Finite Relation Algebras

TL;DR

This paper demonstrates that most nonassociative relation algebras are symmetric and integral, establishes 0-1 laws for their atom structures, and provides improved asymptotic formulas for counting these structures.

Contribution

It introduces new probabilistic and combinatorial results on nonassociative relation algebras, including 0-1 laws and asymptotic enumeration formulas.

Findings

Almost all nonassociative relation algebras are symmetric and integral.

The classes of atom structures of these algebras satisfy 0-1 laws.

New asymptotic formulas for counting relation algebra structures.

Abstract

We will show that almost all nonassociative relation algebras are symmetric and integral (in the sense that the fraction of both labelled and unlabelled structures that are symmetric and integral tends to 1), and using a Fra\"iss\'e limit, we will establish that the classes of all atom structures of nonassociative relation algebras and relation algebras both have 0-1 laws. As a consequence, we obtain improved asymptotic formulas for the numbers of these structures and broaden some known probabilistic results on relation algebras.