
Re: Homomorphism of posets and lattices
Posted:
Sep 18, 2013 8:13 AM


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.

