```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.> > quasiAlso 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
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