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.
There are non-associative groups that are called loops ( in Wikipedia) you can find a very nice diagram of those structures (included below). Otherwise, you can just call them "commutative non-associative groups" or as @Rob Arthan mentioned in his comment "commutative magmas with identity and inverses" in comparison with non-associative algebras or Non-associative-rings). In all cases, I do not know of an existing specific name other than calling them by their properties.
Remark: There is not probably an exact name for what you are looking for, but as far as I know, associativity is a very important property, when removed the operation becomes just an arbitrary function and it's hard to prove interesting theorems without adding either aa generalized form of or even a new kind of associativity. For now, when we study those structures we just call them by their properties and there is not much done yet in that subject, that should be the research field of the next centries.