二元矩阵的表示张量积
Represented Tensor Products of Binary Matroids
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中文总结 AI 辅助
本文分类了二元矩阵表示张量积的正则性、余图性、图性及二元性，指出非自由因子乘积的正则性条件，并证明固定因子时性质不依赖表示域。
中文摘要 AI 辅助
对于在公共域 \\(\F\\) 上可表示的二元矩阵 \\(M,N\\)，其 \\(\F\\)-表示的 Kronecker 积定义了一个矩阵 \\(T_{\F}(M,N)\\)，该矩阵不依赖于所选择的表示。我们分类了这种表示张量积何时是正则的、余图的、图的或二元的。对于简单的非自由因子，正则性恰好发生在（交换后）一个因子是仙人掌矩阵而另一个是外平面矩阵，或者一个是三角仙人掌矩阵而另一个是串并联矩阵。余图性恰好发生在第一种情况。对于具有非空基集的简单因子，图性恰好发生在一个因子是自由的而另一个是图的。对于固定因子，正则性、余图性和图性不依赖于公共表示域，尽管同构类型可能随其特征而变化。至少三个简单非自由因子的乘积是非正则的。
英文摘要
For binary matroids \(M,N\) representable over a common field \(\F\), the Kronecker product of their \(\F\)-representations defines a matroid \(T_{\F}(M,N)\) independent of the chosen representations. We classify when this represented tensor product is regular, cographic, graphic, or binary. For simple nonfree factors, regularity holds exactly when, up to interchange, one factor is a cactus matroid and the other is outerplanar, or one is a triangular cactus matroid and the other is series--parallel. Cographicity holds exactly in the first case. For simple factors with nonempty ground sets, graphicity holds exactly when one factor is free and the other is graphic. For fixed factors, regularity, cographicity, and graphicity are independent of the common representation field, although the isomorphism type may vary with its characteristic. Products of at least three simple nonfree factors are nonregular.
