μ-elements: An extension of essential elements
Published in Preprint, 2025
Joint work with Elena Caviglia and Amartya Goswami.
We introduce and study μ-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 μ-elements are indeed a genuine extension of essential elements. We study preservation of μ-elements under contractions and extensions of quantale homomorphisms. We introduce μ-complements and μ-closedness and study their properties. We determine μ-elements for several distinguished quantales, including ideals of Zn and open subsets of topological spaces. Finally, we provide a complete characterization of μ-elements in modular quantales.
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