Extending semilattices to frames using sites and coverages. (Q2877060)

Meet semilattices are freely extendable to frames. The corresponding free functor \(\mathfrak D\colon\mathbf S\mathrm{Lat}\to\mathbf F\mathrm{rm}\), left adjoint to the forgetful functor \(\mathbf F\mathrm{rm}\to\mathbf S\mathrm{Lat}\), is the down-set functor (where, for each meet semilattice \(S\), \(\mathfrak D(S)\) is the frame of down-sets of \(S\)).NEWLINENEWLINE In the paper under review, the authors say that a \textit{frame extension of a semilattice} \(S\) is a meet semilattice homomorphism \(\tau\colon S\to L\) with \(L\) a frame such that \(\tau\) is one-to-one and \(L\) is join-generated by \(\tau[S]\). Extensions are preordered by declaring, for \(\tau_i\colon S\to L_i\) (\(i=1,2\)), \(\tau_1\leq\tau_2\) if there is a frame homomorphism \(\rho\colon L_2\to L_1\) such that \(\rho\cdot\tau_2=\tau_1\). The natural embedding \(S\to\mathfrak D(S)\) is a frame extension of \(S\) and it is the largest one in the preorder \(\leq\); on the other hand, the smallest one is the extension \(\mathfrak J^e(S)\) of \(S\) respecting all the exact joins in \(S\).NEWLINENEWLINE Then, they present the complete range of frame extensions of a semilattice \(S\): it is precisely the interval \([\mathfrak J^e(S),\mathfrak D(S)]\) in the coframe of all sublocales of \(\mathfrak D(S)\). This is done using a nice technique of sites and coverages that generalizes the one used by \textit{P. T. Johnstone} [in Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge: Cambridge University Press (1982; Zbl 0499.54001)].NEWLINENEWLINE Finally, the Boolean and Heyting cases are discussed. There, the smallest extension is shown to coincide with the Dedekind-MacNeille completion in both cases (in the former case, the formulas are fairly explicit).