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In this paper, we introduce (almost) skew 2-nomial algebras and look for aone-sided or two-sided Gr\"obner basis theory for such algebras at a modestlevel. That is, we establish the existence of a skew multiplicative $K$-basisfor 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 Gr\"obner basis theory(such as algebras of the solvable type [K-RW] and some of their homomorphicimages), a subclass of skew 2-nomial algebras that have a left Gr\"obner basistheory but may not necessarily have a two-sided Gr\"obner basis theory,respectively a subclass of skew 2-nomial algebras that have a right Gr\"obnerbasis theory but may not necessarily have a two-sided Gr\"obner basis theory,are determined such that numerous quantum binomial algebras (which providebinomial solutions to the Yang-baxter equation [Laf], [G-I2]) are involved.

Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras