Norm Inequalities in Zeon Algebras

The zeon (nil-Clifford) algebra Cnnil can be thought of as a commutative analogue of the n-particle fermion algebra and can be constructed as a subalgebra of a Clifford algebra. Combinatorial properties of the algebra make it useful for applications in graph theory and theoretical computer science. In this paper, the zeon p-norms and infinity norms are introduced. The 1-norm is shown to be the only sub-multiplicative p-norm on zeon algebras. Multiplicative inequalities involving the infinity norm (which is not sub-multiplicative) are developed and equivalence of norms in Cnnil is used to establish a number of multiplicative inequalities between p-norms and the infinity norm. As an application of norm inequalities, necessary and sufficient conditions for convergence of the zeon geometric series are established, and the series limit is expressed as a finite sum. The exposition is supplemented by a number of examples computed using Mathematica.