The crux of this thesis is to study the Restricted Quackenbush Problem: if B is a finite algebra of finite type and V(B) is residually finite, must V(B) be residually <N, for some positive integer N? We show that the Restricted Quackenbush Problem is answered affirmatively with respect to unary algebras, a result first shown by Baldwin and Berman in 1975. Then we turn our attention to groupoids and show that all groupoids that generate residually finite varieties must satisfy an identity of the form k(x)≈x∗x, where k(x) is an identity of type {∗} that is not equal to x∗x. Due to this result, we focus on a particular class of idempotent groupoids, called absorbing groupoids. We show that if a certain property holds, with respect to absorbing groupoids, then the Restricted Quackenbush Problem is answered affirmately.
