# zbMATH — the first resource for mathematics

Posets of copies of countable non-scattered labeled linear orders. (English) Zbl 07204024
Summary: We show that the poset of copies $$\mathbb{P} (\mathbb{Q}_n )=\langle \{ f[X]: f\in \text{Emb} (\mathbb{Q}_n ) \},\subset \rangle$$ of the countable homogeneous universal $$n$$-labeled linear order, $$\mathbb{Q}_n$$, is forcing equivalent to the poset $$\mathbb{S} \ast \pi$$, where $$\mathbb{S}$$ is the Sacks perfect set forcing and $$1_{\mathbb{S}} \Vdash$$ “$$\pi$$ is an atomless separative $$\sigma$$-closed forcing”. Under CH (or under some weaker assumptions) $$1_{\mathbb{S}} \Vdash  \pi$$ is forcing equivalent to $$P( \omega )$$/Fin”. In addition, these statements hold for each countable non-scattered $$n$$-labeled linear order $$\mathbb{L}$$ and we have $$\text{ro } \text{sq} \mathbb{P} (\mathbb{L} )\cong \text{ro } \text{sq} \mathbb{P} (\mathbb{Q}_n )\cong \text{ro } \text{sq} (\mathbb{S} \ast \pi )$$.
##### MSC:
 06-XX Order, lattices, ordered algebraic structures
Full Text:
##### References:
 [1] Cameron, P.; Laflamme, C.; Pouzet, M.; Tarzi, S.; Woodrow, R., Overgroups of the Automorphism Group of the Rado Graph, Asymptotic Geometric Analysis, 45-54, Fields Inst Commun., vol. 68 (2013), New York: Springer, New York · Zbl 1266.05052 [2] Fraïssé, R., Theory of Relations, Revised edition, with an Appendix by Norbert Sauer Studies in Logic and the Foundations of Mathematics, vol. 145 (2000), Amsterdam: North-Holland Publishing Co., Amsterdam [3] Jech, Thomas, Descriptive Set Theory, Set Theory, 493-578 (1997), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg [4] Kunen, K.: Set Theory. An Introduction to Independence Proofs. North-Holland (1980) · Zbl 0443.03021 [5] Kurilić, MS, Posets of copies of countable scattered linear orders, Ann. Pure Appl. Logic, 165, 3, 895-912 (2014) · Zbl 1297.06001 [6] Kurilić, MS, Forcing with copies of countable ordinals, Proc. Amer. Math. Soc., 143, 4, 1771-1784 (2015) · Zbl 1386.03065 [7] Kurilić, MS, Different similarities, Arch. Math. Logic, 54, 7-8, 839-859 (2015) · Zbl 1373.03049 [8] Kurilić, MS, Posets of isomorphic substructures of relational structures, Zb. Rad. (Beogr.), 17, 25, 117-144 (2015) · Zbl 06749575 [9] Kurilić, MS; Todorčević, S., Forcing by non-scattered sets, Ann. Pure Appl. Logic, 163, 1299-1308 (2012) · Zbl 1250.03102 [10] Kurilić, MS; Todorčević, S., The poset of all copies of the random graph has the 2-localization property, Ann. Pure Appl. Logic, 167, 8, 649-662 (2016) · Zbl 1432.03059 [11] Kurilić, MS; Todorčević, S., Copies of the Rado graph, Adv. Math., 317, 526-552 (2017) · Zbl 1423.03189 [12] Kurilić, M.S., Todorčević, S.: Copies of ultrahomogeneous tournaments and related structures, submitted [13] Lachlan, AH, Countable homogeneous tournaments, Trans. Amer. Math. Soc., 284, 2, 431-461 (1984) · Zbl 0562.05025 [14] Mwesigye, F.; Truss, JK, Countably categorical coloured linear orders, MLQ Math. Log. Q., 56, 2, 159-163 (2010) · Zbl 1192.03007 [15] Simon, P., Sacks forcing collapses $$\mathfrak{c}$$ c to $$\mathfrak{b}$$ b, Comment Math. Univ. Carolin., 34, 4, 707-710 (1993) · Zbl 0797.03053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.