Measurements of the cross sections for the production of single top quarks and antiquarks in the $t$ channel, and their ratio, are presented for proton-proton collisions at a center-of-mass energy of 13 TeV. The data set used was recorded in 2016 by the CMS detector at the LHC and corresponds to an integrated luminosity of 35.9 fb$^{-1}$. Events with one muon or electron are selected, and different categories of jet and b jet multiplicity and multivariate discriminators are applied to separate the signal from the background. The cross sections for the $t$-channel production of single top quarks and antiquarks are measured to be 130 $\pm$ 1 (stat) $\pm$ 19 (syst) pb and 77 $\pm$ 1 (stat) $\pm$ 12 (syst) pb, respectively, and their ratio is 1.68 $\pm$ 0.02 (stat) $\pm$ 0.05 (syst). The results are in agreement with the predictions from the standard model.