The combinator M and the Mockingbird lattice

04/07/2022
by   Samuele Giraudo, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

04/06/2022

Mockingbird lattices

We study combinatorial and order theoretic structures arising from the f...
02/14/2021

Finite Confluences and Closed Pattern Mining

The purpose of this article is to propose and investigate a partial orde...
11/07/2021

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

An order-theoretic forest is a countable partial order such that the set...
03/27/2013

Efficient Inference on Generalized Fault Diagrams

The generalized fault diagram, a data structure for failure analysis bas...
12/03/2020

Folding and Unfolding on Metagraphs

Typed metagraphs are defined as hypergraphs with types assigned to hyper...
07/06/2022

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

In this paper, we reveal some relations between fuzzy logic and quantum ...
05/28/2013

Matrices of forests, analysis of networks, and ranking problems

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