{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T11:10:02Z","timestamp":1749553802835,"version":"3.41.0"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T00:00:00Z","timestamp":1722297600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T00:00:00Z","timestamp":1722297600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Slovak grant agency VEGA","award":["1\/0152\/22"],"award-info":[{"award-number":["1\/0152\/22"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Order"],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>A distributive lattice with zero is <jats:italic>completely normal<\/jats:italic> if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(x,y)\\mapsto x\\mathbin {\\smallsetminus }y$$<\/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>x<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u21a6<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> satisfying the rules <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$x\\le y\\vee (x\\mathbin {\\smallsetminus }y)$$<\/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:mo>\u2264<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>\u2228<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(x\\mathbin {\\smallsetminus }y)\\wedge (y\\mathbin {\\smallsetminus }x)=0$$<\/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>x<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u2227<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> \u2014 in short a <jats:italic>deviation<\/jats:italic>. In this paper we study the following additional properties of deviations: <jats:italic>monotone<\/jats:italic> (i.e., isotone in\u00a0<jats:italic>x<\/jats:italic> and antitone in\u00a0<jats:italic>y<\/jats:italic>) and <jats:italic>Cevian<\/jats:italic> (i.e., <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$x\\mathbin {\\smallsetminus }z\\le (x\\mathbin {\\smallsetminus }y)\\vee (y\\mathbin {\\smallsetminus }z)$$<\/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:mo>\\<\/mml:mo>\n                    <mml:mi>z<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u2228<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>z<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>). We relate those matters to <jats:italic>finite separability<\/jats:italic> as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-ideals of Abelian <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-groups (which are always completely normal). We prove that for free Abelian <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-groups (and also free <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Bbbk $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>k<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-group with strong unit, of cardinality\u00a0<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\aleph _1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u2135<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, whose principal <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-ideal lattice does not have a monotone deviation.<\/jats:p>","DOI":"10.1007\/s11083-024-09678-6","type":"journal-article","created":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T17:02:12Z","timestamp":1722358932000},"page":"211-229","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Monotone-Cevian and Finitely Separable Lattices"],"prefix":"10.1007","volume":"42","author":[{"given":"Miroslav","family":"Plo\u0161\u010dica","sequence":"first","affiliation":[]},{"given":"Friedrich","family":"Wehrung","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,30]]},"reference":[{"key":"9678_CR1","doi-asserted-by":"crossref","unstructured":"Baker, K.A.: Free vector lattices. 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