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 (Formula presented.) , defined over a field (Formula presented.) 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 (Formula presented.) as far as (Formula presented.). However, only a few infinite families of 3-nets in (Formula presented.) are known to exist whenever (Formula presented.) , or (Formula presented.). Under this condition, the known families are characterized as the only 3-nets in (Formula presented.) 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 (Formula presented.) 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.
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机译：阶数n的3个网是由点和三对成对的不相交线段组成的有限入射结构，每条线段的大小均为n，因此，与不同类别的两条线相交的每个点与三条线中的每条正好入射类。当前在投影平面（表示公式）中的3-net（嵌入）周围的兴趣是由代数几何引起的，该平面在特征p的场（表示公式）上定义。参见Falk和Yuzvinsky（Compos Math 143：1069-1088，2007），Miguel andBuzunÃriz（Graphs Comb 25：469–488，2009），Pereira and Yuzvinsky（Adv Math 219：672–688，2008），Yuzvinsky（ 140：1614–1624，2004）和Yuzvinsky（137：1641–1648，2009）。在（公式表示）中找到3个网络并不难。但是，无论何时（出现公式）或（出现公式），仅存在几个无限的3网络族。在这种情况下，已知族的特征是（公式）中仅有的3个网络，可以由一个小组进行协调。参见Korchmáros等。 （J Algebr Comb 39：939–966，2014）。在本文中，我们处理（表示公式）中的3个网，这些3个网可以通过二联环G进行协调，但不能通过组进行协调。我们证明了关于G的两个结构定理。作为推论，如果G是可交换的，那么G的每个非平凡元素都具有相同的阶数，并且G的指数为2或3，其中有限二阶缔合环的指数为该阶数的最大值其元素。我们还讨论了此类3网络的存在问题。
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