NUCLEAR IDENTIFICATION OF EXTRA LOOPS OF SECOND BOL-MOUFANGTYPE WITH APPLICATIONS TO SECURE INFORMATION ENCODING

Abstract

 Dr´apal and Jedliˇcka identified several loop identities, including both Bol–Moufang and non Bol–Moufang varieties through their nuclei. Among these are the extra identities. Subsequently, George and Jaiyeola developed a generalized nuclear identification scheme for identities of the second Bol–Moufang type. While they discovered twelve new loop identities, their approach did not establish a nuclear identification for
the extra identities. This left open the question of whether extra-type identities admit nuclear identification in the second Bol–Moufang setting. In this paper, we introduced three new loop identities (SBME, SBRE,
SBLE) of second Bol-Moufang type. In particular, we show that an extra loop of second Bol–Moufang type is nuclear-identifiable if it can be expressed as autotopisms αϵ ∗ η(x)αω ∗ ξ(x)ακ ∗ χ(x)αψ ∗ ζ(x) and
satisfies an identity (η, ξ, χ, ζ, ϵ, ω, κ, ψ) such that η = χ ̸ = ξ ̸ = ζ. Furthermore, we show that the newly introduced extra identities of second Bol–Moufang type are equivalent to the extra identities of first
Bol–Moufang type. Beyond the algebraic characterization, we illustrate how nuclear identification codes can inspire secure information encoding schemes, where loop identities serve as human-recognizable keys for
controlled access in sensitive communication environments.