3-nets realizing a diassociative loop in a projective plane

TL;DR

This paper explores special 3-nets in projective planes that are coordinatized by diassociative loops but not by groups, providing structural theorems and discussing their existence.

Contribution

It introduces new structural theorems for 3-nets associated with diassociative loops, expanding understanding beyond group-coordinatized nets.

Findings

If the loop is commutative, all non-trivial elements have the same order.

Such loops have exponent 2 or 3.

The paper discusses conditions for the existence of these 3-nets.

Abstract

A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around $3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$ of characteristic $p$, arose from algebraic geometry. It is not difficult to find $3$-nets in $PG(2,K)$ as far as $0<p\le n$. However, only a few infinite families of $3$-nets in $PG(2,K)$ are known to exist whenever $p=0$, or $p>n$. Under this condition, the known families are characterized as the only $3$-nets in $PG(2,K)$ which can be coordinatized by a group. In this paper we deal with $3$-nets in $PG(2,K)$ which can be coordinatized by a diassociative loop $G$ but not by a group. We prove two structural theorems on $G$. As a corollary, if $G$ is commutative then every non-trivial element of $G$ has the same order, and $G$ has exponent $2$ or $3$. We also discuss the existence problem for such $3$-nets.