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Bi-atomic classes of positive semirings
A subsemiring S of R is called a positive semiring provided that S consists of nonnegative numbers and 1 ∈ S . Here we study factorizations in both the additive monoid ( S , + ) and the multiplicative monoid ( S \ { 0 } , · ) . In particular, we investigate when, for a positive semiring S , both ( S...
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| Published in: | Semigroup forum 2021-08, Vol.103 (1), p.1-23 |
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| Language: | English |
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| container_title | Semigroup forum |
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| creator | Baeth, Nicholas R. Chapman, Scott T. Gotti, Felix |
| description | A subsemiring
S
of
R
is called a
positive semiring
provided that
S
consists of nonnegative numbers and
1
∈
S
. Here we study factorizations in both the additive monoid
(
S
,
+
)
and the multiplicative monoid
(
S
\
{
0
}
,
·
)
. In particular, we investigate when, for a positive semiring
S
, both
(
S
,
+
)
and
(
S
\
{
0
}
,
·
)
have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP
⇒
BFP and FFP
⇒
BFP
⇒
ACCP
⇒
atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible. |
| doi_str_mv | 10.1007/s00233-021-10189-8 |
| format | article |
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S
of
R
is called a
positive semiring
provided that
S
consists of nonnegative numbers and
1
∈
S
. Here we study factorizations in both the additive monoid
(
S
,
+
)
and the multiplicative monoid
(
S
\
{
0
}
,
·
)
. In particular, we investigate when, for a positive semiring
S
, both
(
S
,
+
)
and
(
S
\
{
0
}
,
·
)
have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP
⇒
BFP and FFP
⇒
BFP
⇒
ACCP
⇒
atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.</description><identifier>ISSN: 0037-1912</identifier><identifier>EISSN: 1432-2137</identifier><identifier>DOI: 10.1007/s00233-021-10189-8</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Chains ; Factorization ; Mathematics ; Mathematics and Statistics ; Monoids ; Research Article ; Rings (mathematics)</subject><ispartof>Semigroup forum, 2021-08, Vol.103 (1), p.1-23</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-4f4c36ba9c9b07b331cd9c01b191e1ed395b0003800e51bfffdeacbc73d85d603</citedby><cites>FETCH-LOGICAL-c363t-4f4c36ba9c9b07b331cd9c01b191e1ed395b0003800e51bfffdeacbc73d85d603</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Baeth, Nicholas R.</creatorcontrib><creatorcontrib>Chapman, Scott T.</creatorcontrib><creatorcontrib>Gotti, Felix</creatorcontrib><title>Bi-atomic classes of positive semirings</title><title>Semigroup forum</title><addtitle>Semigroup Forum</addtitle><description>A subsemiring
S
of
R
is called a
positive semiring
provided that
S
consists of nonnegative numbers and
1
∈
S
. Here we study factorizations in both the additive monoid
(
S
,
+
)
and the multiplicative monoid
(
S
\
{
0
}
,
·
)
. In particular, we investigate when, for a positive semiring
S
, both
(
S
,
+
)
and
(
S
\
{
0
}
,
·
)
have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP
⇒
BFP and FFP
⇒
BFP
⇒
ACCP
⇒
atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.</description><subject>Algebra</subject><subject>Chains</subject><subject>Factorization</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Monoids</subject><subject>Research Article</subject><subject>Rings (mathematics)</subject><issn>0037-1912</issn><issn>1432-2137</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-AU8FD56iM5l2mx5V_IIFL3oOTZpIlt1tzXQF_71ZK3jzNDPwfjCPEOcIVwhQXzOAIpKgUCKgbqQ-EDMsSUmFVB-KGQDVEhtUx-KEeQX5hgXNxOVtlO3Yb6Ir3Lpl9lz0oRh6jmP89AX7TUxx-86n4ii0a_Znv3Mu3h7uX--e5PLl8fnuZikdLWiUZSjzYtvGNRZqS4SuaxygzdUefUdNZffdGsBXaEMInW-ddTV1uuoWQHNxMeUOqf_YeR7Nqt-lba40qipLpXSp66xSk8qlnjn5YIYUN236MghmD8RMQEwGYn6AGJ1NNJl42L_k01_0P65v2PxiGw</recordid><startdate>20210801</startdate><enddate>20210801</enddate><creator>Baeth, Nicholas R.</creator><creator>Chapman, Scott T.</creator><creator>Gotti, Felix</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20210801</creationdate><title>Bi-atomic classes of positive semirings</title><author>Baeth, Nicholas R. ; Chapman, Scott T. ; Gotti, Felix</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-4f4c36ba9c9b07b331cd9c01b191e1ed395b0003800e51bfffdeacbc73d85d603</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algebra</topic><topic>Chains</topic><topic>Factorization</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Monoids</topic><topic>Research Article</topic><topic>Rings (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Baeth, Nicholas R.</creatorcontrib><creatorcontrib>Chapman, Scott T.</creatorcontrib><creatorcontrib>Gotti, Felix</creatorcontrib><collection>CrossRef</collection><jtitle>Semigroup forum</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Baeth, Nicholas R.</au><au>Chapman, Scott T.</au><au>Gotti, Felix</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Bi-atomic classes of positive semirings</atitle><jtitle>Semigroup forum</jtitle><stitle>Semigroup Forum</stitle><date>2021-08-01</date><risdate>2021</risdate><volume>103</volume><issue>1</issue><spage>1</spage><epage>23</epage><pages>1-23</pages><issn>0037-1912</issn><eissn>1432-2137</eissn><abstract>A subsemiring
S
of
R
is called a
positive semiring
provided that
S
consists of nonnegative numbers and
1
∈
S
. Here we study factorizations in both the additive monoid
(
S
,
+
)
and the multiplicative monoid
(
S
\
{
0
}
,
·
)
. In particular, we investigate when, for a positive semiring
S
, both
(
S
,
+
)
and
(
S
\
{
0
}
,
·
)
have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP
⇒
BFP and FFP
⇒
BFP
⇒
ACCP
⇒
atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00233-021-10189-8</doi><tpages>23</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Semigroup forum, 2021-08, Vol.103 (1), p.1-23 |
| issn | 0037-1912 1432-2137 |
| language | eng |
| recordid | cdi_proquest_journals_2544228487 |
| source | Springer Link |
| subjects | Algebra Chains Factorization Mathematics Mathematics and Statistics Monoids Research Article Rings (mathematics) |
| title | Bi-atomic classes of positive semirings |
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