Congruences in racks and quandles : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū Campus, Palmerston North, New Zealand
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Date
2022
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Massey University
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Abstract
Racks and quandles are related algebraic structures based on axioms of invertibility
and self-distributivity, and in the case of quandles, an additional idempotence
axiom — thus every quandle is a rack. They have practical application as the
three quandle axioms algebraically encode the Reidemeister moves of knot theory.
However, racks and quandles are interesting and worthy of study in their own right
and that is what we do here. Congruences are a means of distilling patterns of
behaviour within algebraic structures. They allow us to form a quotient that gives
us a coarser view of the structure from which we can discern interesting properties.
Congruences need to respect the operations in the algebraic structure.
Racks, although often defined in terms of only one binary operation, necessarily,
as a result of the invertibility axiom, have two binary operations — a primary
rack operation and an inverse rack operation. We have a rack in the quotient
only when the congruence respects both operations. A congruence that respects
both operations we call a rack congruence or a quandle congruence. Congruences
defined in terms of only one of the binary operations may not preserve the rack
structure in the quotient. This raises the question of whether congruences that
respect only one rack operation — half congruences — can exist. We show they
can by constructing examples of half congruences that do not induce a rack in the
quotient.
For weighted average quandles on Q we completely characterise congruences in
terms of certain subgroups of Q. Depending on the weight, congruences can exhibit
one of three possible behaviours. Weighted average quandles are a special case of
the more general Alexander quandle. For Alexander quandles, we characterise
when a congruence induces an Alexander quandle in the quotient. In weighted
average quandles every congruence comes from a subgroup of Q. In Alexander
quandles, there are additional congruences that do not come from a subgroup. We
give examples of congruences that exhibit that more complex behaviour.
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Keywords
Congruences and residues, Algebra