Closure operators Can a closure operator be not isotone?
The definition below is a pretty standard definition of closure operator:


*

*$A \subseteq I(A)$ ($I$ is extensive)

*$A \subseteq B \implies I(A)\subseteq I(B)$

*$I(I(A))=I(A)$

*$I(\emptyset)=\emptyset$ 
Is it possible to replace property (2) with another weaker property? Is there something in literature about that?
 A: A more standard set of axioms for a closure operation replaces 2. by 
$$I(A \cup B)=I(A) \cup I(B)\tag{2'}$$
and from 2', 2 follows easily. A more general Cech-closure space has 1, 2' and 4 but not 3. This is obeyed by sequential closure (in general topological spaces).
A: I have in mind to set up a theoretical closure that is not isotone.
I am trying to do the relationship of min-bounding hyper-spheres and closures.


*

*For example, we can set the operator $H(A)$ for $A\in \mathcal{P}(\mathbb{R}^n)$ as the minimum hyper-sphere that encloses $A$. 
Easy to see that $H(H(A))$ is idempotent and $A\subseteq H(A)$ extensive, but my trouble is that is not isotone so it does not fullfil axioms of closure operators.


In this case, the space $\mathfrak{C}=\{H(A):A\in\mathcal{P}(\mathbb{R}^n)\}$ together with the inclusion order is not a meet sublattice of $\mathcal{P}(L)$ (case of closures). 
Instead of, I think that under certain restrictions, is possible to define a meet and a join:
$H(A)\wedge H(B)=\bigwedge\{C\in\mathfrak{C}:(A \cap B\subseteq C\}=H(A\cap B)$
and 
$H(A)\vee H(B)=\bigwedge\{C\in\mathfrak{C}: A\cup B \subseteq C \}=H(A\cup B)$
Anyone know where can I found some information and references about something like this?
Thanks.
