Appendix 3: Formal Concept Analysis
Exercise 13 of Chapter 2 is to show that a binary relation R A B induces a
pair of closure operators, described as follows. For X A, let
(X ) = cfw_b B : x R b for all x X .
Similarly, for Y B , let
(Y ) = cfw_a A : a
9. Modular and Semimodular Lattices
To dance beneath the diamond sky with one hand waving free .
The modular law was invented by Dedekind to reect a crucial property of the
lattice of subgroups of an abelian group, or more generally the lattice
10. Finite Lattices and their Congruence Lattices
If memories are all I sing I'd rather drive a truck. Ricky Nelson In this chapter we want to study the structure of finite lattices, and how it is reflected in their congruence lattices. There are differen
11. Geometric Lattices
Many's the time I've 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 statemen
Appendix 2: The Axiom of Choice
In this appendix we want to prove Theorem 1.5. Theorem 1.5. The following set theoretic axioms are equivalent. (1) (Axiom of Choice) If X is a nonempty set, then there is a map : P(X) X such that (A) A for every nonempty A
Appendix 1: Cardinals, Ordinals and Universal Algebra
In these notes we are assuming you have a working knowledge of cardinals and ordinals. Just in case, this appendix will give an informal summary of the most basic part of this theory. We also include a
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
7. Varieties of Lattices
Variety is the spice of life. A lattice equation is an expression p q where p and q are lattice terms. Our intuitive notion of what it means for a lattice L to satisfy p q is that p(x1 , . . . , xn ) = q(x1 , . . . , xn ) whenever
2. Semilattices, Lattices and Complete Lattices
There's nothing quite so fine As an earful of Patsy Cline. Steve Goodman The most important partially ordered sets come endowed with more structure than that. For example, the significant feature about PO(X)
1. Ordered Sets
"And just how far would you like to go in?" he asked. "Not too far but just far enough so's we can say that we've been there," said the first chief. "All right," said Frank, "I'll see what I can do." Bob Dylan In group theory, groups are d
3. Algebraic Lattices
The more I get, the more I want it seems .
In this section we want to focus our attention on the kind of closure operators
and lattices that are associated with modern algebra. A closure operator on a set
X is said to be
4. Representation by Equivalence Relations
No taxation without representation!
So far we have no analogue for lattices of the Cayley theorem for groups, that
every group is isomorphic to a group of permutations. The corresponding representation theorem fo
6. Free Lattices
Freedoms just another word for nothing left to lose .
If x, y and z are elements of a lattice, then x (y (x z ) = x y is always
true, while x y = z is usually not true. Is there an algorithm that, given two lattice
5. Congruence Relations
"You're young, Myrtle Mae. You've got a lot to learn, and I hope you never learn it." Vita in "Harvey" You are doubtless familiar with the connection between homomorphisms and normal subgroups of groups. In this chapter we will est