Computation in Algebraic Hyperstructures

The concept of the relation β plays a central role in the study of hypercompositional structures. In this paper, we extend the definition of β to the general framework of hypergroupoids and develop an algorithm to compute its elements and its transitive closure β★. We then apply this algorithm to determine the β-class of partial identities in a hypergroup, a process equivalent to computing the heart of the given algebraic structure. Furthermore, we propose a more general algorithm that is also applicable to the case of Hv-groups. By extracting the quotient set with respect to β★ and endowing it with an appropriate group structure, we obtain the so-called fundamental group. The identity of this fundamental group can then be computed directly, yielding the heart of the structure.