The analogue, for general partial orders, of commutative idempotent semigroups for semilattices

Question

I know that the variety of commutative idempotent semigroups, along with the binary relation defined by the open formula $x*y=x$, gives rise to the class of partial orders which are meet-semilattices. Inspired by this, I want to ask, is there a variety of magmas and a corresponding binary relation defined by an open formula in $x$ and $y$, which generates precisely the class of partial orders? If so, can someone give such a variety and a binary relation? And if not, can someone prove it?