For finite semidistributive lattices the map $\kappa$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $\kappa$-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the $\kappa$-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend $\kappa$ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll. For hereditary algebras, we show that the extended $\kappa$-map on torsion classes is essentially the same as Ringel's $\epsilon$-map on wide subcategories. Also in hereditary case, we relate the square of $\kappa$ to the Auslander-Reiten translation.

中文翻译：

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动力学组合和扭转类

对于有限半分布格子，映射 $\kappa$ 给出了完全连接不可约元素和完全相遇不可约元素集合之间的双射。在这里，我们在扭转类的上下文中研究 $\kappa$-map。众所周知，艺术代数的扭转类格是半分布的，但一般来说它远非有限。我们表明 $\kappa$-map 在完全连接不可约元素的集合上是明确定义的，即使扭转类的格子是无限的。然后，我们将 $\kappa$ 扩展到扭转类的映射，这些类具有由与第一和第三作者以及 A. Carroll 引入的最小扩展模块相关联的特殊扭转类给出的规范连接表示。对于遗传代数，我们表明，扭转类上的扩展 $\kappa$-map 与 Ringel 在宽子类别上的 $\epsilon$-map 基本相同。同样在遗传情况下，我们将 $\kappa$ 的平方与 Auslander-Reiten 翻译联系起来。