Closure operators on semilattice-ordered semigroups and spectrality of induced operations

Abstract

Let $S$ be a semilattice-ordered semigroup and let $\mathrm{cl}$ be a closure operator on $S$. We consider the space $$X := \{A \in \mathcal{P}(S) \mid A^\mathrm{cl} = A\}$$ of all $\mathrm{cl}$-closed subsets of $S$, endowed with the subspace topology induced by the hull-kernel topology on $\mathcal{P}(S)$. We prove that $X$ is a spectral space and a retrocompact subset of $\mathcal{P}(S)$ if and only if $\mathrm{cl}$ is algebraic. Assuming that $\mathrm{cl}$ is algebraic, we then investigate the operation $$A \star B := (AB)^\mathrm{cl}$$ induced on $X$ by the multiplication on $S$. We obtain finite characterizations of spectrality of the map $\star: X \times X \to X$ in terms of finite subsets of $S$ and finitely generated $\mathrm{cl}$-closed subsets. Analogous criteria are established for left and right translations. We prove that if the underlying poset $(S, \leqslant)$ is well-quasi-ordered and the closure operator $\mathrm{cl}$ is algebraic and order-compatible, then the induced operation $\star$ is spectral. For the identity closure operator, we characterize spectrality of $\star$ by the finite decomposition property. We also show that spectrality of $\star$ implies spectrality of all left and right translations, whereas the converse fails in general, even for an algebraic multiplicative closure operator. Finally, we introduce three closure operators naturally associated with semilattice-ordered semigroups, study their algebraic and multiplicative properties, and apply the general results to the corresponding spectral spaces and induced operations.