Math 222A, Winter 2003

Lattices and Algebraic Systems

MWF 12-1, MS 6118

This course is open to anyone who has completed Math 210A.

The structures studied are of interest for algebra, logic, and combinatorics. The course will be a one-quarter version of what is listed in the catalog for Math 222AB, including partially ordered sets; lattices, distributive lattices, and Boolean algebras; algebraic systems and their laws and structure; and applications. Some sidelights:

• What is Boolean duality?

• What are ultraproducts?

• Why do the two nonabelian groups of order 8, the dihedral and quaternion groups, satisfy exactly the same equations (laws) in spite of seeming so different from one another?

• Do there exist finite algebraic systems whose laws are not finitely describable?

• It is easy to find examples of rings or abelian groups that are not isomorphic to a direct product of ``directly indecomposable'' factors. How does the use of ``subdirect products'' instead make it possible to decompose any kind of algebraic systems into indecomposable pieces?

• What are the five basic kinds of algebraic systems that serve as tags on the structure of arbitrary finite algebraic systems?

Kirby Baker