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Reality-Based Algebras, Generalized Camina-Frobenius Pairs, and the Nonexistence of Degree Maps

2012; Taylor & Francis; Volume: 40; Issue: 4 Linguagem: Inglês

10.1080/00927872.2010.533224

1532-4125

Harvey I. Blau, Gang Chen,

graph theory and CDMA systems

Abstract The notion of a generalized Camina-Frobenius pair is extended to reality-based algebras, and a construction that characterizes such pairs is given. Zero-product sets are defined, and a best-possible upper bound on their size is proved and related to Camina-Frobenius pairs. It is shown that there exist commutative reality-based algebras with zero-product sets and, hence, no degree map, of every dimension at least 4. All such 4-dimensional algebras are constructed explicitly. Key Words: Camina-Frobenius pairDegree mapFusion ringHoheisel algebraLinear characterReality-based algebraTable algebra2000 Mathematics Subject Classification: 16K2016W10 ACKNOWLEDGMENTS Some of the work on this article was done during the visit of the second author at Northern Illinois University from February, 2009 to January, 2010. He appreciates the hospitality of the first author and the Department of Mathematical Sciences of Northern Illinois University. The work of the second author is supported by China Scholarship Council, The Key Project of Chinese Ministry of Education, No. 108099 and the National Science Foundation of China, No. 11001094. Both authors are grateful to William Blair for useful discussions; in particular, for his suggestions that led toward the specific form of Definition 3.1.