Universal Algebra
Code
9623
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Júlia Maria Nunes Loureiro Vaz de Carvalho
Weekly hours
4
Teaching language
Português
Objectives
By the end of this unit the student should have acquired knowledge, skills and competences that allow him to:
1) Understand, apply and work with concepts and results of universal algebra.
2) Be able to read and explain a simple research paper.
Prerequisites
Basic knowledge on some algebraic structures, namely, groups and rings.
Subject matter
1. Elements of Universal Algebra
Algebras and operations
Subalgebras, homomorphisms and direct products
Subalgebras generated by sets
Congruence relations and quotient algebras
Algebras and clones
Isomorphism theorems
Direct products, factor congruences, directly indecomposable algebras
Subdirect products, subdirectly irreducible algebras, simple algebras
Class operators and varieties
Tarski Theorem
Terms, Term algebra, free algebra
Identities, Birkhoff Theorem
Mal''cev conditions
2. Complementation, Boolean algebras
Boolean algebras and Boolean rings
Ideals and filters
Stone duality
Ultraproducts and congruence-distributive varieties
3. Pseudocomplementation, Stone algebras, Heyting algebras
4. Ockham algebras
Bibliography
T.S. Blyth and J.C. Varlet, Ockham Algebras, Oxford University Press, 1994.
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer, 1981.
B.A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge University Press, 2002.
G. Grätzer, General Lattice Theory, Birkhäuser, 1978.
P. Halmos, Lectures on Boolean algebras, Van Nostrand, Princeton, 1963.
R. N. McKenzie, G. F. McNulty, and W. F. Taylor, Algebras, Lattices, Varieties, vol. I, Wadsworth and Brooks, 1987.