Algebra
Lecturer II
Mathematics
At the Mathematics department office
Appointment on Visitation important
Topic: On The Core Of Some Classes Of Generalized Loops
Description:
This study centers on properties of the core of some classes of generalized loops wherein a construction of generalized Bol loop is presented. Several properties of the core of generalized Bol loops are established. Some of these properties inform that the core of generalized Bol loops belongs to the variety of Left Symmetric Left Distributive (LSLD) groupoid and therefore, a rack. A necessary and sufficient condition for the core of half-Bol loops to be medial is given with proof. The core of IP generalized Moufang loop with universal alpha-elasticity is examined.
# | Certificate | School | Year |
---|---|---|---|
1. | Ph.D (Mathematics/Non Associative Algebra and Loop Theory) | Federal University of Agriculture Abeokuta | 2021 |
On Right Automorphic generalized Bol Loops
certain identities under which a Bol loop is Moufang were established by Bol and Pflugfelder. These identities were given to be some two-variable identities. However, Chein proved that the Moufang identity is sometimes forced on a Bol loop not by any of the two-variable identities but rather by giving a pair of elements a choice of two equations to satisfy. Nevertheless, for a particular pair of identities, $(xy)^{-1}=y^{-1}x^{-1}$ or $(yx)^{-1}=x^{-1}y^{-1}$, it is not known, in general, if a Bol loop is Moufang if satisfies these pair of identities. For instance Bol loop of order 8 has the property that for any $x,y,$ either $xy=yx$ or $(xy)^{-1}=y^{-1}x^{-1}$. This shows that antiautomorphic inverse property does not necessarily force a Bol loop to be Moufang. The problem was solved for two special classes of Bol loops, namely right automorphic Bol loops and strongly right alternative loop rings (SRAR-loops) by Orin Chein.
In the present study, we extend some of the results in Chein to generalised Bol loop. To achieve this, we first prove an analogue of generators of inner mapping group of Bol loop for generalised Bol loop using pflugfelder's approach. This makes it plausible to characterize right automorphic generalised Bol loop in terms of the right inner mapping and thus, our motivation to study right automorphic generalised Bol loop with a view to determining when it is a generalised Moufang loop.
ABDULKAREEM ABDULAFEEZ is a Lecturer II at the Department of Mathematics
ABDULKAREEM has a Ph.D in Mathematics/Non Associative Algebra and Loop Theory from Federal University of Agriculture Abeokuta