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The combinator M and the Mockingbird lattice

Published online by Cambridge University Press:  13 October 2022

Samuele Giraudo Open the ORCID record for Samuele Giraudo [Opens in a new window]
Samuele Giraudo*
Affiliation:
LIGM, Université Gustave Eiffel, CNRS, ESIEE Paris, F-77454 Marne-la-Vallée, France
*
Email: samuele.giraudo@univ-eiffel.fr
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Abstract

We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${{\mathbf{M}}}$ . This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule ${{\mathbf{M}}} \mathsf{x}_1 \to \mathsf{x}_1 \mathsf{x}_1$ . We prove that the reflexive and transitive closure of this rewrite relation is a partial order on terms on ${{\mathbf{M}}}$ 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 semi-distributive. 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.

Keywords

Partial orderslatticescombinatory logicrewrite systemstreelike structuresformal power series
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 32 , Issue 3 , March 2022 , pp. 271 - 299
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

This research has been partially supported by the projects CARPLO (ANR-20-CE40-0007) and LambdaComb (ANR-21-CE48-0017) of the Agence nationale de la recherche.

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