Square element

Definition
Suppose $$(S,*)$$ is a magma, i.e., $$S$$ is a set and $$*$$ is a binary operation on $$S$$. Then, an element $$x \in S$$ is termed a square element or square if there exists $$y \in S$$ such that $$x = y * y$$.

Note that $$S$$ may be a group or semigroup, which is the typical context of use; however, the notion of square element makes sense even for non-associative binary operations.