Closure

https://en.wikipedia.org/wiki/Totality https://en.wikipedia.org/wiki/Closure_(mathematics)

The axiom of closure or totality is a property of a operation over a mathematical structure, for example over a set; more precisely, it is a property of the structure and operation co-system

a mathematical structure, e.g. a set, together with an operation over that set

A set is closed under an operation if performance of that operation on members of the set always produces a member of that set.

If combining any two elements of the set gives an element that is also a member of that set, we say that the binary operation ("star", *) closes over the set.

$(\forall x,y \in A) \to (x \star y \in A)$

A binary operation, *, that combines any two elements of the set A must give a result that is also in A.

For example, addition is closed over the natural numbers, but division is not.

The closure is increasing or extensive: the closure of an object contains the object. The closure is idempotent: the closure of the closure equals the closure. The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

Totality

https://en.wikipedia.org/wiki/Totality

(see Closure)

If combining any two elements of the set gives an element that is also a member of that set, we say that the binary operation ("star", *) closes over the set.

$(\forall x,y \in A) \to (x \star y \in A)$

A binary operation, *, that combines any two elements of the set A must give a result that is also in A.

For example, addition is closed over the natural numbers, but division is not.