Universal Algebra and Lattices
Code
8529
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
2
Teaching language
Português
Objectives
Students should acquire basic knowledge on Universal Algebra and Lattice Theory, which will open horizons to future deeper studies in the area and to the study of applications to Theoretical Computer Science.
Prerequisites
Basic knowledge on some algebraic structures, namely, groups and rings.
Subject matter
1. Partially ordered sets. Lattices. Complete lattices. Algebraic lattices. Modular lattices. Distributive lattices. Boolean lattices and Boolean algebras.
2. Algebras. Homomorphisms. Subuniverses and subalgebras. Congruences. Direct products.
3. Homomorphism and Isomorphism theorems. Factor congruences and directly indecomposable algebras. Subdirect products. Subdirectly irreducible algebras and simple algebras.
4. Class operators and varieties. Tarski theorem.
5. Free algebras. Terms and term-algebras. Identities. Birkhoff theorem.
Bibliography
1 – Burris, S. & Sankappanavar, H. P. – A Course in Universal Algebra – Springer Verlag, New York, 1981.
2 – Davey, B. A. & Priestley, H. A. – Introduction to Lattices and Order, Cambridge Mathematical Textbooks, 1990.
3 – Denecke, K. & Wismath, S. L. – Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall/CRC, Boca Raton, Florida, 2002.
4 – Gratzer, G. – Lattice Theory: Foundation, Birkhauser Verlag, Basel, 2011.
5 – McKenzie, R. N. & McNulty, G. F. & Taylor, W. F. – Algebras, Lattices, Varieties Vol. I – Wadsworth & Brooks, California, 1987.