Distributive envelopes and topological duality for lattices via canonical extensions. (Q466890)

In this paper, a topological duality for bounded lattices is established. To this end, a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them is developed. Here are some details. Definition 3.1. Let \(L\) be a lattice. A finite subset \(M\subseteq L\) is \textit{join-admissible} if its join distributes over all meets with elements from \(L\), i.e., if, for all \(a\in L\), \(a\wedge\bigvee M=\bigvee_{m\in M}(a\wedge m)\). We say that a function \(f\colon L_1\to L_2\) between lattices preserves admissible joins if, for each finite join-admissible set \(M\subseteq L_1\), we have \(f(\bigvee M)=\bigvee_{m\in M}f(m)\). Definition 3.2. Let \(L\) be a lattice. An embedding \(\eta _L^{\wedge}\colon L\to D^{\wedge}(L)\) of \(L\) into a distributive lattice \(D^{\wedge}(L)\) which preserves meets and admissible joins is a \textit{distributive }\(\wedge\)-\textit{envelope} of \(L\) if it satisfies the following universal property: For any function \(f\colon L\to D\) into a distributive lattice \(D\) that preserves finite meets and admissible joins, there exists a unique lattice homomorphism \(\widehat f\colon D^{\wedge}(L)\to D\) such that \(\widehat f\circ\eta_L^{\wedge}=f\). The definition of the \textit{distributive }\(\vee\)-\textit{envelope} \(D^{\vee}(L)\) of \(L\) is order dual. Theorem 3.9. Let \(L\) be a lattice. The embedding \(\eta_L^{\wedge}\) of \(L\) into the finitely generated \(a\)-ideals of \(L\) is a distributive \(\wedge\)-envelope of \(L\). Proposition 3.18. For any lattice \(L\), the maps \(u_L\) and \(l_L\) restrict to a Galois connection \(u_L\colon D^{\wedge}(L)\rightleftarrows D^{\vee}(L)^{op}:\!l_L\). The lattice of Galois-closed elements of this Galois connection is isomorphic to \(L\). Via doubly dense adjoint pairs, it is shown that the category \(L_a\) of lattices with admissible homomorphisms is dually equivalent to a full subcategory of TSCP (totally separated compact polarities).