Contributions To The Theory Of Parastrophs And Derivatives Of Loops

ABSTRACT

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.