The combinator ${\bf M}$ and the Mockingbird lattice

TL;DR

This paper explores the combinatorial and order-theoretic structures generated by the basic combinator ${f M}$, revealing that its associated rewrite graph forms lattices with unique properties and introducing new lattice structures on duplicative forests.

Contribution

It establishes that the rewrite relation for ${f M}$ forms a partial order with lattice structures on connected components and introduces novel lattices on duplicative forests.

Findings

The reflexive and transitive closure of the ${f M}$ rewrite relation is a partial order.

Connected components of the rewrite graph are lattices, specifically Hasse diagrams.

New lattices on duplicative forests are introduced, with enumerative properties analyzed.

Abstract

We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${\bf M}$. This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule ${\bf M} x_1 \to x_1 x_1$. We prove that the reflexive and transitive closure of this rewrite relation is a partial order on terms on ${\bf M}$ 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.