μ-elements: An extension of essential elements

We introduce and study μ μ -elements of integral commutative quantales, 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 μ μ -elements are indeed a genuine extension of essential elements. We study preservation of μ μ -elements under contractions and extensions of quantale homomorphisms. We introduce μ μ -complements and μ μ -closedness and study their properties. We determine μ μ -elements for several distinguished quantales, including ideals of Zₙ Z n and open subsets of topological spaces. Finally, we provide a complete characterization of μ μ -elements in modular quantales.