Name: Mitsuyasu Hashimoto Title: $F$-regularity of multi-graded rings Abstract: We study $F$-regularity of $\Bbb Z^n$-graded rings. In particular, we prove that if $A$ and $B$ are standard graded algebras over an algebraically closed field, then the Segre product ring $A\# B$ is strongly $F$-regular if and only if both $A$ and $B$ are strongly $F$-regular. We also prove that if $G$ is a connected reductive group over $\Bbb C$, $S$ is a normal semigroup scheme of finite type over $\Bbb C$, and $\varphi: G\rightarrow S$ is a dominating semigroup homomorphism, then $S$ is of strongly $F$-regular type. Some part of the study utilizes the notion of global $F$-regularity.