NUCLEAR IDENTIFICATION OF EXTRA LOOP IDENTITIES OF SECOND BOL-MOUFANG TYPE WITH APPLICATIONS TO SECURE INFORMATION ENCODING

OLUFEMI  GEORGE

OLUFEMI GEORGE DEPARTMENT OF MATHEMATICS, UNIVERSITY OF LAGOS, AKOKA, LAGOS STATE, NIGERIA.

Keywords: extra loop, nuclear identification, Bol Moufang type loop, first Bol Moufang type loop, second Bol-Moufang type Loop

Abstract

Drpal and Jedlika identified several loop identities, including both BolMoufang and non-BolMoufang 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 BolMoufang 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 BolMoufang setting. In this paper, we introduced three new loop identities (SBME, SBRE, SBLE) of second BolMoufang type. In particular, we show that an extra loop of second BolMoufang type is nuclear- identifiable if it can be expressed as autotopisms α^є ∗ η(x)α^c ∗ ξ(x)αⁿ ∗ χ(x)α^ψ ∗ 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 BolMoufang type are equivalent to the extra identities of first BolMoufang 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.

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