A soft relation approach to approximate the spherical fuzzy ideals of semigroups

Our main objective of this work was to study rough approximations of spherical fuzzy ideals by using soft relations that are free from all the complexities that are faced by many scientists. Furthermore, lower and upper approximations of spherical fuzzy subsemigroups, spherical fuzzy left (right) ideals, spherical fuzzy interior ideals, and spherical fuzzy bi-ideals of semigroups were studied using soft relations. Mainly, we proved, for a spherical fuzzy ideal of the universe, that upper and lower approximations are spherical fuzzy soft ideals but the converse may not hold, as shown by examples. Compatible relations and complete relations are needed for upper approximations and lower approximations, respectively. Also, using examples, we showed that the conditions of complete relations were necessary for lower approximations. Last, a comparison study and conclusions of the introduced technique are given, demonstrating how our work is superior and efficient in contrast to other techniques.