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

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quasi

Posts: 9,898
Registered: 7/15/05
Re: Homomorphism of posets and lattices
Posted: Sep 18, 2013 1:55 PM
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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)).

>Obvious 3. Order embeddings are always injective.

Yes.

>Definition 4. Order isomorphism is a surjective order embedding.

Yes, that's Wiki's definition.

quasi



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