Seminario di algebra e geometria
ore
17:00
presso Seminario II
A rack is a set R together with a binary operation ▷ such that
• For each x, y, z ∈ R, x ▷ (y ▷ z) = (x ▷ y) ▷ (x ▷ z), and
• for each x, y ∈ R, there exists a unique element z ∈ R with x ▷ z = y.
If we have the extra condition x ▷ x = x for each x ∈ R, then R is called a
quandle. For an example, a group G together with the operation x ▷ y = xyx−1
is a quandle.
The study of racks and quandles dates back to 1943 when Takasaki used
a certain algebraic structure to study reflections in finite geometries [?]. Since
then, Racks and quandles have been used in some branches of mathematics
such as knot theory for encoding knot diagrams. In 2015, I. Heckenberger et al.
started the study of racks in a combined perspective of combinatorics and group
theory. Indeed, they considered the lattice of subracks of a rack and obtained
some interesting results [?]. Moreover, they posed some important questions
in the last section of their paper. Two of these questions have been solved in
[?] and [?]. Actually, it has been shown that the lattice of subracks of a rack
is atomic, and this lattice for finite racks is complemented but there are some
infinite racks whose lattices are not complemented.