The rack congruence condition and half congruences in racks

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.