In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for \(m\ge 2\), a set of \(m+1\) partitions of a set \(\Omega \), any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if \(m=2\)), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have \(m+r\) partitions with \(r\ge 2\), any m of which are minimal elements of a Cartesian lattice. If \(m=2\), this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For \(m>2\), things are more restricted. Any \(m+1\) of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality \(m+r\) in the \((m-1)\)-dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.

在三位当前作者和 Csaba Schneider 的早期论文中，表明对于\(m\ge 2\)，一组\(\Omega \) 的一组\(m+1\)分区，其中任何m是笛卡尔格的最小非平凡元素，要么形成拉丁方格（如果\(m=2\)），要么生成与基群上的对角群相关联的维数m的连接半格格。在本文中，我们研究了如果我们有\(m+r\)分区和\(r\ge 2\) 会发生什么，其中任何m是笛卡尔格的最小元素。如果\(m=2\)，这只是一组相互正交的拉丁方。我们考虑所有这些正方形都是群的凯莱表的同位素的情况，并给出一个例子来说明群不一定都是同构的。对于\(m>2\)，事情受到更多限制。任何\(m+1\)分区生成一个连接半格，允许一个对角群在群G 上。可能这些群都是同构的，尽管我们无法证明这一点。在一个额外的假设下，我们证明G必须是阿贝尔的，并且必须有三个无不动点的自同构，其乘积是恒等式。（我们明确地描述了所有具有这种自同构的阿贝尔群。）在这个假设下，该结构给出了一个正交数组，在某些情况下相反。如果该基团是环素数阶p，则该结构精确地对应于基数的弧\（M + R \）在\（（M-1）\）在与所述场维射影空间p的元素，所以所有关于弧的已知结果都适用。更一般地说，q阶有限域上的弧给出了G是q阶基本阿贝尔群的例子. 可以使用p- adic 技术将这些例子提升到非基本阿贝尔群。