Note on š¯‘€-groupoids
Nancy Graham Ā· Proceedings of the American Mathematical Society Ā· 1964
In a recent paper of Tamura, Merkel, and Latimer [2], the following question was raised: Suppose S is a groupoid (cf. [I]) which satisfies: (1) There is at least one left identity in S. (2) If y or z is a left identity of S, then x(yz) -(xy)z for all xCS. (3) For all a, bES there exists xzS such that axx==b. Then does S satisfy: (3') For any xeS there is a unique left identity e (which may depend on x) such that xe =x? A groupoid which satisfies (1), (2), and (3') is defined in [2] to be an M-groupoid. It is the purpose of this note to present an example of a groupoid satisfying (1), (2), and (3), which is not an M-groupoid. It will be shown, however, that every finite groupoid satisfying (1), (2), and (3) is an M-groupoid. Let A be a denumerable set. For simplicity, denote its elements by 1, 2, 3, * . Let * be a binary operation on A which satisfies the following Cayley table: