Multiplicative Bases, Gröbner Bases, and Right Gröbner Bases

Before surveying the results of the paper, we introduce path algebras. Path algebras play a central role in the representation theory of finite-dimensional algebras (Gabriel, 1980; Auslander et al., 1995; Bardzell, 1997) and the theory of Gröbner bases (Bergman, 1978; Mora, 1986; Farkas et al., 1993) has been an important tool in some results (Feustel et al., 1993; Green and Huang, 1995; Bardzell, 1997; Green et al., to appear). In this paper we show that in sense, if there is a Gröbner basis theory for an algebra, that algebra is naturally a quotient of a path algebra; see the survey of results below. To understand the results of the paper, the reader needs to know what a path algebra is. Let Γ be a directed graph with vertex set Γ0 and arrow set Γ1. We usually assume that both Γ0 and Γ1 are finite sets but Γ1 need not be finite in what follows except as noted below. Let B be the set of finite directed paths in Γ, including the vertices viewed as paths of length 0. The path algebra KΓ, has as K-basis B. Multiplication is given by concatenation of paths if they meet or 0. More precisely, if p is a path from vertex v to vertex w and q is a path from vertex x to vertex y, then p · q is the path pq from v to y if w = x or else p · q = 0 if w 6= x. See Auslander et al. (1995) for a fuller description. Note that the free associative algebra on n noncommuting variables is the path algebra with Γ having one vertex and n loops. The loops correspond to the variables and the basis of paths correspond to the words in the variables. Note that B ∪ {0} is a monoid with 0. The multiplicative monoid B∪{0} is finitely generated if and only if Γ1 is a finite set. The only time we must assume that Γ1 is finite is in the case where we assume that B ∪ {0} is finitely generated or when we study finite-dimensional quotients of KΓ. There is a well-established Gröbner basis theory for path algebras; see Farkas et al. (1993), Green (1999). In particular, the basis of paths, B, has many admissible orders; see Section 2 for the definition of an admissible order. For example, there is the lengthlexicographic order where we arbitrarily order the vertices, say v1 < v2 < · · · < vr, and arbitrarily order the arrows all larger than the vertics, say vr < a1 < · · · < as. If p and q