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Topic: Order embedding
Replies: 8   Last Post: Sep 16, 2013 10:01 PM

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Victor Porton

Posts: 520
Registered: 8/1/05
Re: Order embedding
Posted: Sep 16, 2013 7:24 AM
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William Elliot wrote:

> Let X,Y be (partially) ordered sets. Are these definitions correct?
>
> f:X -> Y is order preserving when
> for all x,y, (x <= y implies f(x) <= f(y).


Yes.

> f:X -> Y is an order embedding when
> for all x,y, (x <= y iff f(x) <= f(y)).


Yes.

> f:X -> Y is an order isomorphism when f is surjective
> and for all x,y, (x <= y iff f(x) <= f(y)).


Yes.

> The following are immediate consequences.
>
> Order embedding maps and order isomorphisms are injections.


Yes.

> If f:X -> Y is an order embedding,
> then f:X -> f(X) is an order isomorphism.


Yes.

> Furthermore the composition of two order preserving, order
> embedding or order isomorphic maps is again resp., order


> preserving, order embedding or order isomorphic.

Yes.

> Finally, the inverse of an order isomorphism is an order isomorphism.

Yes.

> That all is the basics of order maps, is it not?
> Or is the more to be included?


Probably all.

There are also Galois connections.



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