Distributive Lattices
Code
10832
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
Total hours
56
Teaching language
Português
Objectives
It is intended that the student acquires knowledge of lattice theory, focusing their study in the important subclasses of distributive lattices and Boolean lattices. It is also intended that the student becomes aware that the lattice structure arises in several areas and realizes that different areas of mathematics are connected, applying notions and results of topology to the study of lattices.
Prerequisites
Basic knowledge on topology, in particular on compact and Hausdorff spaces.
Subject matter
Chapter 1: Lattice (as a poset and as an algebra). Sublattice.
Homomorphism. Ideal and filter. Prime ideal (filter) and maximal ideal (filter).
Congruence relation. Direct product.
Complete lattice and algebraic lattice.
Chapter 2: Distributive lattice. Bounded distributive lattice.
Some properties in distributive lattices.
Caracterizations of distributive lattices, in particular through forbidden substrutures.
Prime ideal theorem.
Boolean lattice and boolean algebra.
Chapter 3: Priestley duality for bounded distributive lattices.
Special cases: Stone duality for Boolean lattices; representation of finite distributive lattices.
Bibliography
1. R. Balbes & P. Dwinger, Distributive Lattices, University of Missouri Press, 1974.
2. T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, 2005.
3. B. Davey & H. A. Priestley, Introduction to Lattices and Order (2nd Edition), Cambridge University Press, 2002.
4. G. Grätzer, Lattice Theory – first concepts and distributive lattices, W. H. Freeman and Company, 1971.
5. G. Grätzer, Lattice Theory: Foundation, Springer, 2011.
Teaching method
Classes consist on an oral explanation of the theory which is illustrated by examples, and the correction of the students'''' resolution of exercises.
Students can ask questions during classes, in weekly scheduled sessions or in special sessions accorded directly with the professor.
Continuous evaluation consists in 3 mid-term tests that substitute the final exam in case of approval. In case of failure, the student must pass the final exam.
Evaluation method
Continuous evaluation consists in 3 mid-term tests that substitute the final exam in case of approval. In case of failure, the student must pass the final exam.