Geodesic
Order of Finite Soft Quasigroups with Application to Egalitarianism
Discussiones Mathematicae. General Algebra and Applications, Tome 42 (2022) no. 1, pp. 135-157

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In this work, a soft set (F, A) was introduced over a quasigroup (Q,·) and the study of finite soft quasigroup was carried out, motivated by the study of algebraic structures of soft sets. By introducing the order of a finite soft quasigroup, various inequality relationships that exist between the order of a finite quasigroup, the order of its soft quasigroup and the cardinality of its set of parameters were established. By introducing the arithmetic mean 𝒜ℳ(F, A) and geometric mean 𝒢ℳ(F, A) of a finite soft quasigroup (F, A), a sort of Lagrange’s Formula |(F, A)| = |A|𝒜ℳ(F, A) for finite soft quasigroup was gotten. Some of the inequalities gotten gave an upper bound for the order of a finite soft quasigroup in terms of the order of its quasigroup and cardinality of its set of parameters, and a lower bound for the order of the quasigroup in terms of the arithmetic mean of the finite soft quasigroup. A chain of inequalities called the Maclaurin’s inequality for any finite soft quasigroup (F, A)_(Q,·) was shown to exist. A necessary and sufficient condition for a type of finite soft quasigroup to be extensible to a finite super soft quasigroup was established. This result is of practical use whenever a larger set of parameters is required. The results therein were illustrated with examples. Application to uniformity, equality and equity in distribution for social living is considered.
Keywords: soft sets, quasigroups, soft quasigroups, soft subquasigroups, arithmetic and geometric mean, inequalities 
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Oyem, A.; Jaiyéọlá, T.G.; Olaleru, J.O. Order of Finite Soft Quasigroups with Application to Egalitarianism. Discussiones Mathematicae. General Algebra and Applications, Tome 42 (2022) no. 1, pp. 135-157. http://geodesic.mathdoc.fr/item/DMGAA_2022_42_1_a6/