We show that in a weak commutative inverse property loop, such as a Bruck loop, if is a right [left] pseudoautomorphism with companion , then [] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing...
A nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is -based, and we exhibit an explicit system of two independent identities. We also show that the original axiom systems of Hickman as well as that of Chajda et al are...
Let be a loop. If is such that for each standard generator of Inn, then does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that...
On the unit sphere in a real Hilbert space , we derive a binary operation such that is a power-associative Kikkawa left loop with two-sided identity , i.e., it has the left inverse, automorphic inverse, and properties. The operation is compatible with the symmetric space structure of . is not a loop, and the right translations which fail to be injective are easily characterized. satisfies the left power alternative and left Bol identities “almost everywhere” but not everywhere....
We give new equations that axiomatize the variety of trimedial quasigroups. We also improve a standard characterization by showing that right semimedial, left F-quasigroups are trimedial.
If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.
In Kepka T., Kinyon M.K., Phillips J.D., , , we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally, we show an equivalence of equational classes...
In Kepka T., Kinyon M.K., Phillips J.D., , J. Algebra (2007), 435–461, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the class of (pointed) F-quasigroups and the class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.
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