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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 2:02 AM
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William Elliot wrote:

>What's your opinion about the following except from a manuscript?

You should credit your source.

Manuscript by who?

Published when and where ?

>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?

Yes, of course.

As your counterexample clearly shows, the author's definition
2.92 is flawed.

>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
>Definition 2.92. Order isomorphism is an surjective order
>embedding (= bijective monotone function).

As noted, the above definition is flawed, but the fix is easy.

Definition 2.92 (possible revision):

An order isomorphism is a bijective function such that both it
and its inverse are monotone functions.

>Order isomorphism preserves properties of posets, such as
>order, joins and meets, etc.

With the above suggested revision of definition 2.92, those
properties _are_ preserved.


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