Date: Sep 20, 2013 6:22 AM
Author: @less@ndro
Subject: Re: Homomorphism of posets and lattices
quasi <quasi@null.set> wrote:

> quasi <quasi@null.set> wrote:

> >

> >Victor Porton wrote:

> >>

> >>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.

> >

> >Not quite right.

> >

> >To have an inverse, it would have to be bijective.

> >

> >Instead, try this (Wikipedia's version):

> >

> >Given posets X,Y, a function f:X -> Y is called an order

> >embedding if (x <= y iff f(x) <= f(y)).

>

> A better choice of letters:

>

> Given posets A,B, a function f:A -> B is called an order

> embedding if (x <= y iff f(x) <= f(y)).

>

> >>Obvious 3. Order embeddings are always injective.

> >

> >Yes.

> >

> >>Definition 4. Order isomorphism is a surjective order embedding.

> >

> >Yes, that's Wiki's definition.

>

> quasi

Also one could note that, once the notion of homomorphism is settled,

the contemporary way of defining isomorphism applies:

| An isomorphism f: A --> B is a homomorphism with a two sided inverse,

| i.e. a homomorphism g: B --> A with f o g = id_B and g o f = id_A.

Then the statement in the above "Definition 4" becomes a description:

Order isomorphism are exactly the surjective order embeddings.

--

Marc