Satellites of an oriented surface link and their local moves

For an oriented surface link F   in R4R4, we consider a satellite construction of a surface link, called a 2-dimensional braid over F, which is in the form of a covering over F. We introduce the notion of an m  -chart on a surface diagram π(F)⊂R3π(F)⊂R3 of F  , which is a finite graph on π(F)π(F) satisfying certain conditions and is an extended notion of an m-chart on a 2-disk presenting a surface braid. A 2-dimensional braid over F is presented by an m  -chart on π(F)π(F). It is known that two surface links are equivalent if and only if their surface diagrams are related by a finite sequence of ambient isotopies of R3R3 and local moves called Roseman moves. We show that Roseman moves for surface diagrams with m-charts can be well-defined.