# The Theorem of Minkowski for Polyhedral Monoids and Aggregated Linear Diophantine Systems

• Achim Bachem
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 157)

## Abstract

We study polyhedral monoids of the form M = {xεZn / Ax ≤ 0} for (m, n) integer matrices with rank m and prove in an elementary and constructive way that M has a finite basis, i.e. every x#x03B5;M is the nonnegative integer linear combination of a finite set of vectors. We show that this theorem holds also for monoids M(N, B)={xɛZ + S / Nx + By=o, yɛZ>n}. We consider the aggregated system GNx+GBy=o where G is an (r,m) aggregation matrix and show how the cardinality of a span of M(GN,GB) and M(N,B) relate to each other. Moreover we show how the group order of the Gomory group derived from M(N,B) changes if we aggregate Nx+By=o to GNx+GBy=o.

## Keywords

Aggregation Matrix Group Order Integer Matrix Finite Basis Aggregate System
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