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Lecturer(s)
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Halaš Radomír, prof. Mgr. Dr.
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Course content
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1. 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. 2. 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. 3. 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. 4. Modullar and distributive lattices. Complements and relative complements, Boolean lattices, generalized Boolean lattices. Correspondence between congruences and ideals. Boolean algebras.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Understand divisibility theory in integral domains and basics of lattice theory.
Comprehension of basics of divisibility theory in integral domains and basics of lattice theory.
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Prerequisites
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unspecified
KAG/ALG1 and KAG/ALG2 and KAG/MALG3
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Assessment methods and criteria
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Oral exam, Written exam
Credit: attendance at seminars, written test. Exam: understanding of basics of divisibility theory and lattice thery, ability to prove crucial statements
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Recommended literature
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Bican, L. (2000). Lineární algebra a geometrie. Praha, Academia.
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Burris S., Sankappanavar H. P. (1981). A Course in Universal Algebra. Springer-Verlag, New York.
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Halaš R., Chajda I. (1999). Cvičení z algebry. VUP Olomouc.
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Hort D., Rachůnek J. (2003). Algebra1. UP Olomouc.
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Chajda. (1991). Algebra III. Teorie svazů a univerzální algebra.. UP Olomouc.
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Jukl M. (2006). Lineární algebra. Univerzita Palackého Olomouc.
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Rachůnek J. (2003). Svazy. VUP Olomouc.
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