Order Structures on Böhm-Like Models

Abstract

We are interested in the question whether the models induced by the infinitary lambda calculus are orderable, that is whether they have a partial order with a least element making the context operators monotone. The first natural candidate is the prefix relation: a prefix of a term is obtained by replacing some subterms by $\bot$. We prove that six models induced by the infinitary lambda calculus (which includes Böhm and Lévy-Longo trees) are orderable by the prefix relation. The following two orders we consider are the compositions of the prefix relation with either transfinite η-reduction or transfinite η-expansion. We prove that these orders make the context operators of the η-Böhm trees and the ∞ η-Böhm trees monotone. The model of Berarducci trees is not monotone with respect to the prefix relation. However, somewhat unexpectedly, we found that the Berarducci trees are orderable by a new order related to the prefix relation in which subterms are not replaced by $\bot$ but by a lambda term O called the ogre which devours all its inputs. The proof of this uses simulation and coinduction. Finally, we show that there are 2^c unorderable models induced by the infinitary lambda calculus where c is the cardinality of the continuum.