Generalized associativity and bisymmetry on quasigroups

Medial topological groupoids

“…A related result is that of ACZ~L, BELOUSOV, and Hosszu [3,Theorem 3] Noting the abundance of surjectivity, the commutativity of o follows by setting x = e2 in the above and then the associativity is immediate. A function f:G-+G' between two topological groupoids is a homomorphism if f(xy) =f(x) f(y) for all x, yeG and is an iseomorphism if, in addition, it is a homeomorphism.…”

Isotopy invariants in quasigroups

“…Aczél in [1] proved that the identities y(z-yx) = (y-zy)x and (xj>-z)j> = x(yzy) are invariant with respect to loop isotopy. Other special results have been obtained by Sade and Hosszu,[2], [13]. A loop is a quasigroup which satisfies x/x=y\y for all x, y.…”

Certain functional equations on groupoids weaker than quasigroups

“…(xlz) 2 (y3z) = x a y (3) where x, y, z, u are taken from certain sets and the integers 1, 2, 3, ..., 6 are binary operations. All three equations have been investigated by several authors under various conditions and a comprehensive bibliography of these may be found in 'Lectures on Functional Equations and Their Applications' by J. Aczdl.…”