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The Lattice of Lambda Theories ( Regular Research Paper )

The lattice Ì of lambda theories is isomorphic to the congruence lattice of the term algebra of the minimal lambda theory ¬. This remark is the starting point for studying the structure of Ì by universal algebraic methods. In this paper we show that Ì satisfies nontrivial lattice quasi-identities, while we conjecture that Ì does not satisfy any nontrivial lattice identity (a lattice identity is trivial if it holds in every lattice and nontrivial otherwise). Further we show that, for every nontrivial lattice identity , there exists a natural number Ò such that fails in the lattice of lambda theories in a language of-terms with Ò constants. We also show that there exists a sublattice of Ì (consisting of all lambda theories extending a suitable lambda theory) which satisfies a restricted form of distributivity, called meet semidistributivity, and a nontrivial congruence identity (i.e., an identity in the language of lattices enriched by the relative product of binary relations).