Editing Racks and quandles (section)

==Racks==
A '''rack''' may be defined as a set <math>\mathrm{R}</math> with a binary operation <math>\triangleleft</math> such that for every <math>a, b, c \in \mathrm{R}</math> the '''self-distributive law''' holds:
:<math>a \triangleleft(b \triangleleft c) = (a  \triangleleft b) \triangleleft(a  \triangleleft c)</math>

and for every <math>a, b \in \mathrm{R},</math> there exists a unique <math>c \in \mathrm{R}</math> such that 
:<math>a \triangleleft c = b.</math>

This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an existential quantifier which is not really necessary. To avoid this, we may write the unique <math>c \in \mathrm{R}</math> such that <math>a \triangleleft c = b</math> as <math>b \triangleright a.</math>  We then have
:<math> a \triangleleft c = b \iff c = b \triangleright a, </math>

and thus 
:<math> a \triangleleft(b \triangleright a) = b,</math>

and
:<math>(a \triangleleft b) \triangleright a = b.</math>

Using this idea, a rack may be equivalently defined as a set <math>\mathrm{R}</math> with two binary operations <math>\triangleleft </math> and <math>\triangleright</math> such that for all <math>a, b, c \in \mathrm{R}\text{:}</math>
#<math>a \triangleleft(b \triangleleft c) =  (a \triangleleft b) \triangleleft(a \triangleleft c)</math> (left self-distributive law)
#<math>(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright(b \triangleright a)</math> (right self-distributive law)
#<math>(a \triangleleft b) \triangleright a = b</math>
#<math>a \triangleleft(b \triangleright a) = b</math>

It is convenient to say that the element <math>a \in \mathrm{R}</math> is acting from the left in the expression <math>a \triangleleft b,</math> and acting from the right in the expression <math>b \triangleright a.</math>  The third and fourth rack axioms then say that these left and right actions are inverses of each other.  Using this, we can eliminate either one of these actions from the definition of rack.  If we eliminate the right action and keep the left one, we obtain the terse definition given initially.

Many different conventions are used in the literature on racks and quandles.  For example, many authors prefer to work with just the ''right'' action.  Furthermore, the use of the symbols <math>\triangleleft</math> and <math>\triangleright</math> is by no means universal: many authors use exponential notation
:<math>a \triangleleft b = {}^a b</math>

and
:<math>b \triangleright a = b^a,</math>

while many others write
:<math>b \triangleright a = b \star a. </math>

Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right as [[automorphism]]s of the rack, with the left action being the inverse of the right one.  In this definition, the fact that each element acts as automorphisms encodes the left and right self-distributivity laws, and also these laws:

:<math>\begin{align}
  a \triangleleft(b \triangleright c) &= (a \triangleleft b) \triangleright(a\ \triangleleft c) \\
  (c \triangleleft b) \triangleright a &= (c \triangleright a) \triangleleft(b \triangleright a)
 \end{align}</math>

which are consequences of the definition(s) given earlier.