Combinatorial approach to Milnor invariants of welded links

For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\bar{\mu }$-invariants. Recently, Chrisman extended Milnor $\bar{\mu }$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $\bar{\mu }$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $\bar{\mu }$-invariants of classical links.