# Algebraic Models For Accounting Systems by Salvador Cruz Rambaud, José García Pérez, Robert A Nehmer,

By Salvador Cruz Rambaud, José García Pérez, Robert A Nehmer, Derek J S Robinson

This ebook describes the development of algebraic versions which characterize the operations of the double access accounting approach. It provides a singular, finished, evidence established remedy of the subject, utilizing such innovations from summary algebra as automata, digraphs, monoids and quotient buildings.

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1 is a free R-module with rank k k (ji − 1) = ( i=1 ji ) − k = n − k. 3. , k = 1; this is because Baln (R) has rank n − 1. 5). The homomorphism θπ is surjective if and only if π is an n-cycle. 3 we present an example. 1). Let n = 5 and choose π to be the permutation (1 2 3)(4 5). 3 a general element of Im(θπ ) should have the form   u1   u2    u = −u1 − u2  . 3, we form the vector   u1  u1 + u 2     v=  0 .  u3  0 Then       u1 0 u1  u1 + u 2   u1   u2         − u1 + u2  = −u1 − u2  = u, 0 θπ (v) = v − π(v) =         u3   0   u3  0 u3 −u3 as predicted.

The set of all such transactions will be denoted by Transn (R) = {τv | v ∈ Baln (R)} . Notice that the zero vector 0 corresponds to the identity function since τ0 (x) = x + 0 = x. Thus τ0 is the identity transaction, which causes no change in the system. Recall that two functions α, β from a set to itself can be combined by using functional composition to yield a new function, the composite α ◦ β, defined by α◦β(x) = α(β(x)). In the case of transactions τv and τw , observe that τv ◦ τw sends x ∈ Baln (R) to (x + w) + v = x + (v + w), as does τw ◦ τv .

U = . . .   .  1 1 1 · · · 1 2 1 1 1 ··· 1 1 It is easy to compute its determinant by using row operations; in fact det(U ) = (−1)k . The matrix U is used to construct a matrix A with the required properties by the following procedure. First divide n − 1 by k − 1 to get a quotient q and a remainder r, both of which are integers; thus n − 1 = (k − 1)q + r 48 Chapter 2. Balance Vectors and 0 ≤ r < k − 1. The (n − 1) × (n − 1) matrix A is to have q blocks U down the main diagonal, with other entries 0 or 1 according to the following scheme:   1 ··· 1 U  ..