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What does " Closed Under Isomorphism " means ? why do we need it ? And how can we use it in " Mathematical Structures" thanks

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    If you can't read PPTX, here's a Google docs viewer link https://docs.google.com/viewer?url=www.cs.tau.ac.il%2F~nachumd%2FASM%2FNondeterminism.pptx2012-03-18

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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 $

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    @jwodder: True, I would probably also use "preserved under isomorphism". Andrew Glass, who taught a course on decision problems in group theory last year, used "closed under isomorphism".2012-03-18
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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 $\mathcal{C}$.

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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.