Date: Sep 19, 2013 12:55 PM
Author: quasi
Subject: Re: Homomorphism of posets and lattices
William Elliot wrote:

>William Elliot wrote:

>> quasi wrote:

>> >

>> > Questions:

>> >

>> > Let X,Y be posets and suppose f:X -> Y and g:Y -> X are

>> > order homomorphisms [order preserving maps].

>> >

>> > (1) If f,g are both injective, must X,Y be order isomorphic?

>>

>> No.

>>

>> > (2) If f,g are both surjective, must X,Y be order isomorphic?

>>

>> No.

>>

>> What happens if both are bijective?

>

>Clearly, for all x,y, (x <= y --> f(x) <= f(y)).

>If f(x) <= f(y): fgg^-1(x) <= fgg^-1(y)

>

>gf(x) <= gf(y)

Presumably you are trying to answer the question:

If X,Y are posets for which there are bijective order

preserving maps f:X -> Y and g:Y -> X, must X,Y be order

isomorphic?

As far as I can see, what you posted above has no impact on

the question.

quasi