  Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary
process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set
of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with
$X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t),
t\ge 0$. In numerous applications such as brain mapping and digital
communication systems, of interest is the approximation of the probability that
the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's
conditions on $X$, in this paper we obtain an exact asymptotic expansion of
this probability as $u$ tends to infinity. Further, we establish the tail
asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel
limit theorem for the minimum order statistics of stationary Gaussian
processes. As a by-product we derive a version of Li and Shao's normal
comparison lemma for the minimum and the maximum of Gaussian random vectors.
