John Fountain and Victoria Gould,
*Relatively free algebras with weak exchange properties*,
J. Aust. Math. Soc. **75** (2003) 355--384.

and papers referred to there.
The closure operators they consider are <*X*> =
subalgebra generated by *X*, and PC(*X*) ("pure
closure of *X*") = { *y* |
<{*y*}> ∩ <*X*> is
nontrivial }, in the latter case restricting
attention to algebras on which PC *is* a closure operator.