subalgebra of the power set of. Hene ( H, C) ( F, ( G, is a soft BCL algebra of the power set of. Thus the union of two soft BCL algebras of the power set of is a soft BCL algebra of the power set of.

338 S. Mahmood Khalil and. Firas Muhamad Jawad al Musawi Example 3.16: In example (3.4), let ( F, and ( G, be two soft sets over {1,,3} where {,{1}{,3}} and B {{},{1,3 }}. Define F : P( ) by F( h) { k P( ) k h k h{,{1, }} for all h and G : B P( ) byg( h) { k P( ) k h k h{,{1,3 }} for all h B. Note that B. Thus, we have F ( ) {,{1, }}, F ({1}) {,{1 },,{1,}}, F () {,{,3}}, G () {,{}} and G ({1,3}) {,{1},{3}, }. Then H ( ) F( ) {,{1, }}, H () F({1 }) {,{1},{}, {1, }}, H () F({,3}) {,{,3}}, H () G({}) {,{}} and H({1,3 }) G ({1,3}) {,{1},{3},{1,3 }} whih are BCL subalgebras of the power set of. Hene, ( H, C) is a soft BCL algebra of the power set of. Remark 3.17:The ondition B is important as if B, then the theorem does not apply. In above example, if {,{},{,3 }} and B {,{1 }}. Then H( ) F( ) G( ) {,{1,},{1,3 }} whih is not a BCL subalgebra of the power set of. Therefore, is not a soft BCL algebra of the power set of. Theorem 3.18: Let ( F, and ( G, be two soft BCL algebras over. If B, then the union ( H, C) ( F, ( G, is a soft BCL algebra of the power set of. Proof: Sine ( H, C) ( F, ( G,, where C B, and H(k) F( k) G( k) for all k C [ by definition (.6)], Note that H : C P( ) is a mapping and so ( H, C) is a soft set over. We have, H( k) F( k) or H( k) G( k) is a BCL subalgebra of the power set of. Hene, ( H, C) ( F, ( G, is a soft BCL algebra of the power set of. Therefore, the intersetion of two soft BCL algebras is a soft BCL algebra. Example 3.19: Consider the algebra in example (3.16) with {,{},{,3 }} and B {,{1}}. Then H ( ) F( ) {,{1, }} or H ( ) G( ) {,{1,3 }}. Note that both are BCL subalgebras of the power set of. Hene, ( H, C) is a soft BCL algebras of the power set of. Theorem 3.0: Let ( P( ),, f ) be a BCL algebra of the power set of with the ondition f h f for any h P( ). If ( F, is a soft BCL algebra of the power set of, then ( F, is a soft d algebra of the power set of. Proof: Straightforward from Definitions [(3.6), (.7)] and remark [()-(3.5)].

Soft BCL-algebras of the power sets 339 Theorem 3.1: If ( P( ),, f ) is a d-algebra of the power set of with the ondition ( C ( C) B and ( F, is a soft d-algebra of the power set of. Then ( F, is a soft BCL algebra of the power set of. Proof: Straightforward from Definitions [(3.6),(.7)] and Lemma[(3.13)]. From [14, Theorem.1] and [1, Theorem 3.5] we an give the following theorems (the proves are diret). Theorem 3.: If ( P( ),, f ) is a BCK algebra of the power set of and ( F, is a soft BCK algebra of the power set of. Then ( F, is a soft BCL algebra of the power set of. Theorem 3.3: If ( P( ),, f ) is a BCL algebra of the power set of. If f h f and k h h g, for any h, k, g P( ), then ( P( ),, f ) is a BCK algebra of the power set of. Theorem 3.4: If ( P( ),, f ) is a BCL algebra of the power set of and ( F, is a soft BCL algebra of the power set of, then ( F, is a soft BCK algebra of the power set of. knowledgements. The author would like to thank from the anonymous reviewers for arefully reading of the manusript and giving useful omments, whih will help to improve the paper. Referenes [1] D. l-kadi and R. Hosny, On BCL-algebra, Journal of dvanes in Mathematis, 3 (013), no., 184-190. [] P. J. llen, Constrution of many d-algebras, Commun. Korean Math. So., 4 (009) 361-366. https://doi.org/10.4134/kms.009.4.3.361 [3] P. J. llen, H. S. Kim and J. Neggers, Deformation of d/bck-lgebras, Bull. Korean Math. So., 48 (011), no., 315-34. https://doi.org/10.4134/bkms.011.48..315 [4] F. Feng, Y. B. Jun and. Zhao, Soft semirings, Computers and Mathematis with ppliations, 56 (008), no. 10, 61-68. https://doi.org/10.1016/j.amwa.008.05.011 [5] Q. P. Hu and. Li, On BCH-algebras, Math. Semin. Notes, Kobe Univ., 11 (1983), 313-30.

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Soft BCL-algebras of the power sets 341 [18] J. Neggers, Y. B. Jun and H. S. Kim, On d-ideals in d-algebra, Mathematia Slovaa, 49 (1999), no. 3, 43-51. [19] J. Neggers, and H. S. Kim, On d-algebra, Mathematia Slovaa, 49 (1999), no. 1, 19-6. [0] Y.H. Liu, new branh of the pure algebra: BCL-lgebras, dvanes in Pure Mathematis, 1 (011), no. 5, 97-99. https://doi.org/10.436/apm.011.15054 Reeived: ugust 7, 017; Published: Otober 30, 017