Table of Contents
International Scholarly Research Notices
Volume 2015, Article ID 242515, 8 pages
http://dx.doi.org/10.1155/2015/242515
Research Article

Tan’s Epsilon-Determinant and Ranks of Matrices over Semirings

Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1

Received 28 November 2014; Accepted 9 January 2015

Academic Editor: Qing-Wen Wang

Copyright © 2015 Preeti Mohindru and Rajesh Pereira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use the ϵ-determinant introduced by Ya-Jia Tan to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.