We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR \emph{homogeneous}. In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR \emph{transitive}. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix \emph{stepwise} to refer to sets of MOLR with this property. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

翻译：我们研究相互正交拉丁矩形（MOLR）的集合，以及自正交拉丁方概念的一种自然变体，该变体适用于更大规模的相互正交拉丁方与MOLR集合，即集合中的每个拉丁矩形均与该集合中的其他矩形同痕。我们将此类MOLR集合称为**齐性集合**。在此过程中，我们对所有满足$k\leq n \leq 7$且$t<n$的$t$个相互正交的$k\times n$拉丁矩形的非同痕集合进行了完整枚举。具体而言，我们跟踪了齐性MOLR集合，以及自同构群在矩形上传递作用的MOLR集合，并将后者称为**传递性集合**。我们逐行构建MOLR集合，并在此过程中记录每个构造步骤中哪些MOLR具有齐性和/或传递性。我们使用前缀**逐级**来指代具有此性质的MOLR集合。MOLR集合与其他离散对象相关联，特别是有限几何与某些正则图。本文观察到，除休斯平面外，所有阶数不超过9的射影平面均可由逐级传递的MOLR构造得出。