All lattices of subracks are complemented

A rack is a set together with a self-distributive bijective binary operation. In this paper, a question due to Heckenberger, Shareshian and Welker is answered. In fact, 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. Further, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a finite rack. Next, we introduce a certain class of racks including all 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.