Abstract
This chapter discusses the indexed Boolean algebras, substitutive i-ideals and formalized theories, and abstract characterization of Lindenbaum algebras. The free Boolean algebra is isomorphic with the Boolean algebra of Boolean functions, having the same number of independent variables as the number of given Boolean indeterminates. The Boolean indeterminates may be used directly as symbols for these variables. There is a further significant property of Boolean algebras, which deserves to be noticed for a more profound insight in the algebraic structures of logic. This property of Boolean algebras is of rare occurrence among other types of algebraic structures, and it need not subsist even in the case of structures for which free structures exist. It is the property of the existence of a finite functionally free structure of the given type. By this, a finite algebraic structure of the given type, that is, a structure satisfying the axioms governing the necessary operations, can be understood, such that every identity satisfied in this structure then holds in all algebraic structures of the given type.
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