A Tychonoff topological space is called a quasi
F
F
-space if each dense cozero-set of
X
X
is
C
∗
{C^{\ast }}
-embedded in
X
X
. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi
F
F
-cover"
Q
F
(
X
)
QF(X)
of a compact space
X
X
as an inverse limit space, and identify the ring
C
(
Q
F
(
X
)
)
C(QF(X))
as the order-Cauchy completion of the ring
C
∗
(
X
)
{C^{\ast }}(X)
. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi
F
F
-cover of an arbitrary Tychonoff space. In this paper the minimal quasi
F
F
-cover of a compact space
X
X
is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of
X
X
. The relationship between
Q
F
(
X
)
QF(X)
and
Q
F
(
β
X
)
QF(\beta X)
is studied in detail, and broad conditions under which
β
(
Q
F
(
X
)
)
=
Q
F
(
β
X
)
\beta (QF(X)) = QF(\beta X)
are obtained, together with examples of spaces for which the relationship fails. (Here
β
X
\beta X
denotes the Stone-Čech compactification of
X
X
.) The role of
Q
F
(
X
)
QF(X)
as a "projective object" in certain topological categories is investigated.