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Some Boolean Algebras with Finitely Many Distinguished Ideals I
Author(s) -
Aragón Regina
Publication year - 1995
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19950410406
Subject(s) - mathematics , ideal (ethics) , infimum and supremum , complete boolean algebra , boolean algebra , stone's representation theorem for boolean algebras , two element boolean algebra , free boolean algebra , order (exchange) , element (criminal law) , parity function , boolean prime ideal theorem , principal ideal , discrete mathematics , combinatorics , pure mathematics , algebra over a field , boolean expression , boolean function , algebra representation , philosophy , prime (order theory) , epistemology , finance , economics , political science , law
We consider the theory Th prin of Boolean algebras with a principal ideal, the theory Th max of Boolean algebras with a maximal ideal, the theory Th ac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th sa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th prin and Th sa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg( T ) be the Lindenbaum‐Tarski algebra with respect to T , and we let intalg(α) be the interval algebra of α. Using rank diagrams, we show that Sentalg(Th prin ) ⋍ intalg(ω 4 ), Sentalg(Th max ) ⋍ intalg(ω 3 ) ⋍ Sentalg(Th ac ), and Sentalg(Th sa ) ⋍ intalg(ω 2 + ω 2 ). For Th max and Th ac 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.