  The mutually commuting $1\times n$ fused single and double-row transfer
matrices of the critical six-vertex model are considered at roots of unity
$q=e^{i\lambda}$ with crossing parameter $\lambda=\frac{(p'-p)\pi}{p'}$ a
rational fraction of $\pi$. The $1\times n$ transfer matrices of the dense loop
model analogs, namely the logarithmic minimal models ${\cal LM}(p,p')$, are
similarly considered. For these $s\ell(2)$ models, we find explicit closure
relations for the $T$-system functional equations and obtain extended sets of
bilinear $T$-system identities. We also define extended $Q$ matrices as linear
combinations of the fused transfer matrices and obtain extended matrix $T$-$Q$
relations. These results hold for diagonal twisted boundary conditions on the
cylinder as well as $U_q(s\ell(2))$ invariant/Kac vacuum and off-diagonal/Robin
vacuum boundary conditions on the strip. Using our extended $T$-system and
extended $T$-$Q$ relations for eigenvalues, we deduce the usual scalar Baxter
$T$-$Q$ relation and the Bazhanov-Mangazeev decomposition of the fused transfer
matrices $T^{n}(u+\lambda)$ and $D^{n}(u+\lambda)$, at fusion level $n=p'-1$,
in terms of the product $Q^+(u)Q^-(u)$ or $Q(u)^2$. It follows that the zeros
of $T^{p'-1}(u+\lambda)$ and $D^{p'-1}(u+\lambda)$ are comprised of the Bethe
roots and complete $p'$ strings. We also clarify the formal observations of
Pronko and Yang-Nepomechie-Zhang and establish, under favourable conditions,
the existence of an infinite fusion limit $n\to\infty$ in the auxiliary space
of the fused transfer matrices. Despite this connection, the
infinite-dimensional oscillator representations are not needed at roots of
unity due to finite closure of the functional equations.
