On quasi-Pr\"{u}fer and UMt domains

In this note we show that an integral domain $D$ of finite $w$-dimension is a quasi-Pr\"{u}fer domain if and only if each overring of $D$ is a $w$-Jaffard domain. Similar characterizations of quasi-Pr\"{u}fer domains are given by replacing $w$-Jaffard domain by $w$-stably strong S-domain, and $w$-strong S-domain. We also give new characterizations of UM$t$ domains.