quasi
Posts:
10,741
Registered:
7/15/05


Re: Homomorphism of posets and lattices
Posted:
Sep 18, 2013 2:18 PM


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

