Research Projects

This project's goal is to investigate epimorphisms and dominions in the categories of semigroups and posemigroups. Surjective morphism is always an epimorphism in every concrete category. In some categories, such as the categories of groups, sets, and vector spaces, the opposite is true. In the case of rings, semigroups, and posemigroups, the opposite is not generally true. Furthermore, the dominion of every group is trivial in the category of all groups is widely recognized. Nonetheless, dominion is generally not trivial in the categories of rings, semigroups and posemigroups. Studying epimorphisms and dominions in these categories is driven by this purpose. J. Isbell and J. M. Howie launched the investigation in the 1960s and T. E. Hall later expanded the findings in this area. Isbell gave the characterization of dominion elements in semigroups. Note that by trivial dominion we mean that the said algebraic object is absolutely closed in the given category. . Further in 1980s N. M. Khan and P. M. Higgins studied varieties of semigrtoups in this direction. Khan and Higgins derive certain saturated varieties of semigroups. In particular, it shows that every epimorphism is surjective from every member of the variety derived by Khan and Higgins. But the determination of all saturated varieties of semigroups is open and challenging problem. Recently Noor Alam and Khan derived certain closed varieties of semigroups. Again the complete determination closed varieties of semigroups is still open. In this project our aim is to settle these problem completely. In 2013, Nasir Sohail studied epimorphisms and dominions in posemigroups and has derived the characterization of dominion elements in posemigroups. In 2019, Ahanger and Shah began working on posemigroups and extended certain classical results of semigroup varieties to posemigroup varieties with the help of Sohail’s result. But the characterization of the class of posemigroups such that every epimorphism is surjective from any member of the class or the posemigroup dominion is trivial of any member of the classs is open problem. Therefore investigating epimorphisms and dominions is a well known theme in these categories. . In 2013, Nasir Sohail studied epimorphisms and dominions in posemigroups and has derived the characterization of dominion elements in posemigroups. In 2019, Ahanger and Shah began working on posemigroups and extended certain classical results of semigroup varieties to posemigroup varieties with the help of Sohail’s result. But the characterization of the class of posemigroups such every epimorphism is surjective from any member of the class or the posemigroup dominion is trivial of any member of the classs is open problem. Therefore investigating epimorphisms and dominions is a well known theme in these categories.