Date: Sep 18, 2013 2:18 PM
Subject: Re: Homomorphism of posets and lattices
quasi <firstname.lastname@example.org> 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.
>>Definition 4. Order isomorphism is a surjective order embedding.
>Yes, that's Wiki's definition.