Figure 1: Directed graphs determining reflexive relations. be a preorder. To define multiplication we will assume X is locally finite, that is, every interval of X is a finite set. (Recall an interval is a subset of the form [ x, y ] = { z ∈ X : xρz and zρy } for some x, y ∈ X .) Throughout the rest of this section ρ is a locally finite relation on a set X and R is an associative ring with unity. Let I ( X, ρ, R ) denote the set of all functions f : X × X → R such that f ( x, y ) = 0 implies xρy . Componentwise operations determine an R -module such that given r ∈ R and f, g ∈ I ( X, ρ, R ) the functions rf, f + g ∈ I ( X, ρ, R ) satisfy ( rf ) ( x, y ) = r ( f ( x, y )) and ( f + g ) ( x, y ) = f ( x, y ) + g ( x, y ) for all x, y ∈ X . The balance property is satisfied by w, x, y, z ∈ X if any of the relations wρx , xρy , yρz , and wρz do not hold, or all four relations hold and wρy if and only if xρz . The relation is balanced if it is reflexive and the balance property is satisfied by all w, x, y, z ∈ X . The relations determined by directed graphs (a), (b), and (c) in Figure 1 are not balanced because they are all missing arrows. Note that a reflexive and transitive relation is balanced but there are balanced relations, such as the one determined by (d) in Figure 1, which are not transitive. In case ρ is balanced we can combine the proof of [1, Proposition 1.2] with the assumption that X is locally finite to construct a ring multiplication with identity 4