e : X × X → R such that e ( x, x ) = 1 and e ( x, y ) = 0 for all x, y ∈ X such that x = y . In this case we call I ( X, ρ, R ) the generalized incidence ring of X with coefficients in R . If ρ is a partial order then I ( X, ρ, R ) is the incidence ring over R . The multiplication is defined so that the product of f, g ∈ I ( X, ρ, R ) is the function given by equation 1 for all x, y ∈ X . This is called convolution . ( fg ) ( x, y ) =      ∑ z ∈ [ x,y ] f ( x, z ) g ( z, y ) if xρy in X 0 otherwise (1) Obviously if the sum in equation 1 is nonzero then xρy in X . In this case [ x, y ] is finite so convolution uniquely determines an element of R . Consider f, g, h ∈ I ( X, ρ, R ) and w, z ∈ X with wρz . A nonzero term of (( fg ) h ) ( w, z ) is determined by y ∈ [ w, z ] and x ∈ [ w, y ]. Thus there are relations wρx , xρy , yρz , wρz , and wρy . Since the relation is balanced these are equivalent to wρx , xρy , yρz , wρz , and xρz , or x ∈ [ w, z ] and y ∈ [ x, z ]. This gives an identical term of ( f ( gh )) ( w, z ). Since the nonzero terms match up over the relations we have ( fg ) h = f ( gh ). For each x, y ∈ X such that xρy there exists e$_{xy}$ ∈ I ( X, ρ, R ) such that e$_{xy}$ ( i, j ) is given by equation 2 for all i, j ∈ X . e$_{xy}$ ( i, j ) =      1 if x = i and y = j 0 otherwise (2) If w ∈ X then wρw since a balanced relation is reflexive. It is easy to show ( e$_{ww}$ ) 2 = e$_{ww}$ for all w ∈ X directly from the definition in 1. Equations 3 and 4 also 5