Basarab loop and its variance with inverse properties

Citation:
Jaiyeola, T, Effiong G.  2018.  Basarab loop and its variance with inverse properties, 2018/12/01. 26:229-238.

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 is a special type of a G-loop. It was shown that a Basarab loop (Q, ·) has the cross inverse property if and only if (Q, ·) is an abelian group or all left (right) translations of (Q, ·) are right (left) regular. In a Basarab loop, the following propertiesare equivalent: flexibility property, right inverse property, left inverse property, inverse property, right alternative property, left alternative property and alternative property. The following were
proved: a Basarab loop is a weak inverse property loop if it is flexible such that the middle inner mapping is contained in a permutation group; a Basarab loop is an automorphic inverse property
loop if a semi-commutative law is obeyed such that the middle inner mapping is contained in a permutation group; a Basarab loop is an anti-automorphic inverse property loop if every element
has a two-sided inverse such that the middle inner mapping is contained in a permutation group; a Basarab loop is a semi-automorphic inverse property loop if the Basarab loop is flexible, the
middle inner mapping is contained in a permutation group such that a semi-cross inverse property holds; a Basarab loop with the m-inverse property such that a permutation condition is true is
a cross inverse property loop if it is flexible. Necessary and sufficient conditions for a Basarab loop to be of exponent 2 or a centrum square were established.

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