<p>We characterise the frame morphisms <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f:L\rightarrow M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>L</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> that lift to frame maps <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{f}:\textsf{S}_b(L)\rightarrow \textsf{S}_b(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>f</mi> <mo>¯</mo> </mover> <mo>:</mo> <msub> <mi mathvariant="sans-serif">S</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi mathvariant="sans-serif">S</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textsf{S}_b(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="sans-serif">S</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the collection of joins of complemented sublocales of a frame <i>L</i>, or equivalently the Booleanization of the collection <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textsf{S}(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">S</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of all its sublocales. We do so by proving that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textsf{S}_b(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="sans-serif">S</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to the Bruns–Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak {c}(a)\cap \mathfrak {o}(b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">c</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi mathvariant="fraktur">o</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a,b\in L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

The Lattice of Smooth Sublocales as a Bruns–Lakser Completion

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2026年5月28日

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Igor Arrieta,
Anna Laura Suarez

摘要

We characterise the frame morphisms $f:L\rightarrow M$ $f : L \to M$ that lift to frame maps $\overline{f}:\textsf{S}_b(L)\rightarrow \textsf{S}_b(M)$ $\bar{f} : S_{b} (L) \to S_{b} (M)$ , where $\textsf{S}_b(L)$ $S_{b} (L)$ is the collection of joins of complemented sublocales of a frame L, or equivalently the Booleanization of the collection $\textsf{S}(L)$ $S (L)$ of all its sublocales. We do so by proving that $\textsf{S}_b(L)$ $S_{b} (L)$ is isomorphic to the Bruns–Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form $\mathfrak {c}(a)\cap \mathfrak {o}(b)$ $c (a) \cap o (b)$ for $a,b\in L$ $a, b \in L$ .