On countably saturated linear orders and certain class of countably saturated graphs

0537538 - MÚ 2022 RIV DE eng J - Journal Article
Kostana, Ziemowit
On countably saturated linear orders and certain class of countably saturated graphs.
Archive for Mathematical Logic. Roč. 60, 1-2 (2021), s. 189-209. ISSN 0933-5846. E-ISSN 1432-0665
R&D Projects: GA ČR GF16-34860L
Institutional support: RVO:67985840
Keywords : countably saturated * homogeneous object * linear order * random graph
OECD category: Pure mathematics
Impact factor: 0.492, year: 2021
Method of publishing: Open access
https://doi.org/10.1007/s00153-020-00742-7

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality c. We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality c, under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fraïssé limit of certain Fraïssé class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to (H(ω1) , ∈ ∪ ∋) , where H(ω1) is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.
Permanent Link: http://hdl.handle.net/11104/0315350