38

Valentin D. Belousov and Zoran M. Stojakovic

and denote {xi ^) = X, (xj*, y'i) = Y, (yl^i) = Z, the equation (14) becomes A,(B(X, Y),Z) = C,(X,D(Y,Z)\

where /4p В, Cp D are G/)-groupoids (see [2]), defined on

Ä , : Qx ö^-^-> Q, Ci : Q'~' xQ->Q,

B : QJ - ' X Q'^^'^Q, DiQ'^'^'xQ'-'-^ Q,

These GD-groupoids satisfy the conditions of Theorem 2 from [2], and by this theorem

B ( X , F) = a-i (y ZoS 7), D(Y, Z) = cp-^ (S Foß Z)

that is,

( 15 ) В(хГ, /i) = а-Чу (^Г VS (хГ, у\)),

( 16 ) i)(x;, >'i) = 9-4S(x;, /ùo^iyUù).

where a, 9 are permutations of the set ß, ß(o) binary group, y and ß quasi- groups of arities \y\=j—i, | ß | = ^ - r and S an infinitary quasigroup of type со + r. Using (15) and (16), we put back В and D in (8) and obtain

( 17 ) А(хГ\ (x.-^ (у(хГ^)о8(х]", y[)),yT+i) =

= С{х\-\ 9-Ч8(^Г. /Ooß(yUO\yT^i),

and if in (17) we substitute the variables x{~^ by elements с^Г^ such that y(c{~^) = e, where e is the unity of the group Q(o), we shall have

A ( x\ - KoL - 4 ( x ; , y[), yr+i)=-K(x\-\ 8(x;, /Oo^iyl^,), j^r+i),

where

^ ( ^1 , y, ys+ù = Cixi , ci ,9 V. J^+O- Hence,

( 18 ) A{xT\(x~^x,yT+î) = K{xr\xo^{/,^i), y^^i).

If in (17) we substitute x^, yl by elements cj*, ^î such that 8(^7, d\) = e, we have

( 19 ) С(хГ\ 9-1 ß(Ai), у7^{)^А{х\'\ a-iy(^r'), ^Г-ы),

and by (18) and (19) it follows

Cix\ - \ <?-'^{yWx\ уТ^г) = К{х\-\ y(xrVß(>'r+i), уТ-,1), i.e.

C ( л : i~^ J, з;Г+1) = ^(л:Г\ уС^Г^ФД^» уТ+ù-