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The homomorphism distinguishing closure is a closure operator

Lax871432.DistinguishingClosureOperator · concepts/Lax871432/DistinguishingClosureOperator.lean · lax-871432

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    Natural Language Statement

    Lemma

    The map F↦cl(F)\mathcal{F} \mapsto \mathrm{cl}(\mathcal{F}) is a closure operator on graph classes ordered by inclusion: for all graph classes F\mathcal{F} and F′\mathcal{F}',

    • cl(F)⊆cl(F′)\mathrm{cl}(\mathcal{F}) \subseteq \mathrm{cl}(\mathcal{F}') if F⊆F′\mathcal{F} \subseteq \mathcal{F}',
    • F⊆cl(F)\mathcal{F} \subseteq \mathrm{cl}(\mathcal{F}), and
    • cl(cl(F))=cl(F)\mathrm{cl}(\mathrm{cl}(\mathcal{F})) = \mathrm{cl}(\mathcal{F}).
    Concept map
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    In the paper

    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Mathlib.Order.Closure
    2import Lax871432.DistinguishingClosure
    3
    4/-!
    5---
    6title: The homomorphism distinguishing closure is a closure operator
    7type: lemma
    8---
    9The map F↦cl(F)\mathcal{F} \mapsto \mathrm{cl}(\mathcal{F}) is a closure operator on graph classes
    10ordered by inclusion: for all graph classes F\mathcal{F} and F′\mathcal{F}',
    11
    12- cl(F)⊆cl(F′)\mathrm{cl}(\mathcal{F}) \subseteq \mathrm{cl}(\mathcal{F}') if
    13 F⊆F′\mathcal{F} \subseteq \mathcal{F}',
    14- F⊆cl(F)\mathcal{F} \subseteq \mathrm{cl}(\mathcal{F}), and
    15- cl(cl(F))=cl(F)\mathrm{cl}(\mathrm{cl}(\mathcal{F})) = \mathrm{cl}(\mathcal{F}).
    16-/
    17
    18open Lax871432.DistinguishingClosure Lax871432.GraphClasses
    19
    20namespace Lax871432.DistinguishingClosureOperator
    21
    22/-- cl\mathrm{cl} is a closure operator: it is monotone, extensive and idempotent. -/
    23axiom isClosureOperator : ∃ c : ClosureOperator GraphClass, ∀ 𝓕, c 𝓕 = cl 𝓕
    24
    25end Lax871432.DistinguishingClosureOperator
    26
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