On $\alpha$-solvable fundamental groups

Abstract

We introduce a specific kind of equivalence relation ${xi^*}_n^{alpha}$ on a fuzzy hypergroup S such that the quotient $S/ {xi^*}_n^{alpha}$, the set of all equivalence classes, is an $alpha$-solvable group. This helps us to introduce the $alpha$-solvable fundamental relation $xi^{*alpha}$. In particular, we obtain an equivalent condition with transitivity of $xi^alpha$.