  The production of single top quarks and top antiquarks via the $t$-channel
exchange of a virtual $W$ boson is measured in proton-proton collisions at a
centre-of-mass energy of 13 TeV at the LHC using $140\,\mathrm{fb^{-1}}$ of
ATLAS data. The total cross-sections are determined to be
$\sigma(tq)=137^{+8}_{-8}\,\mathrm{pb}$ and
$\sigma(\bar{t}q)=84^{+6}_{-5}\,\mathrm{pb}$ for top-quark and top-antiquark
production, respectively. The combined cross-section is found to be
$\sigma(tq+\bar{t}q)=221^{+13}_{-13}\,\mathrm{pb}$ and the cross-section ratio
is $R_{t}=\sigma(tq)/\sigma(\bar{t}q)=1.636^{+0.036}_{-0.034}$. The predictions
at next-to-next-to-leading-order in quantum chromodynamics are in good
agreement with these measurements. The predicted value of $R_{t}$ using
different sets of parton distribution functions is compared with the measured
value, demonstrating the potential to further constrain the functions when
using this result in global fits. The measured cross-sections are interpreted
in an effective field theory approach, setting limits at the 95% confidence
level on the strength of a four-quark operator and an operator coupling the
third quark generation to the Higgs boson doublet: $-0.37 <
C_{Qq}^{3,1}/\Lambda^2 < 0.06$ and $-0.87 < C_{\phi Q}^{3}/\Lambda^2 < 1.42$.
The constraint $|V_{tb}|>0.95$ at the 95% confidence level is derived from the
measured value of $\sigma(tq+\bar{t}q)$. In a more general approach, pairs of
CKM matrix elements involving top quarks are simultaneously constrained,
leading to confidence contours in the corresponding two-dimensional parameter
spaces.
