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What's the name of an algebraic structure that doesn't satisfy the axiom of closure?

For example, if a magma is composed of a set and a operation, which satisfies closure, what would we call the same structure, without the axiom of closure?

Federico Gremmo
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    Closure isn't really an axiom. It's part of the definition of the word, "operation". – Gerry Myerson Aug 02 '15 at 09:44
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    I don't see why we should call something a structure when we don't even know what happens when we perform some 'operation' on the objects in it. – user21820 Aug 02 '15 at 10:55
  • I ain't been sleeping at night for similar reasons: http://math.stackexchange.com/questions/796374/binary-operation-english-terminology – MattAllegro Aug 02 '15 at 14:42
  • @user21820 https://en.wikipedia.org/wiki/Partial_groupoid For that matter, a category is really just a "partial monoid" in a sense. – Noah Schweber Oct 13 '22 at 00:09
  • @NoahSchweber: Ok thanks for the comment 7 years later! Anyway my comment wasn't really serious; it's simply that if we know nothing about what happens then it doesn't have any actual structure, at least not in any usual sense of the word "structure". =) – user21820 Oct 13 '22 at 07:54
  • @user21820 Whoops, sorry, this question got bumped (by a now-deleted answer) and I didn't notice the timestamp. – Noah Schweber Oct 13 '22 at 18:35

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The general term to look for here is partial. A partial function $X\rightarrow Y$ is a function with codomain $Y$ and domain some subset of $X$; when partiality is a possibility, the term "total" is used for partial functions defined everywhere.

There are definitely contexts where partial functions play important roles. Dearest to my heart is computability theory, where one of the earliest fundamental realizations was that the set of partial computable functions is more natural in a sense than the set of total computable functions. (One way to make this precise is the fact that there is a partial computable binary function $p(x,y)$ such that for all partial computable unary functions $q(x)$ there is an $n$ with $\lambda x.p(x,n)\simeq \lambda x.q(x)$, but there is no similarly universal total computable function due to diagonalization; relatedly, the recursion theorem can be thought of as a "failed diagonalization argument" within the set of partial computable functions.)

Partiality shows up in algebraic structures, too. The term partial groupoid is used to refer to a set equipped with a partial binary operation, while partial algebra is the natural universal algebraic generalization; the papers A survey of partial groupoids (Evseev) and Partial algebras - an introductory survey (Burmeister) each seem relevant. Additionally, a category can be thought of as partial algebraic structure, the operation being composition of morphisms (since in general not all pairs of morphisms will be composable).

Noah Schweber
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