On loops with a special property

1. The loop G is defined, as usual, as a multiplicative system having a unit u, and such that in the equation xy=z any two of x, y, z uniquely determine the third. Let the right inverse element of any xEG be x', so that xx'=u. We postulate a special property xr: xy *x' =y for any x and y in G. Such loops are interesting in connection with a generalization [i] of those plane webs whose study inspired the notion of the Moufang loop [2 ]. The problem which has arisen in this respect2 was the independence between the "inverse properties" y y'x=x and xy y' =x (equivalent to those defined in [3 ]) and our property 7r. Since each of the "inverse properties" implies the equality of the right and left inverses of any element, the existence of a loop with the property 7r and different right and left inverses of at least one element will be sufficient to prove the independence. In the study of our loops we use the concept of a cycle of inverses (in short: cycle), i.e. a finite sequence of elements x1, x2, . . ., xn such that x = xk+l mod n. The number n will be called the length of the cycle. A length of 1 or 2 implies identity of right and left inverses in this cycle; groups thus have only cycles of length 1 or 2. Our loops, if finite, consist only of cycles, and every element belongs exactly to one cycle. In the following we shall deal mainly with the cycles and their lengths.