Abstract
IfK is a partition of a setK which is partially ordered by the relationR andR is a collection of pairs of sets ofK such that the sets of each pair are related byR in the sense of Rashevsky, thenR is a relation which partially ordersK. Necessary and sufficient conditions thatK be a chain are obtained, and ifK is a chain under these conditions, it is shown thatK is unique.
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Literature
Birkhoff, G. 1935. “On the Structure of Abstract Algebras.”Proc. Camb. Phil. Soc.,31, 433–454.
Rashevsky, N. 1961. “On Relations Between Sets.”Bull. Math. Biophysics,23, 233–235.
Sommerfield, M. 1963. “Elementary Propositions on Relations Between Sets.”Bull. Math. Biophysics,25, 177–181.
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Leininger, C.W. On chains of related sets. Bulletin of Mathematical Biophysics 28, 103–106 (1966). https://doi.org/10.1007/BF02476393
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DOI: https://doi.org/10.1007/BF02476393