Abstract. We introduce the concept of a weighted quasigroup. We show that, corresponding to any weighted quasigroup, there is a quasigroup from which it can be obtained in a certain natural way, which we term amalgamation. Not all commutative weighted quasigroups can be obtained from commutative quasigroups by amalgamation; however, given a commutative weighted quasigroup whose weights are all even, we give a necessary and sufficient condition for the existence of a commutative quasigroup from which it can be obtained by amalgamation. We also discuss conjugates of weighted quasigroups.

We also introduce the concept of a simplex zeroid, and relate this concept to that of a weighted quasigroup.

Weighted quasigroups.

Suppose that we have a finite set S with a closed binary operation. If a, b ε S, we shall denote the result of this binary operation acting on a and b by ab. If the binary operation has the two properties

1. (i) for each a, b ε S, the equation ax = b is uniquely solvable for x, and

2. (ii) for each a, b ε S the equation ya = b is uniquely solvable for y,

then S is a quasigroup. It is well-known, and easy to see, that S is a quasigroup if and only if its multiplication table is a latin square. The properties (i) and (ii) amount to the assertion that, in the multiplication table, each element of S occurs exactly once in each row and exactly once in each column.