Journal of Symbolic Logic

Elementary Embedding between Countable Boolean Algebras

Abstract

For a complete theory of Boolean algebras $T$, let $M_T$ denote the class of countable models of $T$. For $B_1, B_2 \in M_T$, let $B_1 \leq B_2$ mean that $B_1$ is elementarily embeddable in $B_2$. Theorem 1. For every complete theory of Boolean algebras $T$, if $T \neq T_\omega$, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. $\blacksquare$ We define $T_\omega$. 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 $T_\omega$ to be the theory of all Boolean algebras such that for every $n \in \omega, B^{(n)} \neq \{0\}$. By Tarski , $T_\omega$ 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 $a_i \leq a_j$. 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. $\blacksquare$ Let $M'_0$ denote the class of all countable Boolean algebras. For $B_1, B_2 \in M'_0$, let $B_1 \leq' B_2$ mean that $B_1$ is embeddable in $B_2$. Remark. $\langle M'_0, \leq'\rangle$ is well-quasi-ordered. $\blacksquare$ This follows from Laver's theorem  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.

Article information

Source
J. Symbolic Logic, Volume 56, Issue 4 (1991), 1212-1229.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743809

Mathematical Reviews number (MathSciNet)
MR1136451

Zentralblatt MATH identifier
0742.03010

JSTOR