
Re: Invariance of Sup
Posted:
Feb 16, 2014 1:58 PM


William Elliot <marsh@panix.com> wrote: > On Thu, 13 Feb 2014, quasi wrote: > > William Elliot wrote: > > > > > >Let X,Y be two lattices and f:X > Y a surjective lattice map, > > >i.e., for all x,y, > > > f(x inf y) = f(x) inf f(y), f(x sup y) = f(x) sup f(y), > > >or with shorthand, > > > for all x,y, f(xy) = f(x).f(y), f(x + y) = f(x) + f(y). > > > > > >If X is a complete lattice, is Y a complete lattice? > > > > Not necessarily. In the article > > Quotients of Complete Ordered Sets > > Maurice Pouzet & Ivan Rival > > Algebra Universalis, Vol 17 (1983), pages 393405 > > the authors present this example ... > > > > Let N denote the set of natural numbers. > > Let X be the power set of N. > > > Define a congruence on X by > > A ~ B if (A  B) U (B  A) is finite. > > > Let Y = X/~. Let f be the natural map from X onto Y. > > What map is that? > f:P(N) > P(N)/~, A > { B  A ~ B }?
Yes. Actually, that counterexample is also Exercise V19 in Birkhoff's Lattice Theory.
The above relation is in fact an example of a general construction: let X be a supsemilattice and D be an ideal of X (i.e. D is downwards closed and closed under binary sup) and define
a ~ b iff there exists a d in D with a \/ d = b \/ d.
Then ~ is an equivalence and also a congruence w.r.t. \/. If X happens to be a _distributive_ lattice then ~ is also a congruence w.r.t. /\.
Now the finite subsets form an ideal in P(N) and for subsets A and B of N you have
(A \ B) U (B \ A) finite <==> there is a finite subset F with A U F = B U F
 Marc

