Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
quasi

Posts: 10,399
Registered: 7/15/05
Re: Homomorphism of posets and lattices
Posted: Sep 18, 2013 2:02 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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
>function.
>
>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.

quasi



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.