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Lecturer(s)
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Chajda Ivan, prof. RNDr. DrSc.
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Course content
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The course is an introduction to partially ordered sets and the attention is paid to particular partially ordered sets -- lattices. Basic properties of lattices are introduced: lattices as ordered sets, lattices as algebraic structures, completeness of lattices, properties given by identities (e.g., modularity, distributivity). Further attention is paid to the relationship between ideals and kernels of congruences, complemented lattices, Boolean algebras, and other particular lattices which play important rules in various non-classical logics. 1. Basic notions of lattice theory Partially ordered sets. Types of partial orders. Hasse diagrams. Semilattices and lattices as particular ordered sets and as particular algebraic structures. Significant elements of lattices, join and meet irreducibility. Complete lattices. Algebraic lattices. Distributive lattices and their characterization. Modular lattices. Complemented lattices. Relatively complemented lattices. 2. Congruences and ideals Lattice congruences, tolerance relations, factor lattices. Lattice ideals, lattices of ideals of lattices. The relationship of congruences and ideals on lattice. Prime ideals and maximal ideals on lattices. 3. Boolean lattices and Boolean algebras Boolean lattices. Boolean algebras and their properties. Complete Boolean algebras. Boolean rings. Boolean functions, polynomials, complete disjunctive and conjunctive normal forms. Logical circuits and their relationship to Boolean functions. Minimalization methods. 4. Further topics Lattices of (some) non-classical logics: Pseudo-complements, pseudo-complemented lattices, Glivenko's congruence, relatively pseudo-complemented lattices.
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Learning activities and teaching methods
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Lecture
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Learning outcomes
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Ordered sets, special elements in ordered sets. Lattices and semilattices. Complete lattices. Modular and distributive lattices. Complemented and relatively complemented lattices. Ideals and congruences in lattices. Boolean algebras.
3. Aplication Defines basic properties of posets and lattices.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Written exam
Ordered sets, special elements in ordered sets. Lattices and semilattices. Complete lattices. Modular and distributive lattices. Complemented and relatively complemented lattices. Ideals and congruences in lattices. Boolean algebras.
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Recommended literature
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Birkhoff G. (1984). Lattice Theory. Publ. Amer. Math. Soc., 3rd ed.
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Burris S., Sankappanavar H. P. (1981). A Course in Universal Algebra. Springer-Verlag, New York.
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Davey B. A., Priestley H. A. (2002). Introduction to Lattices and Order. Cambridge University Press (druhé vydání).
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GRÄTZER G. A. (1998). General Lattice Theory. Birkhauser Verlag Basel-Boston-Berlin.
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Chajda I., Glazek K. (2002). A Basic Course on General Algebra. Technical University Press, Zielona Góra.
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Schroeder B. S. W. (2003). Ordered Sets, An introduction. Birkhauser, Boston.
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