On irreducible elements of semilattices (Q1185229)

Let \(S\) be a lower semilattice and let \(\lambda\) be an infinite cardinal. The author constructs an \(S\ell_ \lambda\)-morphism \(h_ \lambda:S\to S^ 0_ \lambda\) which is a \(D\ell_ \lambda\)-reflection of \(S\) if \(\lambda\) is regular. If \(\kappa\) is an infinite regular cardinal, he obtains a \(D\ell_ \kappa\)-reflection of \(S\) if \(S\) satisfies a certain condition. This condition is satisfied if \(S\) is the dual of an algebraic lattice.