On Finitely Star Regular Domains

For an integral domain $R$, we denote by $Star(R)$ (resp. $StarFC(R)$) the set of all star operations (resp. star operations of finite type on $R$). A domain $R$ is a star regular domain if $|Star(T)|\leq |Star(R)|$ for every overring $T$ of $R$. This project is a continuation of a project approved by DSR ``Integral Domains with Finitely many Star Operations of Finite Type \#RG 161001". It aims at studying the notion of finitely star regular domains, that is, integral domains $R$ such that $|StarFC(T)|\leq |StarFC(R)|$ for every overring $T$ of $R$. Precisely, after showing that the notions of Star regular domains and finitely star regular domains are completely different and do not imply each other, we are seeking for a generalization/analogue for many results in the case of star regular domains.