$μ$-elements: An extension of essential elements
Elena Caviglia, Amartya Goswami, Luca Mesiti
TL;DR
${\mu}$-elements extend essential elements from considering a single meet to arbitrary finite meets in integral quantales, unifying lattice-theoretic density with algebraic density notions. The paper develops a comprehensive theory of ${\mu}$-elements, including preservation under quantale homomorphisms, the introduction of ${\mu}$-complements and ${\mu}$-closedness, and complete characterizations in modular quantales and key quantales such as the ideals of ${\mathbb{Z}}_n$ and the frame of open sets of a space. A central finding is that in modular quantales, every ${\mu}$-element is either essential or irreducible, which ties together two fundamental lattice-theoretic notions and yields practical criteria for identifying ${\mu}$-elements across algebraic and topological contexts. These results illuminate structural aspects of lattice-theoretic extensions of the classical ${\mathcal{M}}$-ideals framework and have potential implications for density, irreducibility, and extensions in modules, rings, and topological spaces.
Abstract
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 $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $μ$-elements in modular quantales.