KAG/ALG4 Algebra 4

Lecturers: Radomír Halaš, Jan Kühr
Lecture: 2 hours/week + tutorial 1 hour/week
Credits: 3
Summer semester
Form of course completion: course credit, exam

• Divisibility in integral domains. Units, irreducible and prime elements. Greatest common divisor, least common multiple. Ideal generated by a set, pricipal ideal domains. Euclidean domains, Gaussian domains.
• Partially ordered sets. Mappings of partially ordered sets: monotone, antitone, isomorphic embedding, isomorphism. Distinguished elements: maximal, minimal, greatest, least. Lower and upper cone of a set, directed sets. Supremum and infimum, semilattices. The Zorn lemma.
• Lattices: Partially ordered sets and algebras. Complete lattices, the fixed point theorem. Sublattices. Lattice homomorphisms and congruence relations. Quotient lattices, homomorphism theorem. Ideals (and filters) of lattices. Ideal generated by a set, principal ideals.
• Modullar and distributive lattices. Complements and relative complements, Boolean lattices, generalized Boolean lattices. Correspondence between congruences and ideals. Boolean algebras.