quasi
Posts:
10,671
Registered:
7/15/05


Re: Homomorphism of posets and lattices
Posted:
Sep 18, 2013 1:55 PM


Victor Porton wrote: > >I've corrected the definitions: > >Definition 1. A monotone function (also called order >homomorphism) from a poset A to a poset B is such a function f >that x<=y > fx<=fy. > >Definition 2. Order embedding is an monotone function whose >inverse is also monotone.
Not quite right.
To have an inverse, it would have to be bijective.
Instead, try this (Wikipedia's version):
Given posets X,Y, a function f:X > Y is called an order embedding if (x <= y iff f(x) <= f(y)).
>Obvious 3. Order embeddings are always injective.
Yes.
>Definition 4. Order isomorphism is a surjective order embedding.
Yes, that's Wiki's definition.
quasi

