Graduation Semester and Year

2015

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Minerva Cordero

Abstract

In 2002, A.D Keedwell and V.A Sherbacov introduced the concept of finite m-inverse quasigroups with long inverse cycles. Keedwell and Sherbacov observed that finite m-inverse loops and quasigroups with a long inverse cycle could be useful in the study of cryptology. Keedwell and Sherbacov studied the existence of these algebraic structures by determining if a Cayley table of the elements of such structures could be constructed. They showed that m-inverse loops of order 9 with a long inverse cycle do not exist for m = 2; 4 and 6; thus, there do not exist 2,4, or 6 inverse-quasigroups of order 8. However the investigation of 3 or 7-inverse loops of order 9 and of 3 or 7-inverse quasigroups of order 8 with a long inverse cycle was considered more complicated and was left unanswered. In this paper we attack the unanswered question of the existence of 3 and 7-inverse loops and quasigroups with long inverse cycles. We also investigate the following two problems: (i)The existence of m-inverse loops with a long inverse cycle of orders 11 and 15. (ii)The existence of m-inverse quasigroups with a long inverse cycle of order 12,16 and 20.

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

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