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IPM
30
YEARS OLD

“School of Mathematics”

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Paper   IPM / M / 15520
School of Mathematics
  Title:   The lattice of subracks is atomic
  Author(s):  Dariush Kiani (Joint with A. Saki)
  Status:   Published
  Journal: J. Combin. Theory Ser. A
  Vol.:  162
  Year:  2019
  Pages:   55-64
  Supported by:  IPM
  Abstract:
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.

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