M ay 2 00 9 APPLICATIONS OF THE GRAPH MINOR THEOREM TO ALGEBRAIC STRUCTURES

We use a correspondence between Graphs1 and finite subsets of certain semigroups (the factorial and commutative ones, for example R\{0} if R is a unique factorization domain) and the Graph Minor Theorem to get a characterization of infinite sequences of finite subsets of those semigroups. Especially interesting is the case if N is the regarded semigroup. 1. Some aspects of Graph (Minor) Theory Robertson and Seymour proved the so-called Graph Minor Theorem in twenty groundbreaking papers, published from 1983 till 2004. To state this theorem, we need to introduce the definition of a minor. Definition 1. (see [1]) Let G and H be graphs. We say H is a minor of G, in notation H 4 G, when there is a partition {Vh}h∈V (H) of a subset of the vertex set of G with G[Vh] connected for each h, such that there is a Vh-Vh′ edge if hh ′ ∈ E(H). This definition doesn’t stand there directly, for this you have to read a bit between the lines. An equivalent definition, which can maybe better be used to understand the term of a minor is the following, but it won’t be necessary for the rest of the paper. Definition 2. (see [2]) H 4 G if H can be obtained from G by a sequence of the following operations: • Delete an edge • Delete a vertex • Contract an edge The contraction of an edge is defined in the obvious way. It is easy to see that the minor relation is reflexive and transitive in the class of finite graphs. Now we can state the Graph Minor Theorem. Theorem 3. (see [3]) The finite graphs are well-quasi-ordered by the minor relation 4, i.e. 4 is reflexive and transitive and for an infinite sequence of graphs G1, G2, . . . there are i, j with i < j and Gi 4 Gj . Graphs are always finite in this paper. 2Here N starts at 1.