{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T11:19:08Z","timestamp":1783163948908,"version":"3.54.6"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T00:00:00Z","timestamp":1783123200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T00:00:00Z","timestamp":1783123200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004477","name":"Stellenbosch University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004477","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this paper we introduce and study\n                    <jats:italic>rectangular torsion theories<\/jats:italic>\n                    , i.e. those torsion theories\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$(\\mathbb {C},\\mathcal {T},\\mathcal {F})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathbb {C}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>C<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    a pointed category, where the canonical functor\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathbb {C}\\rightarrow \\mathcal {T}\\times \\mathcal {F}$$<\/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>\u2192<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>F<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is an equivalence of categories. We prove that rectangular torsion theories are pseudo-algebras for a suitable 2-monad on the 2-category of pointed categories. Interestingly, it turns out that this 2-monad is a 2-categorical counterpart of the monad for rectangular bands, i.e., non-empty semigroups satisfying\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$xyz = xz$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . So rectangular torsion theories can be viewed as internal rectangular bands in the 2-category of pointed categories. We also show that a triple\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$(\\mathbb {C},\\mathcal {T},\\mathcal {F})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a rectangular torsion theory if and only if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathbb {C}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>C<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a pointed category with full replete subcategories\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {T},\\mathcal {F}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>F<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that: (i) every morphism between an object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {T}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>T<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and an object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {F}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>F<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    (in either direction) is a zero morphism; (ii) any object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathbb {C}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>C<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    decomposes as a biproduct of an object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {T}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>T<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and an object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {F}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>F<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and, all biproducts of an object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {T}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>T<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    with an object in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {F}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>F<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    exist.\n                  <\/jats:p>","DOI":"10.1007\/s10485-026-09867-x","type":"journal-article","created":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T10:39:07Z","timestamp":1783161547000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Rectangular Torsion Theories"],"prefix":"10.1007","volume":"34","author":[{"given":"Elena","family":"Caviglia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zurab","family":"Janelidze","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luca","family":"Mesiti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,4]]},"reference":[{"key":"9867_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-4049(89)90160-6","volume":"59","author":"R Blackwell","year":"1989","unstructured":"Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory. 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