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- Title
Basarab loop and its variance with inverse properties.
- Authors
Jaiyéọlá, Tèmítope Gboláhàn; Effiong, Gideon Okon
- 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 properties are 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.
- Subjects
AUTOMORPHIC functions; BINARY operations; QUASIGROUPS; GROUPOIDS; PERMUTATION groups
- Publication
Quasigroups & Related Systems, 2018, Vol 26, Issue 2, p229
- ISSN
1561-2848
- Publication type
Article