Graph variety generated by linear terms

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

Graph varieties containing Murskii's groupoid

In this paper varieties are investigated which are generated by graph algebras of undirected graphs and—in most cases—contain Murskii's groupoid (that is the graph algebra of the graph with two adjacent vertices and one loop). Though these varieties are inherently nonfinitely based, they can be finitely based as graph varieties (finitely graph based) like, for example, the varitey generated by Murskii's groupoid. Many examples of nonfinitely based graph varities containing Murskii's groupoid are given, too. Moreover, the coatoms in the subvariety lattice of the graph variety of all undirected graphs are described. There are two coatoms and they are finitely graph based.