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](http://en.wikipedia.org/wiki/Groupoid#Relation_to_groups) & [here](http://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magmas)



Reference request for generalization of groups with out identity element?
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?](https://mathoverflow.net/questions/22579/what-are-the-reasons-for-considering-rings-without-identity)


