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

Semistar dimension of polynomial rings and Pr\"{u}fer-like domains