Orders, reduction graphs and spectra

Abstract

In this paper, we consider Abstract Reduction Systems as the setting where we need to investigate order properties of reduction graphs. To this aim, the main tool is the notion of spectrum, which captures some essential features of reduction sequences. We show that each spectrum is a complete partial order with respect to a suitable information ordering. Then we consider linearly ordered spectra. For those of them which are not well-ordered, we obtain a graph-theoretic characterization in terms of forbidden subgraphs. For well-ordered spectra, we show that they can represent all countable successor ordinals. Then, considering constructive ordinals, we address the well-known problem of knowing which ordinals are lambda representable (that is for which ordinal α there exists a lambda term T such that α is isomorphic to the spectrum of T). We give a partial answer by showing that all successor ordinals a, with α < ε 0, are lambda-representable.