Colorings and doubled colorings of virtual doodles

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.