The combinator M and the Mockingbird lattice

by   Samuele Giraudo, et al.

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 M x_1 → x_1 x_1. 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.


page 1

page 2

page 3

page 4


Mockingbird lattices

We study combinatorial and order theoretic structures arising from the f...

Finite Confluences and Closed Pattern Mining

The purpose of this article is to propose and investigate a partial orde...

Order-theoretic trees: monadic second-order descriptions and regularity

An order-theoretic forest is a countable partial order such that the set...

Efficient Inference on Generalized Fault Diagrams

The generalized fault diagram, a data structure for failure analysis bas...

Folding and Unfolding on Metagraphs

Typed metagraphs are defined as hypergraphs with types assigned to hyper...

Partial Residuated Implications Derived from Partial Triangular Norms and Partial Residuated Lattices

In this paper, we reveal some relations between fuzzy logic and quantum ...

Matrices of forests, analysis of networks, and ranking problems

The matrices of spanning rooted forests are studied as a tool for analys...