3-Nets realizing a diassociative loop in a projective plane

Abstract

A 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 $\mathrm{PG}(2,\mathbb {K})$, defined over a field $\mathbb {K}$ of characteristic p, arose from algebraic geometry; see Falk and Yuzvinsky (Compos Math 143:1069–1088, 2007), Miguel and Buzunáriz (Graphs Comb 25:469–488, 2009), Pereira and Yuzvinsky (Adv Math 219:672–688, 2008), Yuzvinsky (140:1614–1624, 2004), and Yuzvinsky (137:1641–1648, 2009). It is not difficult to find 3-nets in $\mathrm{PG}(2,\mathbb {K})$ as far as $0<p\le n$. However, only a few infinite families of 3-nets in $PG(2,\mathbb {K})$ are known to exist whenever $p=0$, or $p>n$. Under this condition, the known families are characterized as the only 3-nets in $\mathrm{PG}(2,\mathbb {K})$ which can be coordinatized by a group; see Korchmáros et al. (J Algebr Comb 39:939–966, 2014). In this paper we deal with 3-nets in $PG(2,\mathbb {K})$ 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 where the exponent of a finite diassociative loop is the maximum of the orders of its elements. We also discuss the existence problem for such 3-nets.