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Uncountable Homogeneous Partial Orders
Author(s) -
Droste Manfred,
Macpherson Dugald,
Mekler Alan
Publication year - 2002
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200211)48:4<525::aid-malq525>3.0.co;2-3
Subject(s) - uncountable set , countable set , mathematics , homogeneous , isomorphism (crystallography) , automorphism , order (exchange) , combinatorics , set (abstract data type) , discrete mathematics , pure mathematics , computer science , crystallography , chemistry , finance , crystal structure , programming language , economics
A partially ordered set ( P , ≤) is called k ‐homogeneous if any isomorphism between k ‐element subsets extends to an automorphism of ( P , ≤). Assuming the set‐theoretic assumption ⋄(ϰ 1 ), it is shown that for each k , there exist partially ordered sets of size ϰ 1 which embed each countable partial order and are k ‐homogeneous, but not ( k + 1)‐homogeneous. This is impossible in the countable case for k ≥ 4.