If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff gaps and limits". In the paper "R. Frankiewicz, Some remarks on embeddings of boolean algebras and topological spaces II, Fund. Math, 126 (1), 1985, 63-68.", the author showed that it is consistent with MA+¬CH the existence of a boolean algebra with cardinality equal to the continuum which does not embed in ℘(ω)/Fin.

Is it consistent with $MA+\neg CH$ that every boolean algebra with cardinaly equal to the continuum embeds in ℘(ω)/Fin?


In the paper Embedding of Boolean algebras in $Ρ(ω)/$fin the following partial result is proved:

Theorem There is a model of $ZFC$ with arbitrarily large continuum in which each Boolean algebra $B$ of cardinality $\leq 2^{\aleph_0}$ can be embedded into $P(ω)/$fin. In addition, Martin's axiom for $σ$-linked orderings holds in the model.

You may also look at On automorphisms of Boolean algebras embedded in P (ω)/fin, where a model as above is constructed in which the following additional property holds: every automorphism of $B$ extends to an automorphism of $P(ω)/$fin.

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    $\begingroup$ @ClaudiaCorrea Adding to the last paragraph in Mohammad's answer, it is worth mentioning that based on a result of Shelah and Steprāns, Proper Forcing Axiom implies that $\mathcal{P}(\omega)$/Fin has no non-trivial automorphism. See: "PFA implies all automorphisms are trivial." $\endgroup$ – Morteza Azad Jul 29 '18 at 11:26
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    $\begingroup$ @MortezaAzad A result of Velickovic says that $OCA+MA_{\aleph_1}$ is enough. This is even enough for $P(\omega_1}/fin$. $\endgroup$ – Rahman. M Jul 29 '18 at 15:45
  • $\begingroup$ @Rahman.M (+1) Mamnun Rahman jaan! :-) Could you please add a link to that particular work of Velicovic that you cited in your comment? Or maybe it is an unpublished result? $\endgroup$ – Morteza Azad Jul 30 '18 at 1:48
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    $\begingroup$ @MortezaAzad, You are welcome. Here is the link. logique.jussieu.fr/~boban/pdf/OCA-automorphisms.pdf $\endgroup$ – Rahman. M Jul 30 '18 at 7:03

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