Determination of some algebraic properties of basarab loops
| dc.contributor.author | Effiong, Gideon Okon | |
| dc.date.accessioned | 2025-02-24T09:21:27Z | |
| dc.date.available | 2025-02-24T09:21:27Z | |
| dc.date.issued | 2021-12 | |
| dc.description | A Doctor of Philosophy on algebraic properties of basarab loops | |
| dc.description.abstract | Basarab loops are non-associative generalizations of groups and are classified as loops of non Bol-Moufang type. They are G-loops with deep algebraic and structural properties. Not much were known about the form of isotopes, holomorphs, associators, center, and subloops of Basarab loops. This work was to determine some algebraic properties of Basarab loops. The objectives of the study were to construct a Basarab loop, investigate the relationship between Basarab loop and other loops like conjugacy closed loop, abelian inner mapping loop, and Osborn loop, examine the isotopes of a Basarab loop, investigate the holomorphs and associators of a Basarab loop, and characterize some subloops of a Basarab loop. Basarab loop identities were considered and some algebraic properties of loops were investigated. Loop notions such as the use of parentheses, multiplication group, isotopy theory, and holomorphy theory, total multiplication group were examined on a Basarab loop through the governing laws of Basarab loop. Some constructions of Basarab loops were given and some algebraic properties of Basarab loops were determined. The results obtained have shown that the centrum of a Basarab loop is a subloop and it is equal to the center of a Basarab loop, and that a Basarab loop with the left (right) inverse property, or inverse property is an extra loop. Necessary and sufficient conditions for isotopes and principal isotopes of a Basarab loop were determined. It was proved that every principal isotope of a Basarab loop is a Basarab loop. It was proved that any Osborn loop is a Basarab loop if and only if it is a left (right) Basarab loop. Also, the holo morphs of a Basarab loop were investigated by considering a group A(Q) of automorphisms of a loop. Some necessary and sufficient conditions for an A(Q)-holomorph of a loop (Q, ·) to be left (right) Basarab loop, and Basarab loop were established. Some left (right) translation mapping of the holomorph of a left (right) Basarab loop was shown to be left (right) regular. It was shown that an A(Q)-holomorph of a loop (Q, ·) which satisfies the inverse property is a Basarab loop if and only if (Q, ·) is a Basarab loop and every automorphism of Q is nuclear.Some subloops of a Basarab loop which are characterized by permutations were obtained. It was proved that a Basarab loop is a centrum-abelian inner mapping loop. Relationship betweenassociators and inner mappings of a Basarab loop was defined. It was shown that the associator of any three elements of a Basarab loop is contained in the center and centrum of a Basarabloop. This study has presented additional properties of Basarab loops which are now available for applications. Therefore, it is recommended that researchers and cryptographers should usethe properties of Basarab loops determined by this study for further research and applications. | |
| dc.identifier.citation | Effiong, G. O. ( 2021) . Determination of some algebraic properties of basarab loops. (Unpublished Doctoral thesis). Federal University of Technology, Owerri. | |
| dc.identifier.uri | https://repository.futo.edu.ng/handle/20.500.14562/1643 | |
| dc.language.iso | en | |
| dc.publisher | Federal University of Technology, Owerri | |
| dc.rights | Attribution-NonCommercial-NoDerivatives 4.0 International | en |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | |
| dc.subject | Basarab loop | |
| dc.subject | isotopy | |
| dc.subject | holomorphy | |
| dc.subject | associator | |
| dc.subject | inner mapping | |
| dc.subject | mathematics | |
| dc.title | Determination of some algebraic properties of basarab loops | |
| dc.type | Doctoral Thesis |