Let D be an integral domain with quotient field K, A a nonempty set of height-one maximal t-ideals of D, F$({Lambda})={I{subseteq}D|I$ is an ideal of D such that $I{subseteq}P$ for all $P{in}A}$, and $D_F({Lambda})={x{in}K|xA{subseteq}D$ for some $A{in}F({Lambda})}$. In this paper, we prove that if each $P{in}A$ is the radical of a finite type v-ideal (resp., a principal ideal), then $D_{F({Lambda})}$ is a weakly Krull domain (resp., generalized weakly factorial domain) if and only if the intersection $D_{F({Lambda})}={cap}_{P{in}A}D_P$ has finite character, if and only if $F({Lambda})$ is a t-splitting set of ideals, if and only if $F({Lambda})$ is v-finite.