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 Book Id: WPLBN0002097645 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Literature, Education, Authors Community, Mathematics ► Abstract Full Text Search Details...ng identities: 1. (xy) (zx) = (x(yz))x 2. ((xy)z)y = x(y(zy)) 3. x(y(xz) = ((xy)x)z for all x, y, z ∈ L. DEFINITION 2.3.2: Let L be a loo... ...EFINITION 2.3.2: Let L be a loop, L is called a Bruck loop if x(yx)z = x(y(xz)) and (xy) -1 = x -1 y -1 for all x, y, z ∈ L. DEFINITION 2.3.3: A... ...ollowing equality is true: (i) xy = z(yx) or (zy)x (ii) yz = x(zy) or (xz)y (iii) zx = y(xz) or (yx)z then we call the loop L to be a strongly...
 Book Id: WPLBN0002097093 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details... the proof of (2) we shall apply the standard algebraic inequalities 3(xy +xz +yz) (x +y +z) 2 3(x 2 +y 2 +z 2 ): Now, the proof of (2) runs as fol... ... in this case, since T 2 9 16 X x 2 y 2 : Equality occurs when xy sin = xz sin = yz sin and cos( ) = 1, i.e. whenO coincides with ther Ferm... ...ations, we have for example, in a region T = yz sin( + ) 2 xy sin 2 xz sin 2 or T = xy sin 2 + yz sin 2 xz sin( + ) 2 in another region.... ...will be deduced. By (4), (6), (2) this implies that 2(x 2 +y 2 +z 2 ) +xy +xz +yz = constant (10) By (1) and (10) follows xy +xz +yz = constant (11) x... ... p a =x, p b =y, p c =z, when 3p 2p =x +y +z =p. Then p yz(x +y +z), p xz(x +y +z), p xy(x +y +z), p xyz=(x +y +z) are all integers, and since x;y... ...ns on radicals must be perfect squares of integers. Let xy(x +y +z) =t 2 ; xz(x +y +z) =p 2 ; yz(x +y +z) =q 2 ; xyz x +y +z =u 2 : (27) Then by multi... ... xyv 2 = t 2 , xzv 2 = p 2 , yzv 2 =q 2 givexy = t v 2 , wheret =vn 1 ,xz = p v 2 , wherep =vn 2 ,yz = q v 2 , where 50 q = vn 3 (n 1 ;n ... ...y v 3 n 1 n 2 n 3 = u 2 v 5 we get n 1 n 2 n 3 = u 2 v 2 . By xy = n 2 1 , xz = n 2 2 , yz = n 2 3 , xyz = u 2 v 2 , x +y +z = v 2 , we get x = d 1 X ... ... =v 2 with (u;v) = 1. This proves the Lemma. By writing (1) as (x z)(x 2 +xz +z 2 ) =y 2 ; 75 from Lemma we get: x z =du 2 ; x 2 +xz +z 2 =dv 2 ; y...
 Book Id: WPLBN0002097649 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...ing identities (xy)(zx) = (x (yz)) x. ((xy) z) = x (y (xy)). x (y (xz)) = ((xy) x) z for all x, y, z ∈ L. These identities will be ter... ... DEFINITION 2.1.7: A loop L is called a Bruck loop if (x (yx)) z = x (y (xz)) and (xy) -1 = x -1 y -1 for all x, y, z ∈ L. DEFINITION 2.1.8: ... ... y and z in R such that x o y = x + y – xy = 0 and x o z = x + z – xz = 0 but y o z = y + z – yz ≠ 0 and z o y = z + y – zy ≠ 0. Simi... ...; thus the identity reads as (a + x) (a + y) (a + z) = a + (xy) (a + z) + xz (a + y) + yz (a + x) for all a, x, y, z in L) Example 2.4.5: Let Z 2 ... ... where x , y ∈ ZG. Multiplication in ZL is given by (x + yu) (z + wu) = (xz + g o w ∗ y) + (wx + yz ∗ ) u where x, y, z, w ∈ Z G and g o is an e...
 Book Id: WPLBN0002097650 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Technology, Literature, Education, Logic, Authors Community, Mathematics ► Abstract Full Text Search Details...ities: i. (xy) (zx) = (x(yz))x. ii. ((xy)z) y = x(y (zy)). iii. x (y (xz))= ((xy)x)z for all x, y, z ∈ L. BRUCK LOOP: Let L be a loop, L is... ...RUCK LOOP: Let L be a loop, L is called a Bruck loop if (x (yx)) z = x (y (xz)) and (xy) -1 = x -1 y -1 for all x, y, z ∈ L. BOL LOOP: A loop L... ...entities: i. (xy) (zx) = (x(yz))x. ii. ((xy)z) y = x(y(zy)). iii. x(y(xz) = ((xy) x) z for all x, y z in N. We call a near field N a Bruck nea... ...l x, y z in N. We call a near field N a Bruck near-ring if (x(yx) z = x(y (xz)) and (xy) -1 = x -1 y –1 for all x, y , z ∈ N. N a non-associative... ... i. (S, +) is a S-semigroup ii. (N, ) is a semigroup. iii. (x + y) z = xz – 0z + yz for all x, y, z ∈ N. Several properties of S-INR near-ri... ...r the bisemigroup S = S 1 ∪ S 2 = {0, a  a 2 = 0} ∪ {x, y, z xy = z, xz = 0, zx = 0, z 2 = x 2 = y 2 = 0, zy = yz = 0}. Clearly S is a bisemi... ...ies: i. (xy) (zx) = (x (yz))x. ii. ((xy) z) y = x(y(zy)). iii. x (y (xz)) = ((xy) x)z. DEFINITION 4.1.5: A biloop L = L 1 ∪ L 2 . is called ... ...) are Bruck loops i.e. both of them satisfy the identities (x(yz))z = x(y(xz)) and (xy) -1 = x -1 y -1 for all x, y ∈ L 1 and x, y ∈ L 2 . Similar... ...f R 2 . i. (xy) (zx) = (x(yz))x. ii. ((xy)z) y = x(y (zy)). iii. x(y (xz)) = ((xy) x)z; for all x, y, z ∈ R 1 and for all x, y, z ∈ R 2 . ...
 Book Id: WPLBN0002097043 Subjects: Non Fiction, Algebra, Smarandache Collections ► Abstract Full Text Search Details...() 12 12 1 yy12 LLff( x z ) − ∪∪ 98 = ( ) ( ) 12 1121 y111 y2 22 L( f )xz L( f)xz −− ∪ = ( ) ( ) 11 11 1111 2 222 fyxz fx yxz −− −− ∪ = () (... ...12 xy 1 1 xy 2 2 L( f ) ( z )L( f ) ( z ) ∪ = 12 1121 y111 y222 L( f ) ( xz )L( f) ( xz) −− ∪ = f 1 ( ) ( ) 11 11 111 2222 yxz fyxz −− −− ∪ =...
 Book Id: WPLBN0002097647 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...zyx (or yx = zxy). ii. yz = xzy (or zy = xyz). iii. zx = yxz (or xz = yzx). We can define a strongly semi right commutative triple x, y... ...r yx = xyz) or ii. yz = zyx (or zy = yzx) or iii. zx = xzy (or xz = zxy). Similarly we can define strongly semi left commutative rin... ... (or yx = zxy) or ii. yz = xzy (or zy = xyz) or iii. zx = yxz (or xz = yzx). Similarly we define Smarandache strongly semi left commuta... ...or yx = xyz) or ii. yz = zyx (or zy = yzx) or iii. zx = xzy (or xz = zxy). Finally we define a Smarandache strongly semi right (left) ...
 Book Id: WPLBN0002097651 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...y ∈ M, and a given subset Y of a moniod M, we define x –1 y = {z ∈ M / xz = y} xy –1 = {z ∈ M / x = zy} F(Y) = M –1 YM –1 = {m ∈ M/ such tha... ...oop V: L → [0, 1] is called a fuzzy Bruck subloop if P (x (yx) z) = P(x(y(xz))) and P (xy) –1 = P (x –1 y –1 ) for all x, y ∈ L. DEFINITION [1... ... a S-fuzzy C-congruence relation on P ⊂ X if M(µ(x , y), µ(z, ω)) ≤ µ (xz, yω) for all x, y, z, ω ∈ P ⊂ X. DEFINITION 3.1.10: A S-fuzzy subgrou... ...set P such that i. P is a S-na sub-biring of R . ii. (x(yx)) z = x (y(xz)) and (xy) –1 = x –1 y –1 is true for all x, y, z ∈ P. We call (R,... ... ∈ S. A na-bisemiring S is said to be a Bruck bisemiring if (x(yz))z = x(y(xz)) and (xy) –1 = x –1 y –1 for all x, y, z ∈ S. A na-bisemiring S... ... i. (S, +) is a S-semigroup. ii. (N, ) is a semigroup. iii. (x + y)z = xz – 0z + yz for all x, y, z ∈ N. DEFINITION 6.1.28: Let I be a S-left ... ... ∈ P; P ⊂ N where P is an associative ring or P is a near-field such that xz = y or zx = y. DEFINITION 6.2.43: Let N be a non-associative S-semin... ...ts z ∈ P, P ⊂ N where P is an associative seminear-ring such that xz = y or zx = y. DEFINITION 6.2.44: Let N be a S-quasi near-ring (S-qu... ..., y ∈ N there exists z ∈ R; R a ring R ⊂ N (R a semiring R ⊂ N) such that xz = y or yz = x. DEFINITION 6.2.45: Let P be a seminear pseudo ring (S...
 Book Id: WPLBN0002097031 Subjects: Non Fiction, Education, Philosophy-Mathematics ► Abstract Full Text Search Details...expsnatur per w?ain, TZ captam ipii proportionalem, refolvccu x2; q&rum XZ trahcndo corpus fecundurn lun llac in vires XT3 iqiJi~~cljl fili yJX9 ... ...em plano, ~i-y-~-~%., P P?. ’ d perfpi,-uum & quad corpus :-I :-.- - ---~XZ’TZ--...-.---- ‘,/.--------- Se -~ I ----.2?----LC-.. ._~- --_--- .-- (... ...) A? in YG. Sit illud b 6: 82 :,z c L. , complete parallelogrammo ZIN XZ ; dicatur B A?, a; .-BD, o ; NX, c ; & ratio data V’Z ad ZX m vel !ZI ... ...fitas illa efi reciproce at XT: Sunt enim A” & ddA” -?\$A.!A+t;-\$ ipfarum XZ & ZT quadrata. Refifien- - ee XT tia autem in eodem loco G kt ad gr... ... eodem loco G kt ad gravitatem ut 3 S in x ad 4 RR, 2?!w-~2Pe id eit, ut XZ”ad ---L2 VG. Et velocitas ibidem ea ipfa eft, quacum ‘- I - corpus pr... ...ntiatn illam tenuior pa.~celerior fit quain in i@o h-amine i,n ratione 2~ XZ~ ad Z,I x 21 fi+-u I7 &j 12 quamproxi.me, i,d eit in Cubduplicata r,ati...
 Book Id: WPLBN0002097643 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...a and ([Z n (t, t)] x) y = [Z n (t, t)] (xy) for all x, y ∈ Z n (t, t), y (xZ n (t, t)) = (yx) [Z n (t, t)] for all a, x, y ∈ Z n (t, t). The ...
 Book Id: WPLBN0002097119 ► Abstract Full Text Search Details...atisfes the following properties [x,y]=−[y,x] ∀x,y∈E 2n (3) [x,y+z]=[x,y]+[xz] ∀x,y,z∈E 2n , (4) a[x,y]=[ax,y] ∀a∈R, x,y∈E 2n (5) [x,y]=0 ∀y∈E 2n ⇒ x=...
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