Uncountable Strongly Surjective Linear Orders
We call a linear order L strongly surjective if whenever K is a suborder of L then there is a surjective f : L → K so that x ≤ y implies f ( x ) ≤ f ( y ). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo,...
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| Veröffentlicht in: | Order (Dordrecht) 2019-03, Vol.36 (1), p.43-64 |
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| creator | Soukup, Dániel T. |
| description | We call a linear order
L strongly surjective
if whenever
K
is a suborder of
L
then there is a surjective
f
:
L
→
K
so that
x
≤
y
implies
f
(
x
) ≤
f
(
y
). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢
+
implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under
2
ℵ
0
<
2
ℵ
1
or in the Cohen and other canonical models (where
2
ℵ
0
=
2
ℵ
1
); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all. |
| doi_str_mv | 10.1007/s11083-018-9454-7 |
| format | Article |
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L strongly surjective
if whenever
K
is a suborder of
L
then there is a surjective
f
:
L
→
K
so that
x
≤
y
implies
f
(
x
) ≤
f
(
y
). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢
+
implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under
2
ℵ
0
<
2
ℵ
1
or in the Cohen and other canonical models (where
2
ℵ
0
=
2
ℵ
1
); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.</description><identifier>ISSN: 0167-8094</identifier><identifier>EISSN: 1572-9273</identifier><identifier>DOI: 10.1007/s11083-018-9454-7</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Algebra ; Discrete Mathematics ; Lattices ; Mathematics ; Mathematics and Statistics ; Order ; Ordered Algebraic Structures</subject><ispartof>Order (Dordrecht), 2019-03, Vol.36 (1), p.43-64</ispartof><rights>Springer Science+Business Media B.V., part of Springer Nature 2018</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-ddc5df422d0a4b87c221a079026a7592dd71664eb4a64f678010802dfef0fd413</citedby><cites>FETCH-LOGICAL-c316t-ddc5df422d0a4b87c221a079026a7592dd71664eb4a64f678010802dfef0fd413</cites><orcidid>0000-0001-8014-6517</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11083-018-9454-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11083-018-9454-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Soukup, Dániel T.</creatorcontrib><title>Uncountable Strongly Surjective Linear Orders</title><title>Order (Dordrecht)</title><addtitle>Order</addtitle><description>We call a linear order
L strongly surjective
if whenever
K
is a suborder of
L
then there is a surjective
f
:
L
→
K
so that
x
≤
y
implies
f
(
x
) ≤
f
(
y
). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢
+
implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under
2
ℵ
0
<
2
ℵ
1
or in the Cohen and other canonical models (where
2
ℵ
0
=
2
ℵ
1
); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.</description><subject>Algebra</subject><subject>Discrete Mathematics</subject><subject>Lattices</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Order</subject><subject>Ordered Algebraic Structures</subject><issn>0167-8094</issn><issn>1572-9273</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kMFKxDAURYMoWEc_wF3BdfS9NE2apQzqCIVZjLMOaZMMLbUdk1aYv7dDBVeu3ubcex-HkHuERwSQTxERiowCFlTxnFN5QRLMJaOKyeySJIBC0gIUvyY3MbYAkKlcJITu-3qY-tFUnUt3Yxj6Q3dKd1NoXT023y4tm96ZkG6DdSHekitvuujufu-K7F9fPtYbWm7f3tfPJa0zFCO1ts6t54xZMLwqZM0YGpAKmDAyV8xaiUJwV3EjuBeygPl5YNY7D95yzFbkYek9huFrcnHU7TCFfp7UDBUXnDMlZgoXqg5DjMF5fQzNpwknjaDPVvRiRc9W9NmKlnOGLZk4s_3Bhb_m_0M_puZjaQ</recordid><startdate>20190315</startdate><enddate>20190315</enddate><creator>Soukup, Dániel T.</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-8014-6517</orcidid></search><sort><creationdate>20190315</creationdate><title>Uncountable Strongly Surjective Linear Orders</title><author>Soukup, Dániel T.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-ddc5df422d0a4b87c221a079026a7592dd71664eb4a64f678010802dfef0fd413</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Algebra</topic><topic>Discrete Mathematics</topic><topic>Lattices</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Order</topic><topic>Ordered Algebraic Structures</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Soukup, Dániel T.</creatorcontrib><collection>CrossRef</collection><jtitle>Order (Dordrecht)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Soukup, Dániel T.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Uncountable Strongly Surjective Linear Orders</atitle><jtitle>Order (Dordrecht)</jtitle><stitle>Order</stitle><date>2019-03-15</date><risdate>2019</risdate><volume>36</volume><issue>1</issue><spage>43</spage><epage>64</epage><pages>43-64</pages><issn>0167-8094</issn><eissn>1572-9273</eissn><abstract>We call a linear order
L strongly surjective
if whenever
K
is a suborder of
L
then there is a surjective
f
:
L
→
K
so that
x
≤
y
implies
f
(
x
) ≤
f
(
y
). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢
+
implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under
2
ℵ
0
<
2
ℵ
1
or in the Cohen and other canonical models (where
2
ℵ
0
=
2
ℵ
1
); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s11083-018-9454-7</doi><tpages>22</tpages><orcidid>https://orcid.org/0000-0001-8014-6517</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0167-8094 |
| ispartof | Order (Dordrecht), 2019-03, Vol.36 (1), p.43-64 |
| issn | 0167-8094 1572-9273 |
| language | eng |
| recordid | cdi_proquest_journals_2194644296 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Algebra Discrete Mathematics Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures |
| title | Uncountable Strongly Surjective Linear Orders |
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