  We define enhanced presentations of quandles via generators and relations
with additional information comprising signed operations and an order structure
on the set of generators. Such a presentation determines a virtual link diagram
up to virtual moves. We list formal Reidemeister moves in which Tietze moves on
the presented quandle are accompanied by corresponding changes to the order
structure. Omitting the order structure corresponds to replacing virtual
isotopy by welded isotopy.
