  Let $D$ be an integral domain and $\star$ a semistar operation stable and of
finite type on it. In this paper we define the semistar dimension (inequality)
formula and discover their relations with $\star$-universally catenarian
domains and $\star$-stably strong S-domains. As an application we give new
characterizations of $\star$-quasi-Pr\"{u}fer domains and UM$t$ domains in
terms of dimension inequality formula (and the notions of universally
catenarian domain, stably strong S-domain, strong S-domain, and Jaffard
domains). We also extend Arnold's formula to the setting of semistar
operations.
