A universal algebra A is called one-variable equationally compact if every system of equations with constants in A involving a single variable x, every finite subsystem of which has a solution in A, has itself a solution in A. The one-variable equationally compact semilattices with pseudocomplementation
⟨
S
;
∧
,
∗
,
0
⟩
\langle S; \wedge {,^ \ast },0\rangle
which satisfy the partial distributive law
x
∧
(
y
∧
z
)
∗
=
(
x
∧
y
∗
)
∨
(
x
∧
z
∗
)
x \wedge {(y \wedge z)^ \ast } = (x \wedge {y^ \ast }) \vee (x \wedge {z^ \ast })
are characterized, and as a consequence we are able to describe the one-variable compact Stone semilattices. Similar considerations yield a characterization of the one-variable equationally compact Stone algebras, extending a well known result for distributive lattices.