Date: Sep 16, 2013 11:32 AM
Author: David C. Ullrich
Subject: Re: Order embedding
On Mon, 16 Sep 2013 14:24:06 +0300, Victor Porton <porton@narod.ru>

wrote:

>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.

I think no. Surely an embedding is required to be injective.

>

>> f:X -> Y is an order isomorphism when f is surjective

>> and for all x,y, (x <= y iff f(x) <= f(y)).

>

>Yes.

No. An isomorphism must be bijective.

>> 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.