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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.

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    $\begingroup$ You have probably already looked at this but do any of these quasigroups satisfy your requirements? en.wikipedia.org/wiki/Quasigroup#Loop $\endgroup$
    – Nick Bishop
    Commented Dec 29, 2019 at 21:01
  • $\begingroup$ @NickBishop I don't think it's a quasigroup because elements can have more than one inverse $\endgroup$
    – lightning
    Commented Dec 29, 2019 at 21:16
  • $\begingroup$ 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? $\endgroup$
    – Rob Arthan
    Commented Dec 29, 2019 at 23:39
  • $\begingroup$ @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 $\endgroup$
    – lightning
    Commented Dec 30, 2019 at 1:06

1 Answer 1

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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.

enter image description here

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.

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  • $\begingroup$ So binary trees are uninteresting? $\endgroup$
    – Rob Arthan
    Commented Dec 29, 2019 at 23:51
  • $\begingroup$ No, I never said that what I said is "it's hard to prove interesting theorems" which implicitly assumes that there are interesting theorems everywhere, it's just hard to prove them without strong axioms. $\endgroup$
    – Elaqqad
    Commented Dec 30, 2019 at 0:01
  • $\begingroup$ The Op's question is what do we call them, my answer is - maybe not a satisfying one- just saying they are commutative non-associative groups because as far as I know there is no name for those structures at least in abstract algebra (maybe in category theory or functional programming they have a name but I don't know). $\endgroup$
    – Elaqqad
    Commented Dec 30, 2019 at 0:08
  • $\begingroup$ A structure comprising a set equipped with a binary operator is called a magma (see the link in my comment on the question). And I really don't understand why you claim it's hard to prove interesting theorems about binary trees. $\endgroup$
    – Rob Arthan
    Commented Dec 30, 2019 at 0:14
  • $\begingroup$ I never said that either ("hard to prove theorems about binary trees"), Anyways I read your comment and I actually agree with your naming "commutative magmas with identity and inverses" I am including it in the answer if you don't mind. $\endgroup$
    – Elaqqad
    Commented Dec 30, 2019 at 0:40

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