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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Order embedding
Posted: Sep 16, 2013 4:14 AM
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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).

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

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

The following are immediate consequences.

Order embedding maps and order isomorphisms are injections.
If f:X -> Y is an order embedding,
then f:X -> f(X) is an order isomorphism.

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

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

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

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