
Order embedding
Posted:
Sep 16, 2013 4:14 AM


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?

