ELEMENTARY EMBEDDING BETWEEN COUNTABLE BOOLEAN-ALGEBRAS

For a complete theory of Boolean algebras T, let M(T) denote the class of countable models of T. For B1, B2 is-an-element-of M(T), let B1 less-than-or-equal-to B2 mean that B1 is elementarily embeddable in B2. THEOREM 1. For every complete theory of Boolean algebras T, if T not-equal T-omega, then <M(T), less-than-or-equal-to> is well-quasi-ordered. 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 half-arrow-pointing-up-and-to-the-right a is an atomic Boolean algebra and B half-arrow-pointing-up-and-to-the-right 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 is-an-element-of omega, B(n) not-equal {0}. By Tarski [1949], T-omega is complete. Recall that <A, less-than-or-equal-to> is a partial well-quasi-ordering, if it is a partial quasi-ordering and for every {a(i)\i is-an-element-of omega} is contained or equal to A, there are i < j < omega such that a(i) less-than-or-equal-to a(j). THEOREM 2. <M(T)-omega, less-than-or-equal-to> contains a subset M such that the partial orderings <M, less-than-or-equal-to half-arrow-pointing-up-and-to-the-right M> and <P(omega), is contained or equal to> are isomorphic. Let M'0 denote the class of all countable Boolean algebras. For B1, B2 is-an-element-of M'0, let B1 less-than-or-equal-to' B2 mean that B1 is embeddable in B2. REMARK. <M'0, less-than-or-equal-to'> 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.