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Racks and quandles are algebraic structures with a single binary operationthat is right self-distributive and right invertible, and additionallyidempotent in the case of quandles. The invertibility condition is equivalentto the existence of a second binary operation that acts as a right inverse tothe first, so that racks and quandles may also be viewed as algebraicstructures with a pair of (dependent) binary operations.  When forming a quotient rack or quandle it's necessary to take thistwo-operation view, and define a congruence as an equivalence relation on therack or quandle that respects both operations. However, in defining acongruence some authors have omitted the condition on the inverse operation,and defined a congruence as an equivalence relation respecting the primaryoperation only. We show that while respecting the primary operation issufficient in the case of finite racks and quandles, it is not in generalsufficient in the infinite case. We do this by constructing explicit examplesof "half congruences": equivalence relations that respect exactly one of thetwo operations.

The rack congruence condition and half congruences in racks