
Re: Homomorphism of posets and lattices
Posted:
Sep 20, 2013 6:22 AM


quasi <quasi@null.set> wrote: > quasi <quasi@null.set> wrote: > > > >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)). > > A better choice of letters: > > Given posets A,B, a function f:A > B 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
Also one could note that, once the notion of homomorphism is settled, the contemporary way of defining isomorphism applies:
 An isomorphism f: A > B is a homomorphism with a two sided inverse,  i.e. a homomorphism g: B > A with f o g = id_B and g o f = id_A.
Then the statement in the above "Definition 4" becomes a description: Order isomorphism are exactly the surjective order embeddings.
 Marc

