This Thesis investigates the nature of the parastrophs and derivatives of loops both of
Bol-Moufang (Extra, Moufang, Central loops) and non Bol-Moufang (Conjugacy Closed
loops) type in general. Extra loops is the case study. By using Fenyves (1968, 1969)
definition of Extra loops and the results of Goodaire and Robinson (1982, 1990), this work
shows that the parastrophs and derivatives of an Extra loop exist. Taking into consideration
Ken Kunen (1996) results, it has been established here that two of the six parastrophs of any
Extra loop are loops of Bol-Moufang type and precisely Extra loops. The remaining four
could also become Extra loops provided the initial Extra loop is an involution or exponent
two and has the automorphic inverse property. The converse was also found to be true and
detail proofs and statements of those multiple new theorems are provided. Through those
proofs, this study confirms the claim of Fevyves (1968) over the equivalency of the three
Extra identities in loops. A new isotopic invariant for loops is obtained, namely the property
of being Extra. However, the property of being an inverse property loop is not an isotopic
invariant for loops in general. The only class of loops which has inverse property as isotopic
invariant is the class of Moufang loops. In a group (an associative quasigroup, a commutative
Extra loop, an associative loop) and in an Extra loop, the left and the right derivatives with
respect to any element of the initial algebraic structures preserve the algebraic structure.
The Public Access, L (2021). Contributions To The Theory Of Parastrophs And Derivatives Of Loops. Afribary.com: Retrieved May 13, 2021, from https://afribary.com/works/contributions-to-the-theory-of-parastrophs-and-derivatives-of-loops
Library, The Public Access. "Contributions To The Theory Of Parastrophs And Derivatives Of Loops" Afribary.com. Afribary.com, 29 Apr. 2021, https://afribary.com/works/contributions-to-the-theory-of-parastrophs-and-derivatives-of-loops . Accessed 13 May. 2021.
Library, The Public Access. "Contributions To The Theory Of Parastrophs And Derivatives Of Loops". Afribary.com, Afribary.com, 29 Apr. 2021. Web. 13 May. 2021. < https://afribary.com/works/contributions-to-the-theory-of-parastrophs-and-derivatives-of-loops >.
Library, The Public Access. "Contributions To The Theory Of Parastrophs And Derivatives Of Loops" Afribary.com (2021). Accessed May 13, 2021. https://afribary.com/works/contributions-to-the-theory-of-parastrophs-and-derivatives-of-loops