In this paper, a question due to Heckenberger, Shareshian and Welker on racks in Heckenberger et al. (Trans Am Math Soc, 372:1407–1427, 2019) is positively answered. A rack is a set together with a self-distributive bijective binary operation. We show that the lattice of subracks of every finite rack is complemented. Moreover, we characterize finite modular lattices of subracks in terms of complements of subracks. Also, we introduce a certain class of racks including all finite groups with the conjugation operation, called G-racks, and we study some of their properties. In particular, we show that a finite G-rack has the homotopy type of a sphere. Further, we show that the lattice of subracks of an infinite rack is not necessarily complemented which gives an affirmative answer to the aforementioned question. Indeed, we show that the lattice of subracks of the set of rational numbers, as a dihedral rack, is not complemented. Finally, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a rack.

在本文中，Heckenberger等人的货架上的Heckenberger，Shareshian和Welker提出了一个问题。（Trans Am Math Soc，372：1407-1427，2019）得到了肯定的回答。机架是一组具有自分配双射二元运算的集合。我们表明，每个有限机架的子架的格子都是互补的。此外，我们根据子架的互补性来描述子架的有限模块化晶格。此外，我们介绍了包含所有有限组的共轭操作的特定类别的机架，称为G机架，并研究了它们的一些特性。特别是，我们证明了有限的G-rack具有球形的同伦类型。此外，我们显示出无限机架子架的格子不一定是互补的，从而可以对上述问题给出肯定的答案。确实，我们证明了有理数集的子架的格子（作为二面角架）没有得到补充。最后，我们证明，作为布尔代数，伪补充和唯一补充以及分布性与机架子架的晶格等效。