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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

Arturo Magidin
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M.M
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  • I have never heard that phrase used before. – Dustan Levenstein Mar 18 '12 at 17:08
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    Now you heard about it :) – M.M Mar 18 '12 at 17:15
  • Why is this tagged (data-analysis)? What do you mean by mathematical structure? Can you provide definitions? –  Mar 18 '12 at 17:24
  • It was by mistake , I could not change it – M.M Mar 18 '12 at 17:26
  • Set $X$ is closed under an $n$-ary operator $\ast$ if for all $x_1, x_2, \ldots, x_n \in X,$ we have $\ast(x_1, x_2,\ldots, x_n) \in X.$ That is, the result of applying the operator is an element of $X.$ Examples of the non-closure: $\mathbb{Q}$ under $\sqrt{\cdots}.$ We have $2\in \mathbb{Q}$ but $\sqrt{2} \not\in \mathbb{Q}.$ –  Mar 18 '12 at 17:28
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    @Mohammed: It would help if you could provide an excerpt or link to the place where this terminology was used. – Pete L. Clark Mar 18 '12 at 18:00
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    www.cs.tau.ac.il/~nachumd/ASM/Nondeterminism.pptx slid 3 – M.M Mar 18 '12 at 18:11
  • 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.pptx –  Mar 18 '12 at 21:57

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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 $$ \mathcal{B}\text{ is true for }A\in\mathcal{C}\Leftrightarrow\mathcal{B}\text{ is true for every object in }\mathcal{C}\text{ isomorphic to }A $$

MichalisN
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  • Isn't this more commonly called "invariant under isomorphism" or "preserved under isomorphism"? – jwodder Mar 18 '12 at 20:50
  • @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". – MichalisN Mar 18 '12 at 21:09
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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.

Asterion Daedalus
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