  This paper investigates t-reductions of ideals in pullback constructions.
Section 2 examines the correlation between the notions of reduction and
t-reduction in pseudo-valuation domains. Section 3 solves an open problem on
whether the finite t-basic and v-basic ideal properties are distinct. We prove
that these two notions coincide in any arbitrary domain. Section 4 features the
main result, which establishes the transfer of the finite t-basic ideal
property to pullbacks in line with Fontana-Gabelli's result on PvMDs and
Gabelli-Houston's result on v-domains. This allows us to enrich the literature
with new families of examples, which put the class of domains subject to the
finite t-basic ideal property strictly between the two classes of v-domains and
integrally closed domains.
