  Racks and quandles are algebraic structures with a single binary operation
that is right self-distributive and right invertible, and additionally
idempotent in the case of quandles. The invertibility condition is equivalent
to the existence of a second binary operation that acts as a right inverse to
the first, so that racks and quandles may also be viewed as algebraic
structures with a pair of (dependent) binary operations.
  When forming a quotient rack or quandle it is necessary to take this
two-operation view, and define a congruence as an equivalence relation on the
rack or quandle that respects both operations. However, in defining a
congruence some authors have omitted the condition on the inverse operation,
and defined a congruence as an equivalence relation respecting the primary
operation only. We show that while respecting the primary operation is
sufficient in the case of finite racks and quandles, it is not in general
sufficient in the infinite case. We do this by constructing explicit examples
of "half congruences": equivalence relations that respect exactly one of the
two operations.
  Our work also allows us to completely characterise congruences in connected
Alexander quandles.
