# Elementary embedding between countable Boolean algebras

Journal of Symbolic Logic 56 (4):1212-1229 (1991)
Abstract
For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B1 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. ■ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that $B\upharpoonright a$ is an atomic Boolean algebra and $B\upharpoonright s$ is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that $\langle A, \leq\rangle$ is a partial well-quasi-ordering, if it is a partial quasi-ordering and for every $\{a_i\mid i \in \omega\} \subseteq A$ , there are $i < j < \omega$ such that ai ≤ aj. Theorem 2. $\langle M_{T_\omega}, \leq\rangle$ contains a subset M such that the partial orderings $\langle M, \leq \upharpoonright M \rangle$ and $\langle\mathscr{P}(\omega), \subseteq\rangle$ are isomorphic. ■ Let M'0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M'0, let B1 ≤' B2 mean that B1 is embeddable in B2. Remark. $\langle M'_0, \leq'\rangle$ is well-quasi-ordered. ■ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering
Keywords Model theory   Boolean algebras   ordered sets   well-quasi-ordering   better-quasi-orderin
Categories (categorize this paper)
DOI 10.2307/2275469
Options
 Save to my reading list Follow the author(s) My bibliography Export citation Find it on Scholar Edit this record Mark as duplicate Revision history Request removal from index

 PhilPapers Archive Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 23,201 External links Setup an account with your affiliations in order to access resources via your University's proxy server Configure custom proxy (use this if your affiliation does not provide a proxy) Through your library Sign in / register to customize your OpenURL resolver.Configure custom resolver
References found in this work BETA

No references found.

Citations of this work BETA

No citations found.

Similar books and articles

2009-01-28

52 ( #91,817 of 1,940,952 )