Reference request for generalization of groups with out identity element?

In other words what do we call a magma which is associative and has divisibility property but not existence of identity? Or a groupoid when it loses the identity property?

A reference on such objects would be very helpful.

There is a table of various generalizations of groups here & here

• The first one sounds like a cancellative semigroup (assuming that is what you mean by the divisibility property). – Derek Holt Oct 2 '14 at 4:27
• If by divisibility you mean $ax=b$ and $ya=b$ have solutions for all a,b then you have a group. – Benjamin Steinberg Oct 2 '14 at 13:37
• Since you have selected an answer then you may up-vote it as well? – Włodzimierz Holsztyński Jan 10 '15 at 19:01
• I just rolled back an edit which changed perfectly acceptable English idiom to non-standard idiom. It's what do we call, not how do we call – Yemon Choi Jan 10 '15 at 19:35
• Funny, how do we call would be the direct translation from Spanish :) – José Figueroa-O'Farrill Jan 10 '15 at 22:09