The independence of regular identities in groupoids
Lajos Klukovits · Institutional Repositories DataBase (IRDB) · 1989
Let f = g be a groupoid identity in the variables x 1 , .. ., Xn such that each x; (i = I, ... , n) occurs at least in one of f and g.We say that the conditions of this identity f = g are independent over a set A if for each (a 1 , ... , an) E An there exists a binary operation * on A such that in the groupoid A = we have butforany (bi, .. ,,bn)EAn-j(a 1 , ... ,an)\ fA (b;,, ... , b; 1 ) = gA (bi,, ... , biz), It is easy to see, that the conditions of the identity above are non-independent over a set A, if and only if there exists a proper subset B C An such that in every groupoid A =, if for every (b 1 , ... ,bn)EB /A_(b;,, ... ,b;k) = gA(b;,, ... ,bit), then the identity f=g holds in A =.The investigations of identities form independence point of view originated in a problem of L. Redei in which he asked whether the associativity conditions are independent.This original problem was solved by G. Szasz [5): THEOREM.The associativity conditions are independent over a set A, iff card A:2:::4.Later R. Wiegandt and J. Wiesenbauer [7), [8] considered the distributivity of two binary operations from the same point of view.In [2] and [3) we investigated two other groupoid identities: the mediality ((xy) (zu) = (xz) (yu)) and the self-distributivity (x(yz) = (xy) (xz) and (xy)z = (xz) (yz)) and proved the following result.