Strict Refinement for Direct Sums and Graphs

We study direct sums of structures with a one element subuniverse. We give a characterization of direct sums reminescent to that of inner products of groups. The strict refinement property is adapted to direct sums and to restricted Cartesian products of graphs. If a structure has the strict refinement property (for direct products), it has the strict refinement property for direct sums. Connected graphs satisfy the strict refinement property for their restricted Cartesian products. Chang, Jónsson and Tarski introduce in [6] the strict refinement property for relational structures. Some of the ideas also appear in Fell and Tarski [9]. They show that for algebras with the strict refinement property, such as lattices, rings with zero annihilators and perfect groups, if an algebra A is a direct product of directly indecomposable algebras, then not only the directly indecomposable factors are unique up to isomorphism, but also the resulting factor congruence set on A is unique. In [23], Sabidussi defines relations on the edges of graphs that give a representation of certain connected graphs as Cartesian products of finitely many Cartesian indecomposable graphs and again these Cartesian indecomposable factors are unique up to isomorphism and the defined relation itself is unique. Cartesian products of infinite sets of connected nontrivial graphs are not connected. The strict refinement property is not (easily) applicable to Cartesian decompositions of graphs. In the present paper, we study the possibility of strict refinement for direct sums of structures and follow this study with an adaptation of the strict refinement property to graphs. For any set A we denote the identity or diagonal relation {(x, x) : x ∈ A} on A by ∆(A), and sometimes simply by ∆. If α is an equivalence relation on a set A and a ∈ A, a/α is the α-equivalence class of a; i.e., a/α = {x ∈ A : aαx}. If α, β are equivalence relations on a set A, then α ◦ β is the relational composition of α and β; i.e., x(α ◦ β)y iff there is z ∈ A such that xαz and zβy. A set of congruence relations {αi : i ∈ I} on an algebra A is called a direct factor set Received January 24, 1998; revised October 19, 1998. 1980 Mathematics Subject Classification (1991 Revision). Primary 05C99, 08B25; Secondary 68R10.