Complemented lattices of subracks

TL;DR

This paper investigates the structure of subrack lattices, proving they are complemented for finite racks, characterizing modular cases, and exploring properties of G-racks, while also providing counterexamples for infinite racks.

Contribution

It proves that subrack lattices of finite racks are complemented and characterizes modular lattices, introducing G-racks and analyzing their homotopy types.

Findings

Finite rack sublattice is complemented

G-racks have the homotopy type of a sphere

Infinite racks can have non-complemented subrack lattices

Abstract

In this paper, a question due to Heckenberger, Shareshian and Welker on racks in [7] is positively answered. A rack is a set together with a selfdistributive 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 aformentioned 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.