Definition of "closure under isomorphism" What does " Closed Under Isomorphism " means ? 
why do we need it ?
And how can we use it in " Mathematical Structures" 
thanks
 A: Since nobody answered this question I will give it a try.
I encountered closure under isomorphism in the definition of a Markov property in group theory (doesn't correspond to the Markov property I find on Wikipedia). One of the requirements for a property of groups $\mathcal{B}$ to be a Markov property is that it is closed under isomorphism
$$
\mathcal{B}\text{ is true for a group }G\Leftrightarrow\mathcal{B}\text{ is true for every group isomorphic to }G
$$
Most (arguably all) properties one considers in group theory (finite, abelian, ...) are closed under group isomorphism.
So I would generally define a property $\mathcal{B}$ of objects in a category $\mathcal{C}$ to be closed under isomorphism iff 
$$
\mathcal{B}\text{ is true for }A\in\mathcal{C}\Leftrightarrow\mathcal{B}\text{ is true for every object in }\mathcal{C}\text{ isomorphic to }A
$$
A: My first reaction to the phrase is that it would be used in a context such as
The category $\mathcal{C}$ is a subcategory of $\mathcal{D}$. We say that $\mathcal{C}$ is closed under isomorphism iff:
If $f : X \to Y$ is an arrow of $\mathcal{C}$ and $g : Y \to Z$ and $h : W \to X$ are both isomorphisms in $\mathcal{D}$, then $gf$ and $fh$ are both arrows of $\mathcal{C}$.
A: I came across the statement:
For any isomorphism-closed class of finite structures, there is a first-order theory that defines it.
That is:
A property of finite structures is any isomorphism-closed class of structures.
So:
Any class of finite structures closed under isomorphisms is axiomatised by a
first-order theory.
