
Re: Homomorphism of posets and lattices
Posted:
Sep 18, 2013 6:13 AM


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?

