The notion of a meaningless set has been defined for infini tary lambda calculus axiomatically. Standard examples of such sets are sets of terms that have no head normal form, the set of terms without weak head normal form and the set of rootactive terms. In this paper, we study the way the intervals decompose as union of more elementary ones. We also analyse the distribution of the sets of meaningless terms in the lattice by selecting some sets as key vertices and study the cardi nality in the intervals between key vertices. As an application, we prove that the lattice of meaningless sets is neither distributive nor modular. Interestingly, the example translates into a simple counterexample that the lattice of lambda theories is not modular.
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