quasi
Posts:
10,197
Registered:
7/15/05


Re: Homomorphism of posets and lattices
Posted:
Sep 18, 2013 2:02 AM


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

