Signature Gröbner bases, bases of syzygies and cofactor reconstruction in the free algebra

Signature-based algorithms have become a standard approach for computing Gr\"obner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept the setting of noncommutative polynomials free algebra. In paper, we present signature-based algorithm precisely setting. The is an adaptation Buchberger's including signatures. We prove that our correctly enumerates signature basis as well module generated by leading terms generators' syzygies, and terminates whenever ideal admits finite basis. Additionally, adapt well-known criteria eliminating redundant reductions, such syzygy criterion, F5 criterion singular case polynomials. also generalize reconstruction methods from allow recover, partial information about signatures, coordinates elements input polynomials, generators. written toy implementation all Mathematica package OperatorGB compare classical Buchberger