• • • • • • • • ## “School of Mathematics”

Back to Papers Home
Back to Papers of School of Mathematics

Paper   IPM / M / 16306
 School of Mathematics Title: On the lattice of subquandles of a Takasaki quandle Author(s): Dariush Kiani (Joint with A. Saki) Status: Published Journal: Comm. Algebra Year: 2020 Pages: DOI: 10.1080/00927872.2020.1797065 Supported by: IPM
Abstract:
Let A be an abelian group. If for any x,y 2A we define x . y Â¼2xy, then Ã°A,.Ã is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by RÃ°AÃ. Indeed, we characterize the subquandles of T(A) and show that RÃ°AÃ is the lattice of cosets of A if and only if A is an odd abelian group or its 2-Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing jAj and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that RÃ°AÃ is graded if and only if A is an odd abelian group or its 2-Sylow subgroup is cyclic or its 2-Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but RÃ°QÃ is isomorphic to RÃ°Z8Ã: