Reference request for generalization of groups with out identity element?

Question

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 02 '14 at 04:27
If by divisibility you mean $ax=b$ and $ya=b$ have solutions for all a,b then you have a group. — Benjamin Steinberg, Oct 02 '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
@JoséFigueroa-O'Farrill and German and French if I recall correctly. To be honest, I am not keen on such edits even if they are correct — Yemon Choi, Jan 12 '15 at 03:38

score 2 · Accepted Answer · edited Apr 13 '17 at 12:58

I think such structures are called nonunital semigroups. See, for example, ftp://ftp.math.ethz.ch/EMIS/journals/MPRIA/2000/pa100i2/pdf/100210ai.pdf (Non-Unital Semigroup Crossed Products, by N.S. Larsen) and http://www.hindawi.com/journals/aaa/2014/463918/ (Solution of Several Functional Equations on Nonunital Semigroups Using Wilson’s Functional Equations with Involution, by J. Chung and P. K. Sahoo).

Nonunital rings, which are more popular than simply nonunital semigroups, are discussed in answers to this MO question: What are the reasons for considering rings without identity?

Reference request for generalization of groups with out identity element?

1 Answers1