Finite combinatorial generation of metabelian $T$-ideal

In this work, we develop our idea on the construction of a system of combinatorial generators in a $T$-ideal of a free associative algebra, which is a full analogy of a Gröbner–Shirshov basis in a polynomial ideal. We prove a theorem on multilinear monomials that enables us to establish the existence of a finite set of combinatorial generators in a metabelian $T$-ideal.