A loop (Q, ·, \, /) is called a middle Bol loop if it obeys the identity x(yz\x) = (x/z)(y\x). In this paper, some new algebraic properties of a middle Bol loop are established. Four bi-variate mappings fi , gi , i = 1, 2 and four j-variate mappings αj , βj , φj , ψj , j ∈ N are introduced and some interesting properties of the former are found. Neccessary and sufficient conditons in terms of fi , gi , i = 1, 2, for a middle Bol loop to have the elasticity property, RIP, LIP, right alternative property (RAP) and left alternative property (LAP) are establsihed. Also, neccessary and sufficient conditons in terms of αj , βj , φj , ψj , j ∈ N, for a middle Bol loop to have power RAP and power LAP are establsihed. Neccessary and sufficient conditons in terms of fi , gi , i = 1, 2 and αj , βj , φj , ψj , j ∈ N, for a middle Bol loop to be a group, Moufang loop or extra loop are established. A middle Bol loop is shown to belong to some classes of loops whose identiites are of the J.D. Phillips’ RIF-loop and WRIF-loop (generalizations of Moufang and Steiner loops) and WIP power associative conjugacy closed loop types if and only if some identities defined by g1 and g2 are obeyed.
. . , xi) = (. . . (((x1x2)x3)x4) . . . xi1)xi and i(x1, x2, . . . , xi) = x1\(x2\(x3\( xi2\(xi1\xi) ))) i N. The following are true. 1. f1(cid:0)x, n(y, x, x, . . . , x)(cid:1) = n(cid:0)x, x, . . . , x, f1(x, y)(cid:1). 2. f1(cid:0)x, n+1(x, y, x, x, . . . , x)(cid:1) = n+1(cid:0)x, x, x, . . . , x, g1(x, y)(cid:1). 9 3. (Q, ) has the RAP if and only if f1(x, y) = x[(yx2)\x]. 4. (Q, ) has the PRAP if and only if yxn n(cid:0)x, x, . . . , x, f1(x, y)(cid:1) = x. 5. If (Q, ) has the RAP, then (Q, ) is of exponent 2 if and only if f1(x, y) = x(y\x). 6. If (Q, ) has the PRAP, then (Q, ) is of exponent n if and only if y n(cid:0)x, x, . . . , x, f1(x, y)(cid:1) = x. Proof. 1. By Lemma 2.1(b), yx\x = x\(y\x) (yx)Rx = (y\x)Lx RxRx = RxLx Rx = RxLxR1 x (11) By equation (11) x RxLxR1 x = RxRx = RxLxR1 1 1 x = RxL3 x RxLxR xR1 x . x = RxLn R2 x = RxL2 R3 x = R2 R4
id: 24ca7752f8c198e985457371578e116a - page: 9
Thus, for ( ((y x x)x x)x )x \x = (x\ (x\(x \(y\x))) ) (12) | {z n-times } | } {z (n 1)-times Equation (12) implies that f1(cid:0)x, n(y, x, x, . . . , x)(cid:1) = n(cid:0)x, x, . . . , x, f1(x, y)(cid:1). 2. By Lemma 2.1(d), xz\x = x\(x/z) (xz)Rx = (x/z)Lx zLxRx = zLxLx LxRx = LxLx Lx = LxLxR1 x (13) By equation (11) and equation (13), Therefore, LxRn LxRx = LxLxR1 x RxLxR1 xR1 xR1 x = LxLxLxR1 x = LxL2 x RxLxR1 xR1 x . x RxLxR1 xR1 x . x , n 1. Thus, for all y Q, LxR2 LxR3 x = LxL(n+1) x R1 x = LxL2 x = LxL3 x = LxL3 x = LxL4 xR1 x . ( ((xy x)x x)x )x \x = (x\ (x\(x \(x/y))) ) (14) |
id: 89405dc8d1916203203c6a2db6d62cef - page: 10
{z n-times } | {z } (n + 1)-times Equation (14) implies that f1(cid:0)x, n+1(x, y, x, x, . . . , x)(cid:1) = n+1(cid:0)x, x, x, . . . , x, g1(x, y)(cid:1). 3. This follows from 1. when n = 2. 4. This follows from 1. 10 5. This follows from 3. 6. This follows from 4. Lemma 2.3. Let (Q, , \, /) be a loop. The following are equivalent. 1. (Q, , \, /) be a middle Bol loop. 2. x(yz\x) = (x/z)(y\x) for all x, y, z Q. 3. (x/yz)x = (x/z)(y\x) for all x, y, z Q. Proof. From Lemma 2.1(f), x(z\x) = (x/z)x. On another hand, if (x/yz)x = (x/z)(y\x) is true, then x(y\x) = (x/y)x. So, 1., 2. and 3. are equivalent. Theorem 2.2. Let (Q, , \, /) be a middle Bol loop and let f1, g1, f2, g2 : Q2 Q be dened as: f1(x, y) = yx\x or f1(x, y) = x\(y\x) and g1(x, y) = xy\x or g1(x, y) = x\(x/y), f2(x, y) = x/(xy) or f2(x, y) = (x/y)/x and g2(x, y) = x/(yx) or g2(x, y) = (y\x)/x.
id: 5f81ce66440ba021592ce8bc4f833396 - page: 10
Then: (a) x/yx = (y\x)/x. (b) z(yx) = x y(zx) = x and LyLz = I LzLy = I. (c) x/(xz) = (x/z)/x. (d) (yx)u = x y(xu) = x and RuLy = I I = LyRu. (e) f2(x, y) = x/(xy) f2(x, y) = (x/y)/x. (f) g2(x, y) = x/(yx) g2(x, y) = (y\x)/x. (g) The following are equivalent: 1. (Q, /) (Q, \). 2. [x/(xy)]x = [y/(xy)]y. 3. x[(xy)\x] = [y/(xy)]y. 4. [x/(xy)]x = y[(xy)\y]. 5. x[(xy)\x] = y[(xy)\y]. (i) yx z = x xz = [x/(yx)]x y xz = x. (j) (Q, ) is a CIPL if and only if xy1 = [x/(yx)]x. (k) yx z = x xz = g2(x, y) x y xz = x. (l) (Q, ) is a CIPL if and only if xy1 = g2(x, y) x. 11 (m) z xy = x zx = x[(xy)\x] zx y = x. (n) (Q, ) is a CIPL if and only if y1x = x[(xy)\x]. (o) z xy = x zx = x g1(x, y). (p) (Q, ) is a CIPL if and only if y1x = x g1(x, y). (q) z yx = x zx = x[(yx)\x] y zx = x. (r) (Q, ) is a LIPL if and only if y1x = x[(yx)\x]. (s) z yx = x zx = x f1(x, y). (t) (Q, ) is a LIPL if and only if y1x = x f1(x, y). (u) xy z = x xz = [x/(xy)]x.
id: 6c749c3a553c53680867c1094531339b - page: 11