arXiv subjects

Sunday Peter David

Publications and source records attributed to Sunday Peter David.

2 recordsLinked to original sources

arXiv2018

Holomorphic Structure of Middle Bol Loops

A loop $(Q,\cdot,\backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yz\backslash x)=(x/z)(y\backslash x)$. To every right (left) Bol loop corresponds a middle Bol loop via an isostrophism. In this paper, the structure of the holomorph of a middle Bol loop is explored. For some special types of automorphisms, the holomorph of a commutative loop is shown to be a commutative middle Bol loop if and only if the loop is a middle Bol loop and its automorphism group is abelian and a subgroup of both the group of middle regular mappings and the right multiplication group. It was found that commutativity (flexibility) is 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 combined holomorph of a middle Bol loop and its corresponding right (left) Bol loop was shown to be equal to the holomorph of the middle Bol loop if and only if the automorphism group is abelian and a subgroup of the multiplication group of the middle Bol loop. The obedience of an identity dependent on automorphisms was found to be a necessary and sufficient condition the left combined holomorph of a middle Bol loop and its corresponding left Bol loop to be equal to the holomorph of the middle Bol loop.

math.GR↗

arXiv2016

New Algebraic Properties of Middle Bol Loops

A loop $(Q,\cdot,\backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yz\backslash x)=(x/z)(y\backslash x)$. In this paper, some new algebraic properties of a middle Bol loop are established. Four bi-variate mappings $f_i,g_i,~i=1,2$ and four $j$-variate mappings $α_j,β_j,ϕ_j,ψ_j,~j\in\mathbb{N}$ are introduced and some interesting properties of the former are found. Neccessary and sufficient conditons in terms of $f_i,g_i,~i=1,2$, for a middle Bol loop to have the elasticity property, RIP, LIP, right alternative property (RAP) and left alternative property (LAP) are establsihed. Also, neccessary and sufficient conditons in terms of $α_j,β_j,ϕ_j,ψ_j,~j\in\mathbb{N}$, for a middle Bol loop to have power RAP and power LAP are establsihed. Neccessary and sufficient conditons in terms of $f_i,g_i,~i=1,2$ and $α_j,β_j,ϕ_j,ψ_j,~j\in\mathbb{N}$, for a middle Bol loop to be a group, Moufang loop or extra loop are established. A middle Bol loop is shown to belong to some classes of loops whose identiites are of the J.D. Phillips' RIF-loop and WRIF-loop (generalizations of Moufang and Steiner loops) and WIP power associative conjugacy closed loop types if and only if some identities defined by $g_1$ and $g_2$ are obeyed.

math.GR↗