New algebraic properties of middle Bol loop
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Date
2015
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Publisher
Mathematics,Obafemi Awolowo University
Abstract
The objectives of this study were to establish some new algebraic properties of a middle Bol loop; investigate the relationship between a middle Bol loop, right (left) Bol loops and some inverse property loops like WIPLs, CIPLs, AIPLs, SAIPLs, RIPLs and IPLs; establish some new algebraic properties of translations, autotopisms and anti-autotopisms of a middle Bol loop and study the holomorphic structure of a middle Bol loop. This was with a view to preparing a good ground for the reformulation of the 1994 Syrbu’s question on the equivalence of the universal elasticity condition (UEC) and the middle Bol identity (MBI).
Existing literature on the algebraic study of middle Bol loops were surveyed and all accessible materials relevant to the concepts of loops, Moufang loops and Bol loops were also acquired, particularly those that relate them with the concepts of parastophes, extra loop, groups and isotopy. Existing results on middle Bol loops, middle inner mapping Tx and algebraic properties of middle Bol loops by Grecu and Syrbu were employed. The middle
Bol identity was judiciously used to investigate the characterizations of middle Bol loops (Q,·,\,/). The middle inner mapping Tx was used to produce new algebraic properties of middle Bol loops (Q,·). The parastrophes of a middle Bol loop were also used to explore new algebraic properties. The anti-autotopic form of a middle Bol loop was also used to investigate the algebraic properties of autotopisms of middle Bol loops. The structure of the holomorph of a middle Bol loop was also explored.
The result established some new algebraic properties of a middle Bol loop among which were the following near-balanced identities: yx\x = x\(y\x), xz\x = x\(x/z), (yz)\y = y\(y/z), (yz)\z = z\(y\z), x(z\x) = (x/z)x and (x/yz)x = (x/z)(y\x). It was also established that WIP, RIP, LIP and IP were equivalent in a middle Bol loop. Furthermore, commutative WIP, commutative RIP, commutative LIP and commutative IP were equivalent in a middle Bol loop. Two middle Bol loops using a right Bol loop and a ring were constructed. A new method of constructing a middle Bol loop using a non-abelian group and a subgroup of it was developed. The holomorph of a loop was shown to be a middle Bol loop if and only if the loop was a middle Bol loop. For some special autotopisms, it was found that commutativity (flexibility) was a necessary and sufficient condition for holomorphic invariance under the existing isostrophy between middle Bol loops and the corresponding right (left) Bol loops. The right(left) combined holomorph of a middle Bol loop and its corresponding right (left) Bol loop were shown to be equal to the holomorph of the middle Bol loop.
The study concluded that the Syrbu’s open problem can be solved using the algebraic properties of middle Bol loops in this work. And that the following two statements (yx)u = x ⇔ y(xu) = x and (xz)u = x ⇔ (xu)z = x which were found to be true in a middle Bol
loop were useful for cryptosystems.
Description
xi, 126p
Keywords
Cryptosystems, Bol loops, Syrbu, Algebraic
Citation
Peter D.S, (2015). New algebraic properties of middle Bol loop. Obafemi Awolowo University