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 B₁, B₂ ∈ M_T, let B₁ ≤ B₂ mean that B₂ is elementarily embeddable in B₂. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ T_ω, then ‹M_T, ≤› is well-quasi-ordered. ∎ We define T_ω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ 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_ω to be the theory of all Boolean algebras such that for every n ∈ ω, B⁽ⁿ⁾ ≠ {0}. By Tarski [1949], T_ω is complete. Recall that ‹A, < › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {a_i, ⃒ i ∈ ω} ⊆ A, there are i < j < ω such that a_i ≤ a_j. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and 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. ‹M′₀, ≤′› 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.