Selected topics in universal algebra and lattices

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Course Description:

Students will see some of the most exciting results in the theory of ordered sets and other parts of discrete mathematics, including unsolved, open problems that they might be able to solve after taking this course.

• Definitions of poset and order-preserving map; introduction to the fixed point property for posets; dismantlable posets; Tarski\u2019s fixed point theorem; the arithmetic of ordered sets and Hashimoto\u2019s Theorem; Roddy\u2019s theorem on products of posets with the fixed point property
• Introduction to lattices, including distributive and semimodular lattices of finite height; examples of free distributive lattices generated by finite posets; open problems in combinatorics concerning unimodality, including Rota\u2019s conjecture and symmetric chain decompositions of L(m,n)
• Hall's Marriage Theorem, geometric lattices; introduction to matroids; Nash-William\u2019s theorem on sums of matroids
• The Wide Partition Conjecture

Textbooks:

• Davey and Priestley, Introduction to Lattices and Order (2nd edition).
• Bryant and Perfect, Independence Theory in Combinatorics.
• various research papers

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