<p>We introduce and study <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-elements of integral commutative quantales, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-elements are indeed a genuine extension of essential elements. We study preservation of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-elements under contractions and extensions of quantale homomorphisms. We introduce <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-complements and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-closedness and study their properties. We determine <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-elements for several distinguished quantales, including ideals of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {Z}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and open subsets of topological spaces. Finally, we provide a complete characterization of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-elements in modular quantales.</p>

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

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2026年3月6日

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Elena Caviglia,
Amartya Goswami,
Luca Mesiti

摘要

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

\(\mu \)-elements: An extension of essential elements

摘要