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




Re: Order embedding
Posted:
Sep 16, 2013 7:24 AM


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.



