The lattice of subracks is atomic

A rack is a set together with a self-distributive bijective binary operation. In this paper, we give a positive answer to a question due to Heckenberger, Shareshian and Welker. Indeed, we prove that the lattice of subracks of a rack is atomic. Further, by using the atoms, we associate certain quandles to racks. We also show that the lattice of subracks of a rack is isomorphic to the lattice of subracks of a quandle. Moreover, we show that the lattice of subracks of a rack is distributive if and only if its corresponding quandle is the trivial quandle. So the lattice of subracks of a rack is distributive if and only if it is a Boolean lattice.