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Topic: Homomorphism of posets and lattices
Replies: 16   Last Post: Sep 20, 2013 6:22 AM

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Posts: 12,067
Registered: 7/15/05
Re: Homomorphism of posets and lattices
Posted: Sep 18, 2013 2:18 PM
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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.

>>Definition 4. Order isomorphism is a surjective order embedding.
>Yes, that's Wiki's definition.


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