Mathematical Logic Quarterly 41 (4):485-504 (1995)

We consider the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Thprin and Thsa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg be the Lindenbaum-Tarski algebra with respect to T, and we let intalg be the interval algebra of α. Using rank diagrams, we show that Sentalg ⋍ intalg, Sentalg ⋍ intalg ⋍ Sentalg, and Sentalg ⋍ intalg. For Thmax and Thac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic
Keywords Lindenbaum‐Tarski algebras  Rank diagram  Boolean algebra  Elementary invariants  Ideals in Boolean algebras  Completion of Boolean algebras
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DOI 10.1002/malq.19950410406
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Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
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