The lattice of subracks is atomic

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:162

Article

Kiani, D.^1,2  Saki, A.¹ 

[1] Amirkabir Univ Technol, Tehran Polytech, Fac Math & Comp Sci, Dept Pure Math, 424 Hafez Ave, Tehran 15914, Iran

[2] Inst Res Fundamental Sci IPM, Sch Math, POB 19395-5746, Tehran, Iran

关键词: Rack;    Quandle;    Lattice of subracks;    Atomic lattice;    Distributive lattice;   

DOI  :  10.1016/j.jcta.2018.09.010

来源: Elsevier

【 摘 要 】

A rack is a set together with a self-distributive bijective binary operation. In this paper, we give a positive answer to a question due to Heckenberger, Shareshian and Welker. Indeed, we prove that the lattice of subracks of a rack is atomic. Further, by using the atoms, we associate certain quandles to racks. We also show that the lattice of subracks of a rack is isomorphic to the lattice of subracks of a quandle. Moreover, we show that the lattice of subracks of a rack is distributive if and only if its corresponding quandle is the trivial quandle. So the lattice of subracks of a rack is distributive if and only if it is a Boolean lattice. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free