Abstract
A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop
Highlights
Basarab [4] used the result published by Belousov [5] to construct an example of a Basarab loop whose nucleus is an abelian group
After his first publications in 1992 on Basarab loops, no other author researched on properties of Basarab loops until in 1996 and 1997, that he, Basarab studied the relationship between a generalized Moufang loop, Osborn loop, VD-loop, and a Basarab loop, and a special type of a Basarab loop, known as IK-loop respectively
In a Basarab loop (Q, ·) the following are equivalent for all x, y ∈ Q: (i) Txy = TxTy. (ii) L(x, y) ∈ P(Q, ·). (iii) R(x, y) ∈ Λ(Q, ·)
Summary
The authors established the importance of the total inner mapping group by showing that if Q is a middle Bol loop, Inn(Q) T Inn(Q) This result is not necessarily true for a left or right. Basarab [4] used the result published by Belousov [5] to construct an example of a Basarab loop whose nucleus is an abelian group After his first publications in 1992 on Basarab loops, no other author researched on properties of Basarab loops until in 1996 and 1997, that he, Basarab studied the relationship between a generalized Moufang loop, Osborn loop, VD-loop, and a Basarab loop, and a special type of a Basarab loop, known as IK-loop respectively. Some results on the generators of the inner mapping and total inner mapping groups of a Basarab loop are established in order to show that a class of total inner mappings act on a Basarab loop Q by automorphisms if and only if Q is an A-loop and flexible
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