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