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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

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    $\begingroup$ The first one sounds like a cancellative semigroup (assuming that is what you mean by the divisibility property). $\endgroup$ – Derek Holt Oct 2 '14 at 4:27
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    $\begingroup$ If by divisibility you mean $ax=b$ and $ya=b$ have solutions for all a,b then you have a group. $\endgroup$ – Benjamin Steinberg Oct 2 '14 at 13:37
  • $\begingroup$ Since you have selected an answer then you may up-vote it as well? $\endgroup$ – Włodzimierz Holsztyński Jan 10 '15 at 19:01
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    $\begingroup$ 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 $\endgroup$ – Yemon Choi Jan 10 '15 at 19:35
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    $\begingroup$ Funny, how do we call would be the direct translation from Spanish :) $\endgroup$ – José Figueroa-O'Farrill Jan 10 '15 at 22:09
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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?

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