quasi
Posts:
11,911
Registered:
7/15/05


Re: Homomorphism of posets and lattices
Posted:
Sep 19, 2013 12:55 PM


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

