The annihilator of fuzzy subgroups

We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems. In this paper we refer to another type of annihilator which is essential in classical duality theory and extends...

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Published inFuzzy sets and systems Vol. 369; pp. 122 - 131
Main Authors Ardanza-Trevijano, S., Chasco, M.J., Elorza, J.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 15.08.2019
Subjects
B-algebra
Fuzzy annihilator subgroup
Fuzzy least squares
Fuzzy orthogonal complement
Fuzzy subgroup
Pontryagin duality
Pontryagin duality
Fuzzy subgroup
B-algebra
Fuzzy annihilator subgroup
Fuzzy orthogonal complement
Fuzzy least squares
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Abstract We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems. In this paper we refer to another type of annihilator which is essential in classical duality theory and extends the widely applied notion of orthogonal complement in Euclidean spaces. We find that in the natural algebraic duality of a group, a fuzzy subgroup can be recovered after taking the inverse annihilator of its annihilator. We also study the behavior of annihilators with respect to unions and intersections. Some illustrative examples of annihilators of fuzzy subgroups are shown, both with finite and infinite rank.
AbstractList We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems. In this paper we refer to another type of annihilator which is essential in classical duality theory and extends the widely applied notion of orthogonal complement in Euclidean spaces. We find that in the natural algebraic duality of a group, a fuzzy subgroup can be recovered after taking the inverse annihilator of its annihilator. We also study the behavior of annihilators with respect to unions and intersections. Some illustrative examples of annihilators of fuzzy subgroups are shown, both with finite and infinite rank.
Author Chasco, M.J.
Elorza, J.
Ardanza-Trevijano, S.
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10.1016/0165-0114(94)90175-9
10.3792/pja/1195522171
10.1016/0022-247X(81)90164-5
10.1007/BF01057937
10.3792/pja/1195522169
10.1016/0165-0114(94)90265-8
10.1016/S0165-0114(98)00428-X
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Keywords Pontryagin duality
Fuzzy subgroup
B-algebra
Fuzzy annihilator subgroup
Fuzzy orthogonal complement
Fuzzy least squares
Language English
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References Gerla (br0100) 2001; vol. 11
Cignoli, d'Ottaviano, Mundici (br0050) 2013
Imai, Iséki (br0120) 1966; 42
Mundici (br0180) 1986; 31
Sebastian, Babusundar (br0190) 1994; 67
Aslam, Thaheem (br0030) 1991; 36
Jun, Xin (br0140) 2001; 117
Jun, Xin (br0150) 2001; 117
Ajmal (br0010) 1994; 61
Di Nola, Lettieri (br0080) 1994; 53
Dey, Pal (br0070) 2014; 4
Iséki (br0130) 1966; 42
Anderson, Feil (br0020) 1988
Cignoli, Torrens (br0040) 2000; 5
Diamond (br0090) 1988; 46
Hewitt, Ross (br0110) 1979; vol. 115
Yang, Lin (br0200) 2002; 126
Das (br0060) 1981; 84
Mundici (br0170) 1986; 65
Mordeson, Buthani, Rosenfeld (br0160) 2005; vol. 182
Gerla (10.1016/j.fss.2018.11.001_br0100) 2001; vol. 11
Mundici (10.1016/j.fss.2018.11.001_br0180) 1986; 31
Imai (10.1016/j.fss.2018.11.001_br0120) 1966; 42
Cignoli (10.1016/j.fss.2018.11.001_br0040) 2000; 5
Das (10.1016/j.fss.2018.11.001_br0060) 1981; 84
Dey (10.1016/j.fss.2018.11.001_br0070) 2014; 4
Diamond (10.1016/j.fss.2018.11.001_br0090) 1988; 46
Yang (10.1016/j.fss.2018.11.001_br0200) 2002; 126
Aslam (10.1016/j.fss.2018.11.001_br0030) 1991; 36
Mundici (10.1016/j.fss.2018.11.001_br0170) 1986; 65
Sebastian (10.1016/j.fss.2018.11.001_br0190) 1994; 67
Hewitt (10.1016/j.fss.2018.11.001_br0110) 1979; vol. 115
Jun (10.1016/j.fss.2018.11.001_br0140) 2001; 117
Jun (10.1016/j.fss.2018.11.001_br0150) 2001; 117
Iséki (10.1016/j.fss.2018.11.001_br0130) 1966; 42
Di Nola (10.1016/j.fss.2018.11.001_br0080) 1994; 53
Cignoli (10.1016/j.fss.2018.11.001_br0050) 2013
Ajmal (10.1016/j.fss.2018.11.001_br0010) 1994; 61
Mordeson (10.1016/j.fss.2018.11.001_br0160) 2005; vol. 182
Anderson (10.1016/j.fss.2018.11.001_br0020) 1988
References_xml – volume: 67
  start-page: 365
  year: 1994
  end-page: 368
  ident: br0190
  article-title: Existence of fuzzy subgroups of every level-cardinality upto
  publication-title: Fuzzy Sets Syst.
  contributor:
    fullname: Babusundar
– volume: vol. 11
  year: 2001
  ident: br0100
  article-title: Fuzzy Logic: Mathematical Tools for Approximate Reasoning
  publication-title: Trends in Logic
  contributor:
    fullname: Gerla
– volume: 42
  start-page: 19
  year: 1966
  end-page: 22
  ident: br0120
  article-title: On axiom systems of propositional calculi. XIV
  publication-title: Proc. Jpn. Acad.
  contributor:
    fullname: Iséki
– volume: 42
  start-page: 26
  year: 1966
  end-page: 29
  ident: br0130
  article-title: An algebra related with a propositional calculus
  publication-title: Proc. Jpn. Acad.
  contributor:
    fullname: Iséki
– volume: 53
  start-page: 417
  year: 1994
  end-page: 432
  ident: br0080
  article-title: Perfect
  publication-title: Stud. Log.
  contributor:
    fullname: Lettieri
– year: 1988
  ident: br0020
  article-title: Lattice-Ordered Groups: An Introduction, vol. 4
  contributor:
    fullname: Feil
– volume: 117
  start-page: 471
  year: 2001
  end-page: 476
  ident: br0140
  article-title: Fuzzy prime ideals and invertible fuzzy ideals in
  publication-title: Fuzzy Sets Syst.
  contributor:
    fullname: Xin
– volume: 126
  start-page: 389
  year: 2002
  end-page: 399
  ident: br0200
  article-title: Fuzzy least-squares linear regression analysis for fuzzy input–output data
  publication-title: Fuzzy Sets Syst.
  contributor:
    fullname: Lin
– volume: 46
  start-page: 141
  year: 1988
  end-page: 157
  ident: br0090
  article-title: Fuzzy least squares
  publication-title: Inf. Sci.
  contributor:
    fullname: Diamond
– volume: vol. 182
  year: 2005
  ident: br0160
  article-title: Fuzzy Group Theory
  publication-title: Studies in Fuzziness and Soft Computing
  contributor:
    fullname: Rosenfeld
– volume: 36
  start-page: 895
  year: 1991
  end-page: 906
  ident: br0030
  article-title: On certain ideals in
  publication-title: Math. Jpn.
  contributor:
    fullname: Thaheem
– volume: 5
  start-page: 45
  year: 2000
  end-page: 65
  ident: br0040
  article-title: An algebraic analysis of product logic
  publication-title: Mult. Valued Log.
  contributor:
    fullname: Torrens
– volume: 65
  start-page: 15
  year: 1986
  end-page: 63
  ident: br0170
  article-title: Interpretation of
  publication-title: J. Funct. Anal.
  contributor:
    fullname: Mundici
– year: 2013
  ident: br0050
  article-title: Algebraic Foundations of Many-Valued Reasoning, vol. 7
  contributor:
    fullname: Mundici
– volume: 84
  start-page: 264
  year: 1981
  end-page: 269
  ident: br0060
  article-title: Fuzzy groups and level subgroups
  publication-title: J. Math. Anal. Appl.
  contributor:
    fullname: Das
– volume: 31
  start-page: 889
  year: 1986
  end-page: 894
  ident: br0180
  article-title: -algebras are categorically equivalent to bounded commutative
  publication-title: Math. Jpn.
  contributor:
    fullname: Mundici
– volume: 4
  start-page: 21
  year: 2014
  end-page: 37
  ident: br0070
  article-title: Properties of fuzzy inner product spaces
  publication-title: Int. J. Fuzzy Log. Syst.
  contributor:
    fullname: Pal
– volume: vol. 115
  year: 1979
  ident: br0110
  article-title: Abstract Harmonic Analysis I
  publication-title: Grundlehren der Mathematischen Wissenschaften
  contributor:
    fullname: Ross
– volume: 61
  start-page: 329
  year: 1994
  end-page: 339
  ident: br0010
  article-title: Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups
  publication-title: Fuzzy Sets Syst.
  contributor:
    fullname: Ajmal
– volume: 117
  start-page: 463
  year: 2001
  end-page: 469
  ident: br0150
  article-title: Involutory and invertible fuzzy
  publication-title: Fuzzy Sets Syst.
  contributor:
    fullname: Xin
– volume: 5
  start-page: 45
  year: 2000
  ident: 10.1016/j.fss.2018.11.001_br0040
  article-title: An algebraic analysis of product logic
  publication-title: Mult. Valued Log.
  contributor:
    fullname: Cignoli
– volume: 65
  start-page: 15
  issue: 1
  year: 1986
  ident: 10.1016/j.fss.2018.11.001_br0170
  article-title: Interpretation of AF C⁎-algebras in Łukasiewicz sentential calculus
  publication-title: J. Funct. Anal.
  doi: 10.1016/0022-1236(86)90015-7
  contributor:
    fullname: Mundici
– volume: 126
  start-page: 389
  issue: 3
  year: 2002
  ident: 10.1016/j.fss.2018.11.001_br0200
  article-title: Fuzzy least-squares linear regression analysis for fuzzy input–output data
  publication-title: Fuzzy Sets Syst.
  doi: 10.1016/S0165-0114(01)00066-5
  contributor:
    fullname: Yang
– volume: 4
  start-page: 21
  issue: 2
  year: 2014
  ident: 10.1016/j.fss.2018.11.001_br0070
  article-title: Properties of fuzzy inner product spaces
  publication-title: Int. J. Fuzzy Log. Syst.
  contributor:
    fullname: Dey
– volume: vol. 11
  year: 2001
  ident: 10.1016/j.fss.2018.11.001_br0100
  article-title: Fuzzy Logic: Mathematical Tools for Approximate Reasoning
  contributor:
    fullname: Gerla
– volume: 31
  start-page: 889
  year: 1986
  ident: 10.1016/j.fss.2018.11.001_br0180
  article-title: MV-algebras are categorically equivalent to bounded commutative BCK-algebras
  publication-title: Math. Jpn.
  contributor:
    fullname: Mundici
– volume: 46
  start-page: 141
  issue: 3
  year: 1988
  ident: 10.1016/j.fss.2018.11.001_br0090
  article-title: Fuzzy least squares
  publication-title: Inf. Sci.
  doi: 10.1016/0020-0255(88)90047-3
  contributor:
    fullname: Diamond
– volume: 61
  start-page: 329
  issue: 3
  year: 1994
  ident: 10.1016/j.fss.2018.11.001_br0010
  article-title: Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups
  publication-title: Fuzzy Sets Syst.
  doi: 10.1016/0165-0114(94)90175-9
  contributor:
    fullname: Ajmal
– year: 1988
  ident: 10.1016/j.fss.2018.11.001_br0020
  contributor:
    fullname: Anderson
– volume: vol. 182
  year: 2005
  ident: 10.1016/j.fss.2018.11.001_br0160
  article-title: Fuzzy Group Theory
  contributor:
    fullname: Mordeson
– volume: 42
  start-page: 26
  issue: 1
  year: 1966
  ident: 10.1016/j.fss.2018.11.001_br0130
  article-title: An algebra related with a propositional calculus
  publication-title: Proc. Jpn. Acad.
  doi: 10.3792/pja/1195522171
  contributor:
    fullname: Iséki
– volume: 84
  start-page: 264
  issue: 1
  year: 1981
  ident: 10.1016/j.fss.2018.11.001_br0060
  article-title: Fuzzy groups and level subgroups
  publication-title: J. Math. Anal. Appl.
  doi: 10.1016/0022-247X(81)90164-5
  contributor:
    fullname: Das
– volume: 53
  start-page: 417
  issue: 3
  year: 1994
  ident: 10.1016/j.fss.2018.11.001_br0080
  article-title: Perfect MV-algebras are categorically equivalent to abelian ℓ-groups
  publication-title: Stud. Log.
  doi: 10.1007/BF01057937
  contributor:
    fullname: Di Nola
– volume: 42
  start-page: 19
  issue: 6
  year: 1966
  ident: 10.1016/j.fss.2018.11.001_br0120
  article-title: On axiom systems of propositional calculi. XIV
  publication-title: Proc. Jpn. Acad.
  doi: 10.3792/pja/1195522169
  contributor:
    fullname: Imai
– year: 2013
  ident: 10.1016/j.fss.2018.11.001_br0050
  contributor:
    fullname: Cignoli
– volume: vol. 115
  year: 1979
  ident: 10.1016/j.fss.2018.11.001_br0110
  article-title: Abstract Harmonic Analysis I
  contributor:
    fullname: Hewitt
– volume: 67
  start-page: 365
  issue: 3
  year: 1994
  ident: 10.1016/j.fss.2018.11.001_br0190
  article-title: Existence of fuzzy subgroups of every level-cardinality upto ℵ0
  publication-title: Fuzzy Sets Syst.
  doi: 10.1016/0165-0114(94)90265-8
  contributor:
    fullname: Sebastian
– volume: 36
  start-page: 895
  year: 1991
  ident: 10.1016/j.fss.2018.11.001_br0030
  article-title: On certain ideals in BCK-algebras
  publication-title: Math. Jpn.
  contributor:
    fullname: Aslam
– volume: 117
  start-page: 471
  issue: 3
  year: 2001
  ident: 10.1016/j.fss.2018.11.001_br0140
  article-title: Fuzzy prime ideals and invertible fuzzy ideals in BCK-algebras
  publication-title: Fuzzy Sets Syst.
  doi: 10.1016/S0165-0114(98)00428-X
  contributor:
    fullname: Jun
– volume: 117
  start-page: 463
  issue: 3
  year: 2001
  ident: 10.1016/j.fss.2018.11.001_br0150
  article-title: Involutory and invertible fuzzy BCK-algebras
  publication-title: Fuzzy Sets Syst.
  doi: 10.1016/S0165-0114(98)00406-0
  contributor:
    fullname: Jun
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Snippet We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that...
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StartPage 122
SubjectTerms B-algebra
Fuzzy annihilator subgroup
Fuzzy least squares
Fuzzy orthogonal complement
Fuzzy subgroup
Pontryagin duality
Title The annihilator of fuzzy subgroups
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