Invertible Terms in the Lambda Calculus

It is well-known that the set of h -terms modulo &-convertibility is a semi-group with I as identity element and composition 0, defined by M 0 N = BMN, where B = hxyt . x(yz). In [6, pp. 167,168] the question is raised under what conditions an element in this semi-group has an inverse. DezanXiancaglini gave in [8] a characterization of (w.r.t. A&-calculus) invertible terms having a normal form as the ‘finite hereditary permutators’, and she conjectures that these are all the flq-invertible terms, i.e. a term without normal form cannot have an inverse. In this paper we confirm her conjecture. Two proofs are given for this fact. of which the first is more direct. The second proof uses the in itself interesting fact that certain ‘A -trees’ can be represented as Biihm-trees of AI-terms (in fact we prove something more), plus Hyland’s characterization of the equality in the Graph model PO (see [9, m The result on representation of A -trees is further used to characterize the A -terms invertible in D,, Scott’s well-known lattice model (see [12]). Since for this last result a slightly more general form of the main lemma in [S] is needed, we have included a new proof of that lemma.