Distributive and anti-distributive Mendelsohn triple systems

TL;DR

This paper characterizes the existence spectrum of distributive Mendelsohn triple systems using Loeschian numbers, and constructs non-distributive variants, advancing understanding of their algebraic structures and enumeration.

Contribution

It establishes the existence spectrum of distributive Mendelsohn triple systems and provides new constructions of non-distributive systems with maximal non-distributivity.

Findings

Existence spectrum corresponds to Loeschian numbers.

Provides enumeration results for these systems.

Constructs non-distributive Mendelsohn quasigroups with maximal non-distributivity.

Abstract

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.