Counting and Computing Join-Endomorphisms in Lattices

Abstract

Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set ${\mathcal {E}}(L)$ of all join-endomorphisms of a given finite lattice $L$. In particular, we show that when $L$ is $\mathbf {M}_n$, the discrete order of n elements extended with top and bottom, $| {\mathcal {E}}(L) | =n!{\mathcal L}_{n}(-1)+(n+1)^2$ where ${\mathcal L}_{n}(x)$ is the Laguerre polynomial of degree n. We also study the following problem: Given a lattice L of size n and a set $S\subseteq {\mathcal {E}}(L)$ of size m, find the greatest lower bound . The join-endomorphism has meaningful interpretations in epistemic logic, distributed systems, and Aumann structures. We show that this problem can be solved with worst-case time complexity in $O(n+ m\log {n})$ for powerset lattices, $O(mn^2)$ for lattices of sets, and $O(mn + n^3)$ for arbitrary lattices. The complexity is expressed in terms of the basic binary lattice operations performed by the algorithm.

This work has been partially supported by the ECOS-NORD project FACTS (C19M03).