Boolean Algebras for Lambda Calculus

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory Tscr, there is a continuum of lambda theories including Tscr which are omitted by the indecomposable semantics