HOMEWORK 3 - LATTICE THEORY
A closure rule is nullary if it has the form x S, and unary if it
is of the form y S = z S. Prove that if is a collection
of nullary and unary closure rules, then nonempty unions of
closed sets are closed, and hence the lattic

OUTLINE OF CLOSURE OPERATORS
A nite meet semilattice with 1 is a lattice.
A complete lattice is an ordered set in which every subset has a
meet and join.
A complete meet semilattice is an ordered set in which every
subset has a meet.
Every complete me

THREE EASY EXERCISES
1. Flats in Affine Geometry
We want to describe the at containing points a1 , . . . , an in the ane
geometry F n for a eld F . This will be a at of geometric dimension at
most k 1. Show that the following are equivalent for a subset S

SPERNERS LEMMA: SUMMARY
RALPH FREESE
Let P be a nite ordered set and let L be the set of all maximal-sized
antichains.
Theorem 1 (Dilworth). L is a lattice.
Let G = Aut(P ) be the group of automorphisms of P . If A L and
G then (A) L; so G acts on L. Sin

11. Geometric Lattices
Manys the time Ive been mistaken
And many times confused . . . .
Paul Simon
Now let us consider how we might use lattices to describe elementary geometry.
There are two basic aspects of geometry: incidence, involving such statements

2. Semilattices, Lattices and Complete Lattices
Theres nothing quite so ne
As an earful of Patsy Cline.
Steve Goodman
The most important partially ordered sets come endowed with more structure
than that. For example, the signicant feature about PO(X) for

8. Distributive Lattices
Every dog must have his day.
In this chapter and the next we will look at the two most important lattice
varieties: distributive and modular lattices. Let us set the context for our study of
distributive lattices by considering va

1. Ordered Sets
And just how far would you like to go in? he asked.
Not too far but just far enough sos we can say that weve been there, said
the rst chief.
All right, said Frank, Ill see what I can do.
Bob Dylan
In group theory, groups are dened algebrai