{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:23:06Z","timestamp":1740108186745,"version":"3.37.3"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,7,5]],"date-time":"2020-07-05T00:00:00Z","timestamp":1593907200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,5]],"date-time":"2020-07-05T00:00:00Z","timestamp":1593907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["16-34860L"],"award-info":[{"award-number":["16-34860L"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004240","name":"Akademie V\u011bd \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["RVO:67985840"],"award-info":[{"award-number":["RVO:67985840"]}],"id":[{"id":"10.13039\/501100004240","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Arch. Math. Logic"],"published-print":{"date-parts":[[2021,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {c}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>c<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {c}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>c<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fra\u00efss\u00e9 limit of certain Fra\u00efss\u00e9 class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to <jats:inline-formula><jats:alternatives><jats:tex-math>$$(H(\\omega _1),\\in \\cup \\ni )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>\u03c9<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>\u222a<\/mml:mo>\n                    <mml:mo>\u220b<\/mml:mo>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$H(\\omega _1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03c9<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.<\/jats:p>","DOI":"10.1007\/s00153-020-00742-7","type":"journal-article","created":{"date-parts":[[2020,7,5]],"date-time":"2020-07-05T19:02:25Z","timestamp":1593975745000},"page":"189-209","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On countably saturated linear orders and certain class of countably saturated graphs"],"prefix":"10.1007","volume":"60","author":[{"given":"Ziemowit","family":"Kostana","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,7,5]]},"reference":[{"key":"742_CR1","doi-asserted-by":"publisher","first-page":"1534","DOI":"10.1016\/j.topol.2011.05.022","volume":"158","author":"A Avil\u00e9s","year":"2011","unstructured":"Avil\u00e9s, A., Brech, C.: A Boolean algebra and a Banach space obtained by push-out iteration. Topol. Appl. 158, 1534\u20131550 (2011)","journal-title":"Topol. Appl."},{"key":"742_CR2","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/S0166-8641(01)00138-9","volume":"122","author":"A Dow","year":"2002","unstructured":"Dow, A., Hart, K.P.: Applications of another characterization of $$\\beta \\mathbb{N} \\setminus \\mathbb{N}$$. Topol. Appl. 122, 105\u2013133 (2002)","journal-title":"Topol. Appl."},{"key":"742_CR3","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.topol.2004.05.016","volume":"150","author":"A Giarlotta","year":"2005","unstructured":"Giarlotta, A.: The representability number of a chain. Topol. Appl. 150, 157\u2013177 (2005)","journal-title":"Topol. Appl."},{"key":"742_CR4","doi-asserted-by":"publisher","first-page":"77","DOI":"10.4064\/fm-43-1-77-82","volume":"43","author":"L Gillman","year":"1956","unstructured":"Gillman, L.: Some remarks on $$\\eta _\\alpha $$-sets. Fund. Math. 43, 77\u201382 (1956)","journal-title":"Fund. Math."},{"key":"742_CR5","first-page":"97","volume-title":"Ordered Sets","author":"E Harzheim","year":"2005","unstructured":"Harzheim, E.: Ordered Sets, pp. 97\u2013108. Springer, Berlin (2005)"},{"key":"742_CR6","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1007\/BF01451165","volume":"65","author":"F Hausdorff","year":"1908","unstructured":"Hausdorff, F.: Gr\u00fcndzuge einer Theorie geordneten Mengen. Math. Ann. 65, 435\u2013505 (1908)","journal-title":"Math. Ann."},{"key":"742_CR7","volume-title":"A Shorter Model Theory","author":"W Hodges","year":"1997","unstructured":"Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)"},{"key":"742_CR8","unstructured":"Koppelberg, S.: In: Monk, D., Bonnet, R. (eds.) Handbook of Boolean Algebras, vol. 1. North-Hollad, Amsterdam (1989)"},{"issue":"11","key":"742_CR9","doi-asserted-by":"publisher","first-page":"1755","DOI":"10.1016\/j.apal.2014.07.004","volume":"165","author":"W Kubi\u015b","year":"2014","unstructured":"Kubi\u015b, W.: Fr\u00efss\u00e9 sequences: category-theoretic approach to universal homogeneous structures. Ann Pure Appl Logic 165(11), 1755\u20131811 (2014)","journal-title":"Ann Pure Appl Logic"},{"key":"742_CR10","volume-title":"Set Theory, Introduction to Independence Proofs","author":"K Kunen","year":"1980","unstructured":"Kunen, K.: Set Theory, Introduction to Independence Proofs. Elsevier, Amsterdam (1980)"},{"key":"742_CR11","doi-asserted-by":"publisher","first-page":"8","DOI":"10.4064\/fm-56-1-9-20","volume":"56","author":"V Nov\u00e1k","year":"1964","unstructured":"Nov\u00e1k, V.: On the lexicographic dimension of linearly ordered sets. Fund. Math. 56, 8\u201320 (1964)","journal-title":"Fund. Math."},{"key":"742_CR12","doi-asserted-by":"publisher","first-page":"59","DOI":"10.21136\/CPM.1953.117064","volume":"78","author":"M Novotn\u00fd","year":"1953","unstructured":"Novotn\u00fd, M.: On similarity of ordered continua of types $$\\tau $$ and $$\\tau ^2$$. \u010ceskoslovensk\u00e1 Akademie V\u011bd. \u010casopis Pro P\u011bstov\u00e1n\u00ed Matematiky 78, 59\u201360 (1953)","journal-title":"\u010ceskoslovensk\u00e1 Akademie V\u011bd. \u010casopis Pro P\u011bstov\u00e1n\u00ed Matematiky"},{"key":"742_CR13","volume-title":"Linear Orderings","author":"JG Rosenstein","year":"1982","unstructured":"Rosenstein, J.G.: Linear Orderings. Academic Press Inc., Cambridge (1982)"},{"key":"742_CR14","doi-asserted-by":"publisher","first-page":"14","DOI":"10.4064\/fm-5-1-14-19","volume":"5","author":"P Urysohn","year":"1923","unstructured":"Urysohn, P.: Un th\u00e9or\u00e9me sur la puissance des ensembles ordonn\u00e9s. Fund. Math. 5, 14\u201319 (1923)","journal-title":"Fund. Math."},{"issue":"3","key":"742_CR15","first-page":"539","volume":"72","author":"EK van Douwen","year":"1978","unstructured":"van Douwen, E.K., van Mill, J.: Parovi\u010denko\u2019s characterization of $$\\beta \\omega - \\omega $$ implies CH. Proc. Am. Math. Soc. 72(3), 539\u2013541 (1978)","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Archive for Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-020-00742-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00153-020-00742-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-020-00742-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,4]],"date-time":"2021-07-04T23:11:01Z","timestamp":1625440261000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00153-020-00742-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,5]]},"references-count":15,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["742"],"URL":"https:\/\/doi.org\/10.1007\/s00153-020-00742-7","relation":{},"ISSN":["0933-5846","1432-0665"],"issn-type":[{"type":"print","value":"0933-5846"},{"type":"electronic","value":"1432-0665"}],"subject":[],"published":{"date-parts":[[2020,7,5]]},"assertion":[{"value":"19 September 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 June 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}