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.
"Signed ordered knotlike quandle presentations." Algebr. Geom. Topol. 5 (2) 443 - 462, 2005. https://doi.org/10.2140/agt.2005.5.443