Date: Sep 18, 2013 2:18 PM
Author: quasi
Subject: Re: Homomorphism of posets and lattices

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