by Nate Soares May 9 2016 updated Jun 7 2016

A set $~$S$~$ is closed under an operation $~$f$~$ if, whenever $~$f$~$ is fed elements of $~$S$~$, it produces another element of $~$S$~$. For example, if $~$f$~$ is a trinary operation (i.e., a function of three arguments) then "$~$S$~$ is closed under $~$f$~$" means "if $~$x, y, z \in S$~$ then $~$f(x, y, z) \in S$~$".

For example, the set [integer $~$\mathbb Z$~$] is closed under addition (because adding two integers yields another integer), but the set $~$\mathbb Z_5 = \{0, 1, 2, 3, 4, 5\}$~$ is not (because $~$1 + 5$~$ is not in $~$\mathbb Z_5$~$).