Norm Inequalities in Zeon Algebras

  • Theresa Lindell
  • G. Stacey StaplesEmail author


The zeon (“nil-Clifford”) algebra \({\mathcal {C}\ell _n}^\mathrm{nil}\) 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 \({\mathcal {C}\ell _n}^\mathrm{nil}\) 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.


Clifford algebra Zeons Norms Inequalities Series 

Mathematics Subject Classification

15A66 81R05 05A20 



The authors thank the anonymous reviewers for helpful comments.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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