The spectral theory of semitopological semilattices

We characterize the spaces $\mathbf{Spec}\,\mathbb{S}$ of prime closed ideals of an arbitrary semitopological semilattice $\mathbb{S}$. We give an abstract description for the natural homomorphism of such a semilattice $\mathbb{S}$ into the semilattice $\langle T_*,U\rangle$ of the topology $T_*$ of the space $\mathbf{Spec}\,\mathbb{S}$.