Lattices--the distributive inequality

From: Van Jacques (calccurve-test23_at_yahoo.com)
Date: 06/29/04 

Date: 29 Jun 2004 09:56:19 -0700

Consider lattices L as posets with meet(a,b) == a /\ b == glb(a,b),
and join(a,b) == a \/ b == lub(a,b), for any a,b in L.

(I am new to posets and lattices, and would appreciate pointers to any
notes on the web---I don't have a text for this, but I still want to
learn about it.)

I will go ahead and look stupid by asking a very simple question--
I am embarassed to say that I don't see how to proceed.

L.E. means "less than or equal to".

(*) Prove that (a /\ b) \/ (a /\ c) L.E. a /\ (b \/ c)

In words, the join of the meet is L.E. the meet of the join.

With an equal sign, this is the distibutive law for distributive lattices.

We also have the dual to (*), obtained by interchanging /\ with \/,
and changing L.E. to G.E.

Here are some examples of lattices:

1) the power set 2^X == P(X) of X ordered by set-theoretic inclusion,
    including Boolean algebras.
2) the integers Z with L.E. the usual order,

3) the positive integers ordered by divisibility, or dual to this

4) the ideals of Z ordered by set-theoretic inclusion,

5) the subgroups of a groups ordered by set-theoretic inclusion

6) the intermediate fields of a extension E of a field F, again
  ordered by set-theoretic inclusion.

I think all of these are distibutive, though I haven't checked it except for P(X).
Also,

a L.E. b <==> a \/ b = b <==> a /\ b = a

follows from the definitions.

I thought perhaps this could be used to show that (*) is true,
but I still don't see it. I see that the distributive law is true,
but I don't see how to prove the above inequality (*).

Van

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