Let
K
\mathcal {K}
be a class of (universal) algebras of fixed type.
K
t
{\mathcal {K}^t}
denotes the class obtained by augmenting each member of
K
\mathcal {K}
by the ternary discriminator function
(
t
(
x
,
y
,
z
)
=
x
(t(x,y,z) = x
if
x
≠
y
,
t
(
x
,
x
,
z
)
=
z
)
x \ne y,t(x,x,z) = z)
, while
∨
(
K
t
)
\vee ({\mathcal {K}^t})
is the closure of
K
t
{\mathcal {K}^t}
under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to
∨
(
K
t
)
\vee ({\mathcal {K}^t})
where
K
\mathcal {K}
consists of a two-element algebra whose only operations are the two constants. Any equationally defined class (that is, variety) of algebras which is equivalent to some
∨
(
K
t
)
\vee ({\mathcal {K}^t})
is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie, and M. Valeriote, we characterize those locally finite universal classes
K
\mathcal {K}
of unary algebras of finite type for which the first-order theory of
∨
(
K
t
)
\vee ({\mathcal {K}^t})
is decidable.