In this paper we consider several families of abstract algebras including the well-known $BCK$-algebras and several larger classes including the class of $d$-algebras which is a generalization of $BCK$-algebras. For these algebras it is usually difficult and often impossible to obtain a complementation operation and the associated ``de Morgan's laws". In this paper we construct a ``mirror image" of a given algebra which when adjoined to the original algebra permit a natural complementation to take place. The class of $BCK$-algebras is not closed under this operation but the class of $d$-algebras is, thus explaining why it may be better to work with this class rather than the class of $BCK$-algebras. Other classes of interest in this setting are also discussed.