$\mu$-elements: An extension of essential elements

We introduce and study $\mu$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that $\mu$-elements are indeed a genuine extension of essential elements. We study preservation of $\mu$-elements under contractions and extensions of quantale homomorphisms. We introduce $\mu$-complements and $\mu$-closedness and study their properties. We determine $\mu$-elements for several distinguished quantales, including ideals of $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $\mu$-elements in modular quantales.