Looking for Gröbner basis theory for (almost) skew 2-nomial algebras

In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gröbner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative K-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gröbner basis theory (such as algebras of the solvable type (Kandri-Rody and Weispfenning, 1990) and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gröbner basis theory but may not necessarily have a two-sided Gröbner basis theory, respectively a subclass of skew 2-nomial algebras that have a right Gröbner basis theory but may not necessarily have a two-sided Gröbner basis theory, are determined such that numerous quantum binomial algebras (which provide binomial solutions to the Yang–Baxter equation) and their Koszul dual (Gateva-Ivanova and Van den Bergh, 1998; Laffaille, 2000; Gateva-Ivanova, 2009) are involved.