Free Boolean algebras over unions of two well orderings
Given a partially ordered set P there exists the most general Boolean algebra F ˆ ( P ) which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P = P 0 ∪ P 1 , where P 0 , P 1 are well orderings. We call them nearly ordinal...
Gespeichert in:
| Veröffentlicht in: | Topology and its applications 2009-04, Vol.156 (7), p.1177-1185 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 1185 |
|---|---|
| container_issue | 7 |
| container_start_page | 1177 |
| container_title | Topology and its applications |
| container_volume | 156 |
| creator | Bonnet, Robert Faouzi, Latifa Kubiś, Wiesław |
| description | Given a partially ordered set
P there exists the most general Boolean algebra
F
ˆ
(
P
)
which contains
P as a generating set, called the
free Boolean algebra over
P. We study free Boolean algebras over posets of the form
P
=
P
0
∪
P
1
, where
P
0
,
P
1
are well orderings. We call them
nearly ordinal algebras.
Answering a question of Maurice Pouzet, we show that for every uncountable cardinal
κ there are
2
κ
pairwise non-isomorphic nearly ordinal algebras of cardinality
κ.
Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product
(
ω
1
+
1
)
×
(
ω
1
+
1
)
, showing that there are only
ℵ
1
many types. In contrast with the last result, we show that there are
2
ℵ
1
topological types of closed subsets of the Tikhonov plank
(
ω
1
+
1
)
×
(
ω
+
1
)
. |
| doi_str_mv | 10.1016/j.topol.2008.12.012 |
| format | Article |
| fullrecord | <record><control><sourceid>hal_cross</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_00393948v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0166864108003921</els_id><sourcerecordid>oai_HAL_hal_00393948v1</sourcerecordid><originalsourceid>FETCH-LOGICAL-c332t-569f8bc9558b6f9d214bee9e2732a0024109219ed8ddd66dc3c0d9e4e401d5b73</originalsourceid><addsrcrecordid>eNp9kD1PwzAQhi0EEqXwC1iyMiTc2fnywFAq2iJVYoHZcuxLcRXiyg6t-PekFDEy3avT-5x0D2O3CBkClvfbbPA732UcoM6QZ4D8jE2wrmQqOFTnbDK2yrQuc7xkVzFuAQBlxSesWgSi5NH7jnSf6G5DTdAx8XsKyWfvfD_mNhkOPjlQ1yU-WAqu38RrdtHqLtLN75yyt8XT63yVrl-Wz_PZOjVC8CEtStnWjZFFUTdlKy3HvCGSxCvBNQDPESRHSba21palNcKAlZRTDmiLphJTdne6-647tQvuQ4cv5bVTq9laHXcAQgqZ13scu-LUNcHHGKj9AxDU0ZPaqh9P6uhJIVejp5F6OFE0vrF3FFQ0jnpD1gUyg7Le_ct_A7P2cRc</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Free Boolean algebras over unions of two well orderings</title><source>Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals</source><source>Access via ScienceDirect (Elsevier)</source><creator>Bonnet, Robert ; Faouzi, Latifa ; Kubiś, Wiesław</creator><creatorcontrib>Bonnet, Robert ; Faouzi, Latifa ; Kubiś, Wiesław</creatorcontrib><description>Given a partially ordered set
P there exists the most general Boolean algebra
F
ˆ
(
P
)
which contains
P as a generating set, called the
free Boolean algebra over
P. We study free Boolean algebras over posets of the form
P
=
P
0
∪
P
1
, where
P
0
,
P
1
are well orderings. We call them
nearly ordinal algebras.
Answering a question of Maurice Pouzet, we show that for every uncountable cardinal
κ there are
2
κ
pairwise non-isomorphic nearly ordinal algebras of cardinality
κ.
Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product
(
ω
1
+
1
)
×
(
ω
1
+
1
)
, showing that there are only
ℵ
1
many types. In contrast with the last result, we show that there are
2
ℵ
1
topological types of closed subsets of the Tikhonov plank
(
ω
1
+
1
)
×
(
ω
+
1
)
.</description><identifier>ISSN: 0166-8641</identifier><identifier>EISSN: 1879-3207</identifier><identifier>DOI: 10.1016/j.topol.2008.12.012</identifier><language>eng</language><publisher>Elsevier B.V</publisher><subject>Compact distributive lattices ; Computer Science ; Discrete Mathematics ; Poset algebras ; Superatomic Boolean algebras ; Well quasi orderings</subject><ispartof>Topology and its applications, 2009-04, Vol.156 (7), p.1177-1185</ispartof><rights>2008 Elsevier B.V.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c332t-569f8bc9558b6f9d214bee9e2732a0024109219ed8ddd66dc3c0d9e4e401d5b73</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.topol.2008.12.012$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>230,314,780,784,885,3550,27924,27925,45995</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00393948$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Bonnet, Robert</creatorcontrib><creatorcontrib>Faouzi, Latifa</creatorcontrib><creatorcontrib>Kubiś, Wiesław</creatorcontrib><title>Free Boolean algebras over unions of two well orderings</title><title>Topology and its applications</title><description>Given a partially ordered set
P there exists the most general Boolean algebra
F
ˆ
(
P
)
which contains
P as a generating set, called the
free Boolean algebra over
P. We study free Boolean algebras over posets of the form
P
=
P
0
∪
P
1
, where
P
0
,
P
1
are well orderings. We call them
nearly ordinal algebras.
Answering a question of Maurice Pouzet, we show that for every uncountable cardinal
κ there are
2
κ
pairwise non-isomorphic nearly ordinal algebras of cardinality
κ.
Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product
(
ω
1
+
1
)
×
(
ω
1
+
1
)
, showing that there are only
ℵ
1
many types. In contrast with the last result, we show that there are
2
ℵ
1
topological types of closed subsets of the Tikhonov plank
(
ω
1
+
1
)
×
(
ω
+
1
)
.</description><subject>Compact distributive lattices</subject><subject>Computer Science</subject><subject>Discrete Mathematics</subject><subject>Poset algebras</subject><subject>Superatomic Boolean algebras</subject><subject>Well quasi orderings</subject><issn>0166-8641</issn><issn>1879-3207</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><recordid>eNp9kD1PwzAQhi0EEqXwC1iyMiTc2fnywFAq2iJVYoHZcuxLcRXiyg6t-PekFDEy3avT-5x0D2O3CBkClvfbbPA732UcoM6QZ4D8jE2wrmQqOFTnbDK2yrQuc7xkVzFuAQBlxSesWgSi5NH7jnSf6G5DTdAx8XsKyWfvfD_mNhkOPjlQ1yU-WAqu38RrdtHqLtLN75yyt8XT63yVrl-Wz_PZOjVC8CEtStnWjZFFUTdlKy3HvCGSxCvBNQDPESRHSba21palNcKAlZRTDmiLphJTdne6-647tQvuQ4cv5bVTq9laHXcAQgqZ13scu-LUNcHHGKj9AxDU0ZPaqh9P6uhJIVejp5F6OFE0vrF3FFQ0jnpD1gUyg7Le_ct_A7P2cRc</recordid><startdate>20090401</startdate><enddate>20090401</enddate><creator>Bonnet, Robert</creator><creator>Faouzi, Latifa</creator><creator>Kubiś, Wiesław</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope></search><sort><creationdate>20090401</creationdate><title>Free Boolean algebras over unions of two well orderings</title><author>Bonnet, Robert ; Faouzi, Latifa ; Kubiś, Wiesław</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c332t-569f8bc9558b6f9d214bee9e2732a0024109219ed8ddd66dc3c0d9e4e401d5b73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Compact distributive lattices</topic><topic>Computer Science</topic><topic>Discrete Mathematics</topic><topic>Poset algebras</topic><topic>Superatomic Boolean algebras</topic><topic>Well quasi orderings</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bonnet, Robert</creatorcontrib><creatorcontrib>Faouzi, Latifa</creatorcontrib><creatorcontrib>Kubiś, Wiesław</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Topology and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bonnet, Robert</au><au>Faouzi, Latifa</au><au>Kubiś, Wiesław</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Free Boolean algebras over unions of two well orderings</atitle><jtitle>Topology and its applications</jtitle><date>2009-04-01</date><risdate>2009</risdate><volume>156</volume><issue>7</issue><spage>1177</spage><epage>1185</epage><pages>1177-1185</pages><issn>0166-8641</issn><eissn>1879-3207</eissn><abstract>Given a partially ordered set
P there exists the most general Boolean algebra
F
ˆ
(
P
)
which contains
P as a generating set, called the
free Boolean algebra over
P. We study free Boolean algebras over posets of the form
P
=
P
0
∪
P
1
, where
P
0
,
P
1
are well orderings. We call them
nearly ordinal algebras.
Answering a question of Maurice Pouzet, we show that for every uncountable cardinal
κ there are
2
κ
pairwise non-isomorphic nearly ordinal algebras of cardinality
κ.
Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product
(
ω
1
+
1
)
×
(
ω
1
+
1
)
, showing that there are only
ℵ
1
many types. In contrast with the last result, we show that there are
2
ℵ
1
topological types of closed subsets of the Tikhonov plank
(
ω
1
+
1
)
×
(
ω
+
1
)
.</abstract><pub>Elsevier B.V</pub><doi>10.1016/j.topol.2008.12.012</doi><tpages>9</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0166-8641 |
| ispartof | Topology and its applications, 2009-04, Vol.156 (7), p.1177-1185 |
| issn | 0166-8641 1879-3207 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_00393948v1 |
| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; Access via ScienceDirect (Elsevier) |
| subjects | Compact distributive lattices Computer Science Discrete Mathematics Poset algebras Superatomic Boolean algebras Well quasi orderings |
| title | Free Boolean algebras over unions of two well orderings |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-04T01%3A53%3A28IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-hal_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Free%20Boolean%20algebras%20over%20unions%20of%20two%20well%20orderings&rft.jtitle=Topology%20and%20its%20applications&rft.au=Bonnet,%20Robert&rft.date=2009-04-01&rft.volume=156&rft.issue=7&rft.spage=1177&rft.epage=1185&rft.pages=1177-1185&rft.issn=0166-8641&rft.eissn=1879-3207&rft_id=info:doi/10.1016/j.topol.2008.12.012&rft_dat=%3Chal_cross%3Eoai_HAL_hal_00393948v1%3C/hal_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rft_els_id=S0166864108003921&rfr_iscdi=true |