Some characterizations of generalized middle Bol loops

Benard Osoba^a, Tèmítópé Gbóláhàn Jaíyéolá^b, Abednigo Orobosa Isere^c, Yakub Tunde Oyebo<sup>d

a</sup> Department of Physical Science, College of Natural and Applied Sciences, Bells University of Technology, Ota, Ogun State, Nigeria
^b Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria. Department of Mathematics, University of Lagos, Akoka, Nigeria
^c Department of Mathematics, Ambrose Alli University, Ekpoma 310001, Nigeria
^d Department of Mathematics, Lagos State University, Ojo, Lagos State, Nigeria

Abstract: In this work, the two identities that characterize middle Bol loop (MBL) were separately generalized to define generalized middle Bol loops (GMBL1 and GMBL2) and their relationship was unraveled for possible equivalence. Thereafter, autotopic characterizations of identities GMBL1 and GMBL2 in a loop were carried out. Banking on these, the Bryant-Schneider group characterization of generalized middle Bol loop was carried out. Furthermore, an autotopic-split and right (left) pseudo-automorphic characterization of generalized middle Bol loop were carried out. Furthermore, a right (left) pseudo-automorphic characterization of the isotopy-isomorphy of generalized middle Bol loop was also carried out. This is a sort of 'G-generalized middle Bol loop' characterization based on pseudo-automorphic companionship of elements of GMBL as already established for 'G-loops'.

Keywords and phrases: generalized middle Bol loop, autotopisms, Bryant-Schneider group, pseudo-automorphism, crypto-automorphism.

MSC: 20N05, 08A05

Received: 15.01.2024

Language: English

DOI: 10.56415/basm.y2024.i3.p12