Algebraic structures without the axiom of closure 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?
 A: 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).
