A Certain Structure of Bipolar Fuzzy Subrings

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bip...
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Autor*in:	Hanan Alolaiyan [verfasserIn] Muhammad Haris Mateen [verfasserIn] Dragan Pamucar [verfasserIn] Muhammad Khalid Mahmmod [verfasserIn] Farrukh Arslan [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism

Übergeordnetes Werk:	In: Symmetry - MDPI AG, 2009, 13(2021), 8, p 1397
Übergeordnetes Werk:	volume:13 ; year:2021 ; number:8, p 1397

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/sym13081397

Katalog-ID:	DOAJ023999306

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650		4	\|a bipolar fuzzy set
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allfields	10.3390/sym13081397 doi (DE-627)DOAJ023999306 (DE-599)DOAJ18f8406f24f24bb684916ef3fb6b62ec DE-627 ger DE-627 rakwb eng QA1-939 Hanan Alolaiyan verfasserin aut A Certain Structure of Bipolar Fuzzy Subrings 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism Mathematics Muhammad Haris Mateen verfasserin aut Dragan Pamucar verfasserin aut Muhammad Khalid Mahmmod verfasserin aut Farrukh Arslan verfasserin aut In Symmetry MDPI AG, 2009 13(2021), 8, p 1397 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:13 year:2021 number:8, p 1397 https://doi.org/10.3390/sym13081397 kostenfrei https://doaj.org/article/18f8406f24f24bb684916ef3fb6b62ec kostenfrei https://www.mdpi.com/2073-8994/13/8/1397 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 13 2021 8, p 1397
spelling	10.3390/sym13081397 doi (DE-627)DOAJ023999306 (DE-599)DOAJ18f8406f24f24bb684916ef3fb6b62ec DE-627 ger DE-627 rakwb eng QA1-939 Hanan Alolaiyan verfasserin aut A Certain Structure of Bipolar Fuzzy Subrings 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism Mathematics Muhammad Haris Mateen verfasserin aut Dragan Pamucar verfasserin aut Muhammad Khalid Mahmmod verfasserin aut Farrukh Arslan verfasserin aut In Symmetry MDPI AG, 2009 13(2021), 8, p 1397 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:13 year:2021 number:8, p 1397 https://doi.org/10.3390/sym13081397 kostenfrei https://doaj.org/article/18f8406f24f24bb684916ef3fb6b62ec kostenfrei https://www.mdpi.com/2073-8994/13/8/1397 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 13 2021 8, p 1397
allfields_unstemmed	10.3390/sym13081397 doi (DE-627)DOAJ023999306 (DE-599)DOAJ18f8406f24f24bb684916ef3fb6b62ec DE-627 ger DE-627 rakwb eng QA1-939 Hanan Alolaiyan verfasserin aut A Certain Structure of Bipolar Fuzzy Subrings 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism Mathematics Muhammad Haris Mateen verfasserin aut Dragan Pamucar verfasserin aut Muhammad Khalid Mahmmod verfasserin aut Farrukh Arslan verfasserin aut In Symmetry MDPI AG, 2009 13(2021), 8, p 1397 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:13 year:2021 number:8, p 1397 https://doi.org/10.3390/sym13081397 kostenfrei https://doaj.org/article/18f8406f24f24bb684916ef3fb6b62ec kostenfrei https://www.mdpi.com/2073-8994/13/8/1397 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 13 2021 8, p 1397
allfieldsGer	10.3390/sym13081397 doi (DE-627)DOAJ023999306 (DE-599)DOAJ18f8406f24f24bb684916ef3fb6b62ec DE-627 ger DE-627 rakwb eng QA1-939 Hanan Alolaiyan verfasserin aut A Certain Structure of Bipolar Fuzzy Subrings 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism Mathematics Muhammad Haris Mateen verfasserin aut Dragan Pamucar verfasserin aut Muhammad Khalid Mahmmod verfasserin aut Farrukh Arslan verfasserin aut In Symmetry MDPI AG, 2009 13(2021), 8, p 1397 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:13 year:2021 number:8, p 1397 https://doi.org/10.3390/sym13081397 kostenfrei https://doaj.org/article/18f8406f24f24bb684916ef3fb6b62ec kostenfrei https://www.mdpi.com/2073-8994/13/8/1397 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 13 2021 8, p 1397
allfieldsSound	10.3390/sym13081397 doi (DE-627)DOAJ023999306 (DE-599)DOAJ18f8406f24f24bb684916ef3fb6b62ec DE-627 ger DE-627 rakwb eng QA1-939 Hanan Alolaiyan verfasserin aut A Certain Structure of Bipolar Fuzzy Subrings 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism Mathematics Muhammad Haris Mateen verfasserin aut Dragan Pamucar verfasserin aut Muhammad Khalid Mahmmod verfasserin aut Farrukh Arslan verfasserin aut In Symmetry MDPI AG, 2009 13(2021), 8, p 1397 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:13 year:2021 number:8, p 1397 https://doi.org/10.3390/sym13081397 kostenfrei https://doaj.org/article/18f8406f24f24bb684916ef3fb6b62ec kostenfrei https://www.mdpi.com/2073-8994/13/8/1397 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 13 2021 8, p 1397
language	English
source	In Symmetry 13(2021), 8, p 1397 volume:13 year:2021 number:8, p 1397
sourceStr	In Symmetry 13(2021), 8, p 1397 volume:13 year:2021 number:8, p 1397
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism Mathematics
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container_title	Symmetry
authorswithroles_txt_mv	Hanan Alolaiyan @@aut@@ Muhammad Haris Mateen @@aut@@ Dragan Pamucar @@aut@@ Muhammad Khalid Mahmmod @@aut@@ Farrukh Arslan @@aut@@
publishDateDaySort_date	2021-01-01T00:00:00Z
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callnumber-first	Q - Science
author	Hanan Alolaiyan
spellingShingle	Hanan Alolaiyan misc QA1-939 misc bipolar fuzzy set misc bipolar fuzzy subring misc bipolar fuzzy ideal misc bipolar fuzzy homomorphism misc bipolar fuzzy isomorphism misc Mathematics A Certain Structure of Bipolar Fuzzy Subrings
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topic_title	QA1-939 A Certain Structure of Bipolar Fuzzy Subrings bipolar fuzzy set bipolar fuzzy subring bipolar fuzzy ideal bipolar fuzzy homomorphism bipolar fuzzy isomorphism
topic	misc QA1-939 misc bipolar fuzzy set misc bipolar fuzzy subring misc bipolar fuzzy ideal misc bipolar fuzzy homomorphism misc bipolar fuzzy isomorphism misc Mathematics
topic_unstemmed	misc QA1-939 misc bipolar fuzzy set misc bipolar fuzzy subring misc bipolar fuzzy ideal misc bipolar fuzzy homomorphism misc bipolar fuzzy isomorphism misc Mathematics
topic_browse	misc QA1-939 misc bipolar fuzzy set misc bipolar fuzzy subring misc bipolar fuzzy ideal misc bipolar fuzzy homomorphism misc bipolar fuzzy isomorphism misc Mathematics
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title	A Certain Structure of Bipolar Fuzzy Subrings
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title_full	A Certain Structure of Bipolar Fuzzy Subrings
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title_sort	certain structure of bipolar fuzzy subrings
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title_auth	A Certain Structure of Bipolar Fuzzy Subrings
abstract	The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.
abstractGer	The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.
abstract_unstemmed	The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mi<α</mi<<mo<,</mo<<mi<β</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.
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title_short	A Certain Structure of Bipolar Fuzzy Subrings
url	https://doi.org/10.3390/sym13081397 https://doaj.org/article/18f8406f24f24bb684916ef3fb6b62ec https://www.mdpi.com/2073-8994/13/8/1397 https://doaj.org/toc/2073-8994
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A Certain Structure of Bipolar Fuzzy Subrings

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