We introduce a notion of ‘hereditarily antisymmetric’ operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case.

中文翻译：

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遗传的反对称算子代数

我们引入了“遗传反对称”算子代数的概念，并在有限维中证明了它们的结构定理。我们还在有限维数中刻画了那些算子代数，这些算子代数可以做成上三角形，并证明 Dilworth 和 Mirsky 定理的矩阵类似物用于有限偏序。在无限维情况下获得了一些部分结果。