Purely existentially positively definable subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a purely existentially positively definable subgroup if $$H$$ can be defined as the subset of $$G$$ comprising precisely those elements that satisfy a first-order logic condition where all the variables are existentially quantified and there are no negation symbols (so the only conditions we have are equality conditions, no inequality conditions).