Relative Gröbner–Shirshov bases for algebras and groups

The notion of a relative Gröbner–Shirshov basis for algebras and groups is introduced. The relative composition lemma and relative (composition-)diamond lemma are established. In particular, it is shown that the relative normal forms of certain groups arising from Malcev's embedding problem are the irreducible normal forms of these groups with respect to their relative Gröbner–Shirshov bases. Other examples of such groups are given by showing that any group $G$ in a Tits system $(G,B,N,S)$ has a relative ($B$-)Gröbner–Shirshov basis such that the irreducible words are the Bruhat words $G$.