A commutativity condition for subsets in quandles — a generalization of antipodal subsets

Abstract

In this paper, we introduce some commutativity condition for subsets in quandles, which we call the s-commutativity. Note that quandles can be regarded as a generalization of symmetric spaces, and the notion of s-commutative subsets is a generalization of antipodal subsets. We study maximal s-commutative subsets in quandles, and show that they have some nice properties. As one example, any maximal s-commutative subsets in quandles are subquandles. We also determine maximal s-commutative subsets in spheres, projective spaces, and dihedral quandles. In these quandles, maximal s-commutative subsets turn out to be unique up to automorphisms.