The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra

Abstract

By a graph we mean a pair (X, R) where X is a non-void set and R ⊂ X × X. A mapping f: X → Y is called a compatible map (or morphism) from (X, R) into (Y, S) if 2f(R) ⊂ S, where 2f: X2 → Y2 is defined by 2f((x1, x2)) = (f(x1),f(x2)). The set of all compatible maps from (X, R) into itself forms a monoid (semigroup with a unit element) under composition, which is denoted by M(X, R). A graph (X1, R1) is said to be a full subgraph of (X, R) if X1 ⊂ X and R1 = R ∩ (X1 × X1). A graph (X, R) is said to be without loops if (x, x) ∉ R for all x ∈ X.