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On the lattice of subquandles of a Takasaki quandle
Communications in Algebra ( IF 0.7 ) Pub Date : 2020-07-30 , DOI: 10.1080/00927872.2020.1797065
A. Saki 1 , D. Kiani 1, 2
Affiliation  

Abstract Let A be an abelian group. If for any we define , then 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 . Indeed, we characterize the subquandles of T(A) and show that 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 and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that 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 is isomorphic to

中文翻译:

论高崎乱子的子乱子的格子

摘要 设 A 为阿贝尔群。如果我们定义了 ,则 是一个称为高崎 quandle 的 quandle,用 T(A) 表示。在本文中,我们关注有限高崎量子 T(A) 的子量子点格,用 表示。实际上,我们刻画了 T(A) 的子群,并证明了它是 A 的陪集格当且仅当 A 是奇阿贝尔群或其 2-Sylow 子群是循环群。此外,我们确定了 T(A) 的子量子点格的同伦类型,它是维度为 d 的 r 个球体的楔形,其中 r 是素数相除的个数,d 是通过应用有限阿贝尔群的基本定理来确定的. 此外,我们证明当且仅当 A 是奇阿贝尔群或其 2-Sylow 子群是循环的或其 2-Sylow 子群是初等阿贝尔群时,它是分级的。而且,我们证明了一个有限的 Takasaki quandle 的 subquandles 的格是由一个唯一的 Takasaki quandle 决定的。换句话说,我们证明了两个有限的 Takasaki quandles T(A) 和 T(B) 的 subquandles 的格是同构的当且仅当 A 和 B 是同构的。然而,不是高崎四面体的四面体的子四面体的格子可以与高崎四面体的子面体的格子同构。我们通过提供一个 8 阶的 quandle Q 来证明这一点,它不是 Takasaki quandle,而是同构于 不是 Takasaki quandle 的 quandle 的 subquandles 的格子可以同构为 Takasaki quandle 的 subquandles 格子。我们通过提供一个 8 阶的 quandle Q 来证明这一点,它不是 Takasaki quandle,而是同构于 不是 Takasaki quandle 的 quandle 的 subquandles 的格子可以与 Takasaki quandle 的 subquandles 的格子同构。我们通过提供一个 8 阶的 quandle Q 来证明这一点,它不是 Takasaki quandle,而是同构于
更新日期:2020-07-30
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