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

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Victor Porton

Posts: 520
Registered: 8/1/05
Re: Homomorphism of posets and lattices
Posted: Sep 18, 2013 8:13 AM
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William Elliot wrote:

> What's your opinion about the following except from a manuscript?
>
> I consider the Definition 2.92 to be in error unless the domain
> of the function is a linear order. For example, the identity
> function from a two point antichain { a,b } to the chain a < b.
>
> Comments? Are not antichains preserved by order isomorphisms?
>
> 2,1,12 Homomorphism of posets and lattices
>
> Definition 2.90. A monotone function (also called order homomorphism)
> from a poset A to a poset B is such a function f that x <= y --> f(x) <=
> f(y).
>
> Definition 2.91. Order embedding is an injective monotone function.
>
> Definition 2.92. Order isomorphism is an surjective order embadding
> (= bijective monotone function).
>
> Order isomorphism preserves properties of posets, such as
> order, joins and meets, etc.
>
> Definition 2.93.


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.

Obvious 3. Order embeddings are always injective.

Definition 4. Order isomorphism is a surjective order embedding.



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