Lambda Congruences and Extensionality

Abstract: In this work we provide alternative formulations of the concepts of lambda theory and extensional theory without introducing the notion of substitution and the sets of all, free and bound variables occurring in a term. We also clarify the actual role of $\alpha$-renaming and $\eta$-extensionality in the lambda calculus: both of them can be described as properties of extensionality for certain classes of terms.