The combinator M and the Mockingbird lattice

Samuele Giraudo LIGM, Université Gustave Eiffel, CNRS, ESIEE Paris, F-77454 Marne-la-Vallée, France. samuele.giraudo@univ-eiffel.fr ABSTRACT. We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator M. This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule Mx1 → x1x1. We prove that the reflexive and transitive closure of this rewrite relation is a partial order on terms on M and that all connected components of its rewrite graph are Hasse diagram of lattices. This last result is based on the introduction of new lattices on duplicative forests, which are sorts of treelike structures. These lattices are not graded, not self-dual, and not semidistributive. We present some enumerative properties of these lattices like the enumeration of their elements, of the edges of their Hasse diagrams, and of their intervals. These results are derived from formal power series on terms and on duplicative forests endowed with particular operations. CONTENTS Introduction 2 1. Terms, rewrite relations, and combinatory logic systems 4 1.1. Terms, compositions, and rewrite relations 4 1.2. Combinatory logic systems 6 2. The Mockingbird lattice 9 2.1. The combinator M and its poset 9 2.2. Duplicative lattices 11 2.3. Mockingbird lattices 14 3. Enumerative properties 16 3.1. Formal power series 16 3.2. Maximal, minimal, and isolated terms 17 3.3. Shortest and longest saturated chains 20 3.4. Elements, covering pairs, and intervals 21 Open questions and future work 28 References 28