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We show that if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textbf{S}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>S<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is atomic then\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${{\\,\\mathrm{{\\textbf {Cl}}}\\,}}(\\textbf{S})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mspace\/>\n                              <mml:mi>Cl<\/mml:mi>\n                              <mml:mspace\/>\n                            <\/mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a complete atomic Boolean algebra. If\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textbf{S}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>S<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textbf{S}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>S<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a complete pseudocomplemented lattice then\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${{\\,\\mathrm{{\\textbf {Cl}}}\\,}}(\\textbf{S})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mspace\/>\n                              <mml:mi>Cl<\/mml:mi>\n                              <mml:mspace\/>\n                            <\/mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a complete Boolean algebra. For pseudocomplemented posets a similar result holds if the subset of pseudocomplements forms a complete lattice satisfying a certain compatibility condition.\n                  <\/jats:p>","DOI":"10.1007\/s11083-025-09696-y","type":"journal-article","created":{"date-parts":[[2025,1,25]],"date-time":"2025-01-25T03:46:41Z","timestamp":1737776801000},"page":"577-592","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Induced Orthogonality in Semilattices with 0 and in Pseudocomplemented Lattices and Posets"],"prefix":"10.1007","volume":"42","author":[{"given":"Ivan","family":"Chajda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Miroslav","family":"Kola\u0159\u00edk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Helmut","family":"L\u00e4nger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,25]]},"reference":[{"key":"9696_CR1","unstructured":"Dacey Jr, J.C.: Orthomodular Spaces. 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