On a certain class of integral domains with finitely many overrings.

An integral domain is called {\em Globalized multiplicatively pinched-Dedekind domain $($GMPD domain$)$} if every nonzero noninvertible ideal can be written as $JP_1\cdots P_k$ with $J$ invertible ideal and $P_1,...,P_k$ distinct ideals which are maximal among the nonzero noninvertible ideals, cf. \cite{DumII}. The GMPD domains with only finitely many overrings have been recently studied in \cite{SU}. In this paper we continue to investigate the overring-theoretic properties of GMPD domains. We study the effect of quasi-local overrings on the properties of GMPD domains. Moreover, we consider the structure of the partially ordered set of prime ideals (ordered under inclusion) in a GMPD domain.