Date: Sep 18, 2013 6:13 AM
Author: William Elliot
Subject: Re: Homomorphism of posets and lattices

On Wed, 18 Sep 2013, quasi wrote:

> If X,Y are posets, a function f:X -> Y is called an order
> homomorphism if x <= y implies f(x) <= f(y).
>
> If X,Y are posets, a bijective function f:X -> Y is called an
> order isomorphism if both f and f^(-1) are order homomorphisms.
>
> Posets X,Y are said to be order isomorphic if there exists an
> order isomorphism f:X -> Y.
>
> Questions:
>
> Let X,Y be posets and suppose f:X -> Y and g:Y -> X are
> order homomorphisms.
>
> (1) If f,g are both injective, must X,Y be order isomorphic?
>

No. X = Rx{0,1}; (a,b) <= (r,s) iff a <= r, b = s; Y = R
(x,0) -> arctan x, (x,1) -> pi + arctan x; y -> (y,0).

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

No. X = R - (0,1); Y = R
x -> x if x <= 0
x -> x - 1 if 1 <= x

y -> y if y <= 0
y -> max{ 1,y } if 0 < y

What happens if both are bijective?