  A monotone Boolean circuit is composed of OR gates, AND gates and input gates
corresponding to the input variables and the Boolean constants. It is
$q$-multilinear if for each its output gate $o$ and for each prime implicant
$s$ of the function computed at $o$, the arithmetic version of the circuit
resulting from the replacement of OR and AND gates by addition and
multiplication gates, respectively, computes a polynomial at $o$ which contains
a monomial including the same variables as $s$ and each of the variables in $s$
has degree at most $q$ in the monomial. First, we study the complexity of
computing semi-disjoint bilinear Boolean forms in terms of the size of monotone
$q$-multilinear Boolean circuits. In particular, we show that any monotone
$1$-multilinear Boolean circuit computing a semi-disjoint Boolean form with $p$
prime implicants includes at least $p$ AND gates. We also show that any
monotone $q$-multilinear Boolean circuit computing a semi-disjoint Boolean form
with $p$ prime implicants has $\Omega(\frac p {q^4})$ size. Next, we study the
complexity of the monotone Boolean function $Isol_{k,n}$ that verifies if a
$k$-dimensional Boolean matrix has at least one $1$ in each line (e.g., each
row and column when $k=2$), in terms of monotone $q$-multilinear Boolean
circuits. We show that that any $\Sigma_3$ monotone Boolean circuit for
$Isol_{k,n}$ has an exponential in $n$ size or it is not $(k-1)$-multilinear.
