A new semilattice of function algebras and its Boolean form on a lattice of groups (Q1207701)

A semigroup compactification of a separately continuous semigroup \(S\) consists of a compact right topological semigroup \(X\) and a continuous homomorphism \(\psi: S\to X\) with the properties (i) \(\psi(S)\) is dense in \(X\) and (ii) \(x\mapsto \psi(s)x\) is continuous for each \(s\in S\). The compactification has the local joint continuity property with respect to \(A\subseteq S\) if \((s,x)\mapsto\psi(s)x\) is jointly continuous on \(S\times X\) at every point \((a,x)\) of \(A\times X\). The largest compactification of \(S\) is the \(lmc\)-compactification, and a function \(f\) on \(S\) is said to be in \(LMC(S)\) if it has a continuous extension to this compactification. A function \(f\in LMC(S)\) is said to belong to \(LLC(S,A)\) for a subset \(A\subseteq S\) if the map \(s\mapsto L_ sf\) which sends \(f\) to its left translate is continuous at each point of \(A\) from \(S\) to \(LMC(S)\) with the uniform norm. If \(LLC(S,A)\) is left translation invariant, then it is isomorphic with the space of all continuous functions on the largest compactification of \(S\) which has the local joint continuity property with respect to \(A\). The algebra \(LLC(S,S)\) is just the familiar \(LUC(S)\). Given any set \(A\subseteq S\), there is a largest set, here denoted by \(M(A)\), for which \(LLC(S,A)=LLC(S,M(A))\). Then \(LLC(S,A)\subseteq LLC(S,B)\) if and only if \(M(B)\subseteq M(A)\), and the correspondence between the function algebras \(LLC(S,A)\) and the sets \(M(A)\) is a meet semilattice anti-isomorphism (more precisely, the intersection of the algebras determined by the sets \(A\) and \(B\) corresponds to \(M(A\cup B)\)). The paper goes on to consider the particular case in which \(S\) is a direct product \(\prod_{\alpha\in I}S_ \alpha\) of semigroups \(S_ \alpha\), each of which is a group with zero. In case \(S\) is locally compact, the collection of all algebras \(LLC(S,A)\) is lattice isomorphic with the power set of \(I\) with its usual operations.