A Semigroup of Theories and Its Lattice of Idempotent Elements
On the set of all first-order theories T (σ) of similarity type σ, a binary operation {·} is defined by the rule T · S = Th({ A × B | A |= T and B |= S }) for any theories T , S ∈ T (σ). The structure 〈 T ( σ ); ⋅〉 forms a commutative semigroup, which is called a semigroup of theories. We prove that...
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| Veröffentlicht in: | Algebra and logic 2021-03, Vol.60 (1), p.1-14 |
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| creator | Bekenov, M. I. Nurakunov, A. M. |
| description | On the set of all first-order theories
T
(σ) of similarity type σ, a binary operation {·} is defined by the rule
T
·
S
= Th({
A
×
B
|
A
|=
T
and
B
|=
S
}) for any theories
T
,
S
∈
T
(σ). The structure 〈
T
(
σ
); ⋅〉 forms a commutative semigroup, which is called a semigroup of theories. We prove that a semigroup of theories is an ideal extension of a semigroup
S
T
∗
by a semigroup
S
T
. The set of all idempotent elements of a semigroup of theories forms a complete lattice with respect to the partial order ≤ defined as
T
≤
S
iff
T
·
S
=
S
for all
T
,
S
∈
T
(σ). Also the set of all idempotent complete theories forms a complete lattice with respect to ≤, which is not necessarily a sublattice of the lattice of idempotent theories. |
| doi_str_mv | 10.1007/s10469-021-09623-1 |
| format | Article |
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T
(σ) of similarity type σ, a binary operation {·} is defined by the rule
T
·
S
= Th({
A
×
B
|
A
|=
T
and
B
|=
S
}) for any theories
T
,
S
∈
T
(σ). The structure 〈
T
(
σ
); ⋅〉 forms a commutative semigroup, which is called a semigroup of theories. We prove that a semigroup of theories is an ideal extension of a semigroup
S
T
∗
by a semigroup
S
T
. The set of all idempotent elements of a semigroup of theories forms a complete lattice with respect to the partial order ≤ defined as
T
≤
S
iff
T
·
S
=
S
for all
T
,
S
∈
T
(σ). Also the set of all idempotent complete theories forms a complete lattice with respect to ≤, which is not necessarily a sublattice of the lattice of idempotent theories.</description><identifier>ISSN: 0002-5232</identifier><identifier>EISSN: 1573-8302</identifier><identifier>DOI: 10.1007/s10469-021-09623-1</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Analysis ; Mathematical Logic and Foundations ; Mathematics ; Mathematics and Statistics ; Semigroups ; Web services</subject><ispartof>Algebra and logic, 2021-03, Vol.60 (1), p.1-14</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2021</rights><rights>COPYRIGHT 2021 Springer</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c383t-10ff61229d50b942d3f37d02f2acbb6a26ddc2dd8b24213d002e7aef31fa71383</citedby><cites>FETCH-LOGICAL-c383t-10ff61229d50b942d3f37d02f2acbb6a26ddc2dd8b24213d002e7aef31fa71383</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10469-021-09623-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10469-021-09623-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Bekenov, M. I.</creatorcontrib><creatorcontrib>Nurakunov, A. M.</creatorcontrib><title>A Semigroup of Theories and Its Lattice of Idempotent Elements</title><title>Algebra and logic</title><addtitle>Algebra Logic</addtitle><description>On the set of all first-order theories
T
(σ) of similarity type σ, a binary operation {·} is defined by the rule
T
·
S
= Th({
A
×
B
|
A
|=
T
and
B
|=
S
}) for any theories
T
,
S
∈
T
(σ). The structure 〈
T
(
σ
); ⋅〉 forms a commutative semigroup, which is called a semigroup of theories. We prove that a semigroup of theories is an ideal extension of a semigroup
S
T
∗
by a semigroup
S
T
. The set of all idempotent elements of a semigroup of theories forms a complete lattice with respect to the partial order ≤ defined as
T
≤
S
iff
T
·
S
=
S
for all
T
,
S
∈
T
(σ). Also the set of all idempotent complete theories forms a complete lattice with respect to ≤, which is not necessarily a sublattice of the lattice of idempotent theories.</description><subject>Algebra</subject><subject>Analysis</subject><subject>Mathematical Logic and Foundations</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Semigroups</subject><subject>Web services</subject><issn>0002-5232</issn><issn>1573-8302</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNqNUU1LAzEQDaJgrf4BTwuet04mu8nuRSilaqHgwXoO6SapW7qbmmwP_nvTrlAFKTKHYSbvI8wj5JbCiAKI-0Ah42UKSFMoObKUnpEBzQVLCwZ4TgYAgGmODC_JVQjrOJa8gAF5GCevpqlX3u22ibPJ4t04X5uQqFYnsy4kc9V1dWX2bzNtmq3rTNsl041pYg_X5MKqTTA3331I3h6ni8lzOn95mk3G87RiBetSCtZyiljqHJZlhppZJjSgRVUtl1wh17pCrYslZkiZjp81QhnLqFWCRokhuet1t9597Ezo5NrtfBstJeYZExllgh9RK7Uxsm6t67yqmjpUcsxLLnJAIU6jCshzWh4cR3-gYsUj1JVrja3j_pfs_wg_HLAnVN6F4I2VW183yn9KCnKfquxTlTFVeUhV0khiPSlEcLsy_niIE6wvhdugIA</recordid><startdate>20210301</startdate><enddate>20210301</enddate><creator>Bekenov, M. I.</creator><creator>Nurakunov, A. M.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20210301</creationdate><title>A Semigroup of Theories and Its Lattice of Idempotent Elements</title><author>Bekenov, M. I. ; Nurakunov, A. M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c383t-10ff61229d50b942d3f37d02f2acbb6a26ddc2dd8b24213d002e7aef31fa71383</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algebra</topic><topic>Analysis</topic><topic>Mathematical Logic and Foundations</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Semigroups</topic><topic>Web services</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bekenov, M. I.</creatorcontrib><creatorcontrib>Nurakunov, A. M.</creatorcontrib><collection>CrossRef</collection><jtitle>Algebra and logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bekenov, M. I.</au><au>Nurakunov, A. M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Semigroup of Theories and Its Lattice of Idempotent Elements</atitle><jtitle>Algebra and logic</jtitle><stitle>Algebra Logic</stitle><date>2021-03-01</date><risdate>2021</risdate><volume>60</volume><issue>1</issue><spage>1</spage><epage>14</epage><pages>1-14</pages><issn>0002-5232</issn><eissn>1573-8302</eissn><abstract>On the set of all first-order theories
T
(σ) of similarity type σ, a binary operation {·} is defined by the rule
T
·
S
= Th({
A
×
B
|
A
|=
T
and
B
|=
S
}) for any theories
T
,
S
∈
T
(σ). The structure 〈
T
(
σ
); ⋅〉 forms a commutative semigroup, which is called a semigroup of theories. We prove that a semigroup of theories is an ideal extension of a semigroup
S
T
∗
by a semigroup
S
T
. The set of all idempotent elements of a semigroup of theories forms a complete lattice with respect to the partial order ≤ defined as
T
≤
S
iff
T
·
S
=
S
for all
T
,
S
∈
T
(σ). Also the set of all idempotent complete theories forms a complete lattice with respect to ≤, which is not necessarily a sublattice of the lattice of idempotent theories.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10469-021-09623-1</doi><tpages>14</tpages></addata></record> |
| fulltext | fulltext |
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| ispartof | Algebra and logic, 2021-03, Vol.60 (1), p.1-14 |
| issn | 0002-5232 1573-8302 |
| language | eng |
| recordid | cdi_proquest_journals_2543741376 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Algebra Analysis Mathematical Logic and Foundations Mathematics Mathematics and Statistics Semigroups Web services |
| title | A Semigroup of Theories and Its Lattice of Idempotent Elements |
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