You should be interested in the discussion in
Evans, Gareth A.; Wensley, Christopher D., Complete involutive rewriting systems, J. Symb. Comput. 42, No. 11-12, 1034-1051 (2007). ZBL1128.68044.
on the notion of noncommutative involutive systems, which in the commutative case are a well known modification of the Gröbner basis theory. Evans' thesis is available at math.RA/0602140.
I can't help on the proposed applications.
Ronnie Brown
