Vol. 35, No. 2, 1970

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Generalized Bol functional equation

Valentin Danilovich Belousov and Palaniappan L. Kannappan

Vol. 35 (1970), No. 2, 259–265

Each identity in a group or in a quasigroup induces a generalized identity (functional equation) in a class of quasigroups. Generalized associativity, generalized bisymmetry and generalized distributivity are examples of such generalized identities. From the left Bol identity

x(y(xz)) = (x(yx ))z

on a quasigroup, we obtain a generalized Bol identity on a class of quasigroups:

A1(x,A2(y,A3(x,z))) = A4(A5(x,A6(y,x )),z),

where the Ai’s are quasigroup operations on a set Q. The general solution of this generalized Bol functional equation is obtained by reducing it to another functional equation

P(x,y+ S (x,z)) = P (x,y + α(x)) + z

where P and S are quasigroup operations on Q and α(x) = S(x,0). If the operations in the last functional equation are considered on real numbers (or groups), then the solution of this equation is obtained.

Mathematical Subject Classification
Primary: 20.95
Received: 5 February 1970
Published: 1 November 1970
Valentin Danilovich Belousov
Palaniappan L. Kannappan