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Banschewského funkce na komplementárních modulárních svazech
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: Banaschewski function on countable complemented modular lattices Author: Samuel Mokriš Department: Department of Algebra Supervisor of the bachelor thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: A Banaschewski function on a bounded lattice L is an antitone self-map on L that picks a complement for each element of L. On any at most countable complemented modular lattice L, there exists a Banaschewski function with a Boolean range M. Moreover, such M is a maximal Boolean sublattice of L and is uniquely determined up to isomorphism. In the thesis we give a negative answer to the related question whether all maximal Boolean sublattices in an arbitrary countable complemented modular lattice are isomorphic and whether every max- imal Boolean sublattice in an arbitrary countable complemented modular lattice L is the range of some Banaschewski function on L. We also generalize the coun- terexample to greater cardinalities; for a given infinite cardinal κ we construct a complemented modular lattice L of cardinality κ and maximal Boolean sublat- tices B and E of L such that B is not the range of any Banaschewski function on L, that there exists a Banaschewski function on L of range E, and that B is not isomorphic to E. Keywords: complemented modular lattice, Banaschewski function, von...
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Banschewského funkce na komplementárních modulárních svazech
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: Banaschewski function on countable complemented modular lattices Author: Samuel Mokriš Department: Department of Algebra Supervisor of the bachelor thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: A Banaschewski function on a bounded lattice L is an antitone self-map on L that picks a complement for each element of L. On any at most countable complemented modular lattice L, there exists a Banaschewski function with a Boolean range M. Moreover, such M is a maximal Boolean sublattice of L and is uniquely determined up to isomorphism. In the thesis we give a negative answer to the related question whether all maximal Boolean sublattices in an arbitrary countable complemented modular lattice are isomorphic and whether every max- imal Boolean sublattice in an arbitrary countable complemented modular lattice L is the range of some Banaschewski function on L. We also generalize the coun- terexample to greater cardinalities; for a given infinite cardinal κ we construct a complemented modular lattice L of cardinality κ and maximal Boolean sublat- tices B and E of L such that B is not the range of any Banaschewski function on L, that there exists a Banaschewski function on L of range E, and that B is not isomorphic to E. Keywords: complemented modular lattice, Banaschewski function, von...
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