Completely weakly, quasi-regular semigroups characterized by soft union Quasi ideals, (generalized) bi-ideals and semiprime ideals

Abstract

In this paper, certain kinds of regularities of semigroups are studied by correlating soft set theory. Completely, weakly and quasiregular semigroups are characterized by soft union quasiideals, soft union (generalized) bi-ideals and soft union semiprime ideals of a semigroup. It is proved that if every soft union quasi-ideal of a semigroup is soft union semiprime, then every quasi-ideal of a semigroup is semiprime and thus, if every quasi-ideal of a semigroup is semiprime, then the semigroup is completely regular. Also, it is obtained that the case when every soft union quasi-ideal (bi-ideal, generalized bi-ideal, respectively) of a semigroup is soft union semiprime is equivalent to the case when every quasi-ideal (bi-ideal, generalized bi-ideal, respectively) of a semigorup is semiprime, where the semigroup is completely semi group. Similar characterizations are obtained for weakly and quasi-regular semigroups. By these characterizations, we intent to bring a new perspective to the regularities of semigroup theory via soft set theory. Further study can be focused on soft union tri quasi-ideals, soft union bi-quasi ideals, soft union lateral bi-quasi-ideals and soft union lateral tri-quasi ideals of a semigroup.