We ascertain conditions and structures on categories and semigroups which admit the construction of pseudo-products and trace products respectively, making their connection as precise as possible. This topic is modelled on the ESN Theorem and its generalisation to ample semigroups. Unlike some other variants of ESN, it is self-dual (two-sided), and the condition of commuting projections is relaxed. The condition that projections form a band (are closed under multiplication) is shown to be a very natural one. One-sided reducts are considered, and compared to (generalised) D-semigroups. Finally the special case when the category is a groupoid is examined.

中文翻译：

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迹和伪产品：具有投影带的限制类半群

我们确定了分别允许构造伪积和痕量积的范畴和半群的条件和结构，使它们的联系尽可能精确。本主题以 ESN 定理及其对充足半群的推广为模型。与 ESN 的其他一些变体不同，它是自对偶的（两侧），并且放宽了通勤投影的条件。投影形成带（在乘法下闭合）的条件被证明是非常自然的。考虑单边归约，并与（广义）D 半群进行比较。最后，检查类别是 groupoid 时的特殊情况。