# What's the name for a datatype which has every property of a group except commutativity instead of associativity?

I have a set X and an action Y where:

• Closure: for every element x1 and x2 in X, x1Yx2 is also in X
• Identity: there is an element e in X where for all x in X, eYx = xYe = e
• There is at least one inverse of every element in X: for all x1 in X, exists some x2 such that x1Yx2 = x2Yx1 = e. However, there may be more than one such inverse
• There is commutativity (x1Yx2 always equals x2Yx1), but there is no associativity: x1Y(x2Yx3) does not always equal (x1Yx2)Yx3.

What is such an object called, and where can I find literature relating to it?

Thanks!

Note that a similar question has been asked (A set which satisfies all conditions for a Group except associativity), but I didn't see an answer which provided a standard definition for exactly my conditions.

• You have probably already looked at this but do any of these quasigroups satisfy your requirements? en.wikipedia.org/wiki/Quasigroup#Loop Commented Dec 29, 2019 at 21:01
• @NickBishop I don't think it's a quasigroup because elements can have more than one inverse Commented Dec 29, 2019 at 21:16
• A structure comprising a set equipped with a binary operator is called a magma or groupoid. So the structures you are interested in may be called commutative magmas with identity and inverses. Why do you ask about such structures? Do you have any interesting examples in mind? Commented Dec 29, 2019 at 23:39
• @RobArthan this is really getting into the deep end, but I'm a deep learning researcher who read a paper proving theorems on deep learning for abelian groups and was wondering if it could extend to my domain Commented Dec 30, 2019 at 1:06