A Class of Self-Distributive Quasigroup and its Parastrophs

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Ogunrinade, S.O
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The study of quasigroups dated back to over two centuries, and despite its antiquity, the area remained relatively undeveloped. In the wake of development, new results arising from research in this area set new problems that lead to interesting mathematics whose very existence had not been well explored. A quasigroup is distributive when it satis es the left and the right distributive laws. A distributive quasigroup (Q; ) equipped with the key laws known as left-right sided quasigroup is given consideration in this work with special attention to its parastrophs. The problems of parastrophy and isotopy-isomorphy properties of left-right sided quasigroup arose with the questions: Are parastrophs of left-right sided quasigroup (Q; ) also left-right sided quasigroup? What type of quasigroup can be obtained by universal left-right sided quasgroup? What additional condition must be imposed on a left(right) sided quasigroup to make it a left(right) Bruck loop? Is the holomorph of a left-right sided quasigroup also a left-right sided quasigroup? It is the purpose of this work to give a rmative answers to the questions raised. The relationship of the given quasigroup (Q; ) with the ve associated operations ( ; ; ; =; n) called conjugates or parastrophs and the left(right) principal isotope of the quasigroup are used to generate examples of left-right sided quasigroup which are parastrophic to the given quasigroup (Q; ). It is shown that the parastrophs of a left-right sided quasigroup are left-right sided quasigroups. Also, the left sided quasigroup is left universal while the right sided quasigroup is right universal. Furthermore, a left sided quasigroup is isotopic to a left Bruck loop and a right sided quasigroup is isotopic to a right Bruck loop. Finally, it is established that if (Q; ) is a left-right sided quasigroup, then A(Q)-holomorph (H; ) is a distributive groupoid if and only if A(Q) (that is, a submonoid of endomorphism semigroup Q of the distributive quasigroup) is left-right sided quasigroup and x (yz) = (x y)z(yz) x = (y x) (z x) hold for all , , 2 A(Q) and for all x; y; z 2 Q. Therefore, viii results in this work are interesting in the sense that a new class of distributive quasigroup with useful algebraic properties is constructed. This class of quasigroup generalises those existing in the literature. Thus, our results improve and extend many celebrated results in this area of research.
A Thesis Submitted to the School of Postgraduate Studies, University of Lagos
Quasigroup , Holomorphy , Parastrophy , Isotopy-isomorphy
Ogunrinade, S.O (2017). A Class of Self-Distributive Quasigroup and its Parastrophs. A Thesis Submitted to University of Lagos School of Postgraduate Studies Phd Thesis, 98p.