AI 中文总结
本文研究模糊拉丁方，将其定义为模糊置换矩阵的线性组合，确定了消失情形下四项模糊拉丁方的强条件，并完成了非消失情形下六项模糊拉丁方的计算机辅助分类。
AI 中文摘要
阶为$n$的拉丁方可视为将$n \times n$全1矩阵划分为若干置换矩阵求和项的分划。本文考虑一种放宽情形，即允许矩阵求和项由更短的置换诱导得到。对于$\sigma \in S_k$，“模糊置换矩阵”$P_\sigma^{\uparrow n}$由$k \times k$置换矩阵$P_\sigma$的所有$\binom{n}{k}^2$个保序嵌入组合而成，嵌入到$n \times n$矩阵中。我们将模糊拉丁方定义为$n \times n$模糊置换矩阵$P_\sigma^{\uparrow n}$的线性组合，其结果等于一个常数矩阵。我们研究这类对象的多个方面，包括相关的向量空间维数及少量项的模糊拉丁方的计数。特别地，我们确定了“消失”情形（常数矩阵为全零矩阵）下四项模糊拉丁方的强条件；还报告了非消失情形下六项模糊拉丁方的计算机辅助分类结果。
英文摘要
A latin square of order $n$ can be viewed as a partition of the $n \times n$ all-ones matrix into permutation matrix summands. Here, we consider a relaxation in which the matrix summands are allowed to be induced from shorter permutations. For $σ\in S_k$, the `fuzzy permutation matrix' $P_σ^{\uparrow n}$ arises from combining all $\binom{n}{k}^2$ order-preserving embeddings of the $k \times k$ permutation matrix $P_σ$ into an $n \times n$ matrix. We define a fuzzy latin square as a linear combination of $n \times n$ fuzzy permutation matrices $P_σ^{\uparrow n}$ equaling a constant matrix. We study various aspects of these objects, including certain relevant vector space dimensions and a census of fuzzy latin squares with a small number of terms. In particular, we determine strong conditions on four-term fuzzy latin squares in the `vanishing' case (when the constant matrix is all zeros). We also report on a computer-assisted classification of six-term fuzzy latin squares in the non-vanishing case.
