The variety of lambda abstraction algebras does not admit n-permutable congruences for all n

In this section we summarize deenitions and results concerning the lambda calculus and the theory of lambda abstraction algebras. Our main references will be 10] and 11] for lambda abstraction algebras and Barendregt's book 1] for lambda calculus. Lambda calculus. The untyped lambda calculus was introduced by Church as a foundation for logic. Although the appearance of paradoxes caused the program to fail, a consistent part of the theory turned out to be successful as a theory of \functions as rules" (formalized as terms of the lambda calculus) that stresses the computational process of going from argument to value. Every object is at the same time a function and an argument; in particular a function can be applied to itself contrary to the usual notion of function in set theory. The two primitive notions of the lambda calculus are application, the operation of applying a function to an argument (expressed as juxtaposition of terms), and lambda (functional) abstraction, the process of forming a function from the \rule" that deenes it. The set F I (C) of ordinary terms of lambda calculus over an innnite set I of variables and a set C of constants is constructed as usual 1]: every variable x 2 I and every constant c 2 C is a-term; if t and s are-terms, then so are (st) and x:t for each variable x 2 I. We will write F I for F I (;), the set of-terms without constants. An occurrence of a variable x in a-term is bound if it lies within the scope of a lambda abstraction x; otherwise it is free. A-term without free variables is said to be closed. A-term s is free for x in t if no free occurrence of x in t lies within the scope of a lambda abstraction with respect to a variable that occurs free in s. ts=x] is the result of substituting s for all free occurrences of x in t subject to the usual provisos about renaming bound variables in t to avoid capture of free variables in s. The above proviso is empty if s is free for x in t.