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Categor Struct"],"published-print":{"date-parts":[[2026,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We characterise the frame morphisms\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$f:L\\rightarrow M$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mo>\u2192<\/mml:mo>\n                            <mml:mi>M<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    that lift to frame maps\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\overline{f}:\\textsf{S}_b(L)\\rightarrow \\textsf{S}_b(M)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mover>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mo>\u00af<\/mml:mo>\n                            <\/mml:mover>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2192<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>M<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textsf{S}_b(L)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is the collection of joins of complemented sublocales of a frame\n                    <jats:italic>L<\/jats:italic>\n                    , or equivalently the Booleanization of the collection\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textsf{S}(L)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of all its sublocales. We do so by proving that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textsf{S}_b(L)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is isomorphic to the Bruns\u2013Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathfrak {c}(a)\\cap \\mathfrak {o}(b)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>\u2229<\/mml:mo>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$a,b\\in L$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mi>L<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s10485-026-09873-z","type":"journal-article","created":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T12:56:35Z","timestamp":1779972995000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Lattice of Smooth Sublocales as a Bruns\u2013Lakser Completion"],"prefix":"10.1007","volume":"34","author":[{"given":"Igor","family":"Arrieta","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anna Laura","family":"Suarez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,28]]},"reference":[{"key":"9873_CR1","unstructured":"Arrieta, I.: A study of localic subspaces, separation, and variants of normality and their duals. 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