Universal Algebra and Lattices
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.
General characterization
Code
8529
Credits
6.0
Responsible teacher
Herberto de Jesus da Silva
Hours
Weekly - 4
Total - Available soon
Teaching language
Português
Prerequisites
Basic knowledge on some algebraic structures, namely, groups and rings.
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.
Teaching method
Classes consist on an oral explanation of the theory which is illustrated by examples and the resolution of some exercises.
Evaluation method
There are three mid-term tests. These tests can substitute the final exam if the student has grade, at least, 7.5 in the third one and CT is, at least, 9.5. CT is the arithmetic mean of the non-rounded grades of the tests.
To be approved in final exam, the student must have a minimum grade of 9.5 in it.
More detailed rules are available in the portuguese version.
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.