About partially ordered sets
DOI:
https://doi.org/10.35819/remat2024v10i1id7008Keywords:
partially ordered set, totally ordered set, finite setAbstract
During classes, it’s common for intriguing questions to arise regarding the presented content. This article was prompted by the following inquiries: Considering a finite set U equipped with a partial order G contained in U x U, what would be the largest (and smallest) number of elements in G? Is there a relationship between this quantity of elements and the nature of the pair (U, G) as a totally ordered set? This article demonstrates that (U, G) is totally ordered if, and only if, (U, G) is partially ordered and G contains n(n + 1)/2 elements, where n represents the cardinality of U.
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References
O'CONNOR, J. J.; ROBERTON, E. F. A history of set theory. Escócia: School of Mathematics and Statistics, University of St Andrews, 1996. Disponível em: https://mathshistory.st-andrews.ac.uk/HistTopics/Beginnings_of_set_theory. Acesso em: 26 jan. 2024.
HALMOS, P. R. Naive Set Theory. Princeton, New Jersey: D. Van Nostrand Company, 1960.
HRBACEK, K.; JECH, T. Introduction to Set Theory. 3. ed. New York: Marcel Dekker, 1999.
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