Note on M-grotjpoids
نویسنده
چکیده
In a recent paper of Tamura, Merkel, and Latimer [2], the following question was raised: Suppose 5 is a groupoid (cf. [l]) which satisfies: (1) There is at least one left identity in S. (2) If y or z is a left identity of 5, then x(yz) =■= (xy)z for all xES. (3) For all a, bES there exists x£5 such that ax = 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 Af-groupoid. It is the purpose of this note to present an example of a groupoid satisfying (1), (2), and (3), which is not an ikf-groupoid. It will be shown, however, that every finite groupoid satisfying (1), (2), and (3) is an Af-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:
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