Definition: In mathematics, a pupilate is a type of binary relation where every element in the domain has at least one preimage. This means that for any two elements x and y in the domain, there exists some element z such that (x, z) ∈ R. In other words, a binary relation R on a set A is said to be a pupilate if it satisfies the following property: ∀x∈A, ∃y∉A: (x, y) ∈ R.