Define the "model theoretic" notion of a closure function as follows:

**Definition (1):** Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the following properties:

$(1)~\forall A\subseteq D~~~~A\subseteq cl(A)$

$(2)~\forall A,B\subseteq D~~~~A\subseteq B\Longrightarrow cl(A)\subseteq cl(B)$

$(3)~cl(cl(A))=cl(A)$

We say that $cl$ is a "good" closure on $D$ if it has the following property too:

$(4)~\forall A\subseteq D~~~~cl(A)=\bigcup_{B\in P_{<\omega}(A)}cl(B)$

and a "pregeometry" if we have:

$(5)~\forall A\subseteq D~~~~\forall a,b\in D~~~~~~a\in cl(A\cup \lbrace b\rbrace)\setminus cl(A)\Longrightarrow b\in cl(A\cup \lbrace a\rbrace)$

There are many trivial good closures on a given set which are not related to any structure, but even there are some non-trivial natural good closures on the domain of an arbitrary $\mathcal{L}$-structure $\mathcal{M}$ like well known "algebraic closure" ($acl_{\mathcal{M}}$), "definable closure" ($dcl_{\mathcal{M}}$) and "structural closure" ($scl_{\mathcal{M}}$) which is defined as follows:

$\forall A\subseteq Dom(\mathcal{M})~~~~~scl_{\mathcal{M}}(A):=Dom(\langle A\rangle_{\mathcal{M}})$

and $\langle A\rangle_{\mathcal{M}}$ is the substructure of $\mathcal{M}$ generated by the set $A$.

Now the main question is about the "essential" properties of these closure functions on an arbitrary structure. In the other words is "goodness" the unique essential property of $acl_{\mathcal{M}}$, $dcl_{\mathcal{M}}$ and $scl_{\mathcal{M}}$ on an arbitrary $\mathcal{L}$ - structure $\mathcal{M}$? Precisely:

**Question (1):** Let $D$ be an arbitrary non-empty set, and $cl:P(D)\longrightarrow P(D)$ be a good closure function on $D$, is there a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $\mathcal{M}$ such that: $Dom(\mathcal{M})=D$ and $scl_{\mathcal{M}}=cl$?

**Question (2):** What is the answer of question (1) for $acl_{\mathcal{M}}$ and $dcl_{\mathcal{M}}$?

**Remark (1):** Note that producing a negative answer for the above questions needs finding a property $P$ different from "being a good closure" and proving that the functions $acl$, $dcl$ and $scl$ have the property $P$ on "any" structure and so there is no such language and structure for an arbitrary function $cl:P(D)\longrightarrow P(D)$ because it is possible that such function have the property $\neg P$.So one can re ask the above questions by the following re stating:

"Let $D$ be an arbitrary non-empty set, and $cl:P(D)\longrightarrow P(D)$ be a good closure function with property $P$ on $D$, is there a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $\mathcal{M}$ such that: $Dom(\mathcal{M})=D$ and $scl_{\mathcal{M}}=cl$ ($acl_{\mathcal{M}}=cl$ or $dcl_{\mathcal{M}}=cl$)?

So we can go further and try to characterize "all" essential properties of $scl, acl, dcl$ on an arbitrary structure and ask the following question:

**Question (3):** What is the property $P$ such that for any non-empty set $D$ and any function $cl:P(D)\longrightarrow P(D)$ which is a good closure function with property $P$, the answer of the question (1) or (2) be true?

In the other direction It is well known that $acl_{\mathcal{M}}$ is a pregeometry on "strongly minimal" structures. So:

**Question (4):** Is there a known type of $\mathcal{L}$-structures which the functions $scl$ or $dcl$ be a pregeometry on them?