Abstract
Lambda abstraction algebras (LAA’s) are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. Like combinatory algebras they can be denned by true identities and thus form a variety in the sense of universal algebra, but they differ from combinatory algebras in several important respects. The most natural LAA’s are obtained by coordinatizing environment models of the lambda calculus. This gives rise to two classes of LAA’s of functions of infinite arity: functional LAA’s (FLAA) and point-relativized functional LAA’s (RLAA). It is shown that RLAA is a variety and is the smallest variety including FLAA. Dimension-complemented LAA’s constitute the widest class of LAA’s that can be represented as an algebra of functions and are known to have a natural intrinsic characterization. We prove that every dimension-complemented LAA is isomorphic to RLAA. This is the crucial step in showing that RLAA is a variety.Keywords and phraseslambda calculuscylindric algebraspolyadic algebrasabstract substitutionrepresentation theorems
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