Polygroupoid, Polyquasigroup, Polyloop and their Nuclei

TL;DR

Addresses extending nuclei theory from classical groupoids to hyperstructures by proposing polygroupoid, polyquasigroup, and polyloop. Defines four nuclei families $N^1_{\\lambda}, N^2_{\\lambda}, N^3_{\\lambda}, N^4_{\\lambda}$ (and corresponding $\\mu$ and $\\rho$) and proves inclusion chains such as $N^1_{\\lambda} ⊆ N^2_{\\lambda} ⊆ N^4_{\\lambda}$, showing higher nuclei generalize the first. Provides concrete Examples 3.1–3.5 to illustrate nontrivial behavior and to validate the theory. Concludes that these results extend and unify nuclei concepts within hyperstructures, with potential implications for hyperalgebraic modeling in geometry, codes, and other domains.

Abstract

In this paper, new hyper-algebraic structures called polygroupoid, polyquasigroup and polyloop were introduced with concrete examples given. The first, second, third and fourth left (middle, right) nuclei of polygroupoid were introduced and studied. It was shown that first left (middle, right) nuclei of a polygroupoid is contained in second, third and fourth left (middle, right) nuclei of the polygroupoid. Hence, the second, third and fourth left (middle, right) nuclei of a polygroupoid generalize the first left (middle, right) nuclei of the polygroupoid. Examples of polyquasigroups and polyloops that satisfy the above results were provided.