Algebraic Models For Accounting Systems

Algebraic Models for Accounting Systems This page intentionally left blank Algebraic Models for Accounting Systems Salvador Cruz Rambaud José García Pérez University of Almeria, Spain Robert A Nehmer Oakland University, USA Derek J S Robinson University of Illinois at Urbana-Champaign, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ALGEBRAIC MODELS FOR ACCOUNTING SYSTEMS Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4287-11-1 ISBN-10 981-4287-11-3 Printed in Singapore. “The imagination . . . gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors.” Samuel Taylor Coleridge, “The Statesman’s Manual”, 1816. This page intentionally left blank Preface In recent years there has been no shortage of applications of mathematics to economics, mainly through the use of methods from statistics, probability and risk analysis. It is much harder to find significant applications of abstract algebra to the area. However, the rise of the information sciences has clearly displayed the opportunities for applying what used to be considered the purest of pure mathematics. It is now commonplace for students of computer science to take the time to acquire a basic knowledge of algebra. It is not hard to see why algebra should be enjoying popularity: in the analysis of complex systems of all kinds the power and precision of algebraic concepts, and sometimes just algebraic notation, can be an enormous aid. On the other hand, one can search the literature in accounting theory and find few attempts to make use of algebra, and what there is tends to be at quite a modest level. The object of the present work is to make the case for applying algebra to the study of accounting systems by finding algebraic concepts which are able to reflect accurately the workings of real life systems. The benefits of such a study are diverse: the demand of algebra for precision compels us to question and make exact everyday ideas and processes in order to express them in abstract form. It also serves to provide tools to analyze accounting systems. The concepts which appear most frequently in the present study are: column vectors with zero sum, the so called balance vectors, which reflect the perfect balance of an accounting system; directed graphs to show the flow of value through the system; automata to model the comput