Closure operators

Question

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

1. $A \subseteq I(A)$ ($I$ is extensive)

2. $A \subseteq B \implies I(A)\subseteq I(B)$

3. $I(I(A))=I(A)$

4. $I(\emptyset)=\emptyset$

Is it possible to replace property (2) with another weaker property? Is there something in literature about that?

Answer 1

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).

Answer 2

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.