## 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

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Looney, Carl Edward, "On M-inverse Loops And Quasigroups Of Order N With A Long Inverse Cycle." (2015). *Mathematics Dissertations*. 88.

https://mavmatrix.uta.edu/math_dissertations/88

## Comments

Degree granted by The University of Texas at Arlington