topological action of a discrete semi-group

Given a discrete semi-group $s$, we define a topological action $theta$ of s on a locally compact space $x$. additionally, there is an action α of $s$ on the $c^∗$-algebra $c0(x)$, which is introduced in relation to $θ$. we explore the topological independence of $theta$ and the impact of $theta$ on α. finally, we discuss the concept of a semi-partial dynamical system $(c0(x), s, alpha)$ and examine some of its properties.