The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity \(\le 1\). Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.

中文翻译：

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入射幺半群：自同构和复杂性

研究了关联代数的代数幺半群结构。我们表明，乘法结构本身就决定了关联代数的代数自同构。我们提出了一个公式，该公式根据偏序集的 zeta 多项式表示关联幺半群相对于其最大环面的两侧作用的复杂性。此外，我们描述了有限（连接）偏序，其关联幺半群具有复杂度\(\le 1\)。最后，我们确定了依从阶对星偏序的入射幺半群的覆盖关系。