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Cryptography lies at the heart of most technologies deployed today for secure communications. At the same time, mathematics lies at the heart of cryptography, as cryptographic constructions are based on algebraic scenarios ruled by group or number theoretical laws. Understanding the involved algebraic structures is, thus, essential to design robust cryptographic schemes. This Special Issue is concerned with the interplay between group theory, symmetry and cryptography. The book highlights four exciting areas of research in which these fields intertwine: postquantum cryptography, coding theory, computational group theory and symmetric cryptography. The articles presented demonstrate the relevance of rigorously analyzing the computational hardness of the mathematical problems used as a base for cryptographic constructions. For instance, decoding problems related to algebraic codes and rewriting problems in nonabelian groups are explored with cryptographic applications in mind. New results on the algebraic properties or symmetric cryptographic tools are also presented, moving ahead in the understanding of their security properties. In addition, postquantum constructions for digital signatures and key exchange are explored in this Special Issue, exemplifying how (and how not) group theory may be used for developing robust cryptographic tools to withstand quantum attacks.
cryptography  noncommutative cryptography  oneway functions  NPCompleteness  key agreement protocol  group theory  symmetry  Engel words  alternating group  WalnutDSA  digital signatures  postquantum cryptography  cryptanalysis  braid groups  algorithms in groups  groupbased cryptography  Reed–Solomon codes  key equation  Berlekamp–Massey algorithm  Sugiyama et al. algorithm  euclidean algorithm  numerical semigroup  Weierstrass semigroup  semigroup ideal  errorcorrecting code  algebraicgeometry code  lightweight cryptography  permutation group  block cipher  generalized selfshrinking generator  tmodified selfshrinking generator  pseudorandom number generator  statistical randomness tests  cryptography  pseudorandom permutation  block cipher  ideal cipher model  beyond birthday bound  provable security  group key establishment  group theory  provable security  protocol compiler
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Following the tremendous reception of our first volume on topological groups called ""Topological Groups: Yesterday, Today, and Tomorrow"", we now present our second volume. Like the first volume, this collection contains articles by some of the best scholars in the world on topological groups. A feature of the first volume was surveys, and we continue that tradition in this volume with three new surveys. These surveys are of interest not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. Wellknown researchers who contributed to this volume include Taras Banakh, Michael Megrelishvili, Sidney A. Morris, Saharon Shelah, George A. Willis, O'lga V. Sipacheva, and Stephen Wagner.
descriptive set theory  polish group topologies  rightangled Artin groups  topological group  paratopological group  topological semigroup  absolutely closed topological group  topological group of compact exponent  locally compact group  endomorphism  scale  tree  Neretin’s group  Thompson’s group  padic Lie group  large set in a group  vast set  syndetic set  thick set  piecewise syndetic set  Boolean topological group  arrow ultrafilter  Ramsey ultrafilter  coarse structure  coarse space  ballean  varieties of coarse spaces  space of closed subgroups  Chabauty topology  Vietoris topology  Bourbaki uniformity  Gromov’s compactification  group representation  matrix coefficient  semigroup compactification  tame function  dynamical system  continuous inverse algebra  character  maximal ideal  fixed point algebra  extension  pseudocompact  strongly pseudocompact  pcompact  selectively sequentially pseudocompact  pseudo?bounded  nontrivial convergent sequence  separable  free precompact Boolean group  reflexive group  maximal space  ultrafilter space  topological group  Lie group  compact topological semigroup  Hspace  mapping cylinder  fibre bundle  separable topological group  subgroup  product  isomorphic embedding  quotient group  free topological group
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This book is a collection of 12 innovative research papers in the field of hypercompositional algebra, 7 of them being more theoretically oriented, with the other 5 presenting strong applicative aspects in engineering, control theory, artificial intelligence, and graph theory. Hypercompositional algebra is now a wellestablished branch of abstract algebra dealing with structures endowed with multivalued operations, also called hyperoperations, having a set as the result of the interrelation between two elements of the support set. The theoretical papers in this book are principally related to three main topics: (semi)hypergroups, hyperfields, and BCKalgebra. Heidari and Cristea present a natural generalization of breakable semigroups, defining the breakable semihypergroups where every nonempty subset is a subsemihypergroup. Using the fundamental relation ? on a hypergroup, some new properties of the
breakable semigroup  semihypergroup  hyperideal  semisymmetry  intuitionistic fuzzy soft hyper BCK ideal  intuitionistic fuzzy soft weak hyper BCK ideal  intuitionistic fuzzy soft sweak hyper BCKideal  intuitionistic fuzzy soft strong hyper BCKideal  mpolar fuzzy hypergraphs  mpolar fuzzy equivalence relation  level hypergraphs  granular computing  application  qrung picture fuzzy graphs  edge regular  perfect edge regular  square qrung picture fuzzy graphs  qrung picture fuzzy line graphs  ego networks  clustering protocols  quasiautomaton  quasimultiautomaton  semihypergroup  UWSN  timevarying artificial neuron  ordered group  transposition hypergroup  linear differential operator  multisets  functions on multiset  selection operation  submultiset  Hvstructures  Hvring  fundamental equivalence relation  Hvideal  multiset  fuzzy multiset  fuzzy multiHvideal  hypergroup  hyperring  hyperfield  (hyper)homography  singlepower cyclic hypergroup  invertible subhypergroup  lower approximation  upper approximation  rough set  hypergroup  semihypergroup  1hypergroup  fundamental relation  height  lower BCKsemilattice  relative annihilator  semiprime closure operation  minimal prime decomposition  minimal prime factor
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This book contains 37 papers by 73 renowned experts from 13 countries around the world, on following topics: neutrosophic set; neutrosophic rings; neutrosophic quadruple rings; idempotents; neutrosophic extended triplet group; hypergroup; semihypergroup; neutrosophic extended triplet group; neutrosophic extended triplet semihypergroup and hypergroup; neutrosophic offset; uninorm; neutrosophic offuninorm and offnorm; neutrosophic offconorm; implicator; prospector; nperson cooperative game; ordinary singlevalued neutrosophic (co)topology; ordinary singlevalued neutrosophic subspace; ?level; ordinary singlevalued neutrosophic neighborhood system; ordinary singlevalued neutrosophic base and subbase; fuzzy numbers; neutrosophic numbers; neutrosophic symmetric scenarios; performance indicators; financial assets; neutrosophic extended triplet group; neutrosophic quadruple numbers; refined neutrosophic numbers; refined neutrosophic quadruple numbers; multigranulation neutrosophic rough set; nondual; two universes; multiattribute group decision making; nonstandard analysis; extended nonstandard analysis; monad; binad; left monad closed to the right; right monad closed to the left; pierced binad; unpierced binad; nonstandard neutrosophic mobinad set; neutrosophic topology; nonstandard neutrosophic topology; visual tracking; neutrosophic weight; objectness; weighted multiple instance learning; neutrosophic triangular norms; residuated lattices; representable neutrosophic tnorms; De Morgan neutrosophic triples; neutrosophic residual implications; infinitely ?distributive; probabilistic neutrosophic hesitant fuzzy set; decisionmaking; Choquet integral; emarketing; Internet of Things; neutrosophic set; multicriteria decision making techniques; uncertainty modeling; neutrosophic goal programming approach; shale gas water management system.
singlevalued neutrosophic soft number and its operations  SVN soft weighted arithmetic averaging operator  SVN soft weighted geometric averaging operator  decisionmaking  optimization solution  sampling plan  producer’s risk’  consumer’s risk  sample size  decision making  neutrosophic set  Qneutrosophic set  Qneutrosophic soft set  soft set  Neutrosophic compound orthogonal neural network  Neutrosophic number  Neutrosophic function  Function approximation  triangular neutrosophic cubic fuzzy number  Einstein tnorm  arithmetic averaging operator  Multiattribute decision making  numerical application  neutrosophic statistical interval method  classical statistics  fuzzy logic  producer’s risk  consumer’s risk  neutrosophic cubic weighted geometric operator (NCWG)  neutrosophic cubic ordered weighted geometric operator (NCOWG)  neutrosophic cubic Einstein weighted geometric operator (NCEWG)  neutrosophic cubic Einstein ordered weighted geometric operator (NCEOWG)  multiattribute decision making  singlevalued neutrosophic set  aggregation  TOPSIS  singlevalued neutrosophic linguistic set  distance measure  combined weighted average  MAGDM  lowcarbon supplier selection  weighted geometric operator  weighted average operator  neutrosophic cubic sets  multiattribute decisionmaking (MADM)  neutrosophic cubic hybrid weighted arithmetic and geometric aggregation operator (NCHWAGA)  neutrosophic sets  cubic sets  soft sets  neutrosophic cubic soft sets  neutrosophic cubic soft expert system  multicriteria decisionmaking  neutrosophic extended triplet group  generalized neutrosophic extended triplet group  quasicompletely regular semigroup  clifford semigroup  prostate cancer  neutrosophic statistics  dietary fat level  neutrosophic regression  neutrosophic correlation  single valued neutrosophic sets  logarithmic operational laws  logarithmic aggregation operators  MCGDM problems  complex neutrosophic set  complex neutrosophic soft expert set  fuzzy parameterized single valued neutrosophic soft expert set  single valued neutrosophic set  soft expert set  decision making  membership function  neutrosophic set  neutrosophic soft rough  relations  rough set approximation  soft set  neutrosophic time series  triangular neutrosophic number  neutrosophic logical relationship  neutrosophic logical relationship groups  smart port  simplified neutrosophic set  singlevalued neutrosophic set  exponential similarity measure  port evaluation  nonstandard analysis  extended nonstandard analysis  open and closed monads to the left/right  pierced and unpierced binads  MoBiNad set  infinitesimals  infinities  nonstandard reals  standard reals  nonstandard neutrosophic lattices of first type (as poset) and second type (as algebraic structure), nonstandard neutrosophic logic  extended nonstandard neutrosophic logic  nonstandard arithmetic operations  nonstandard unit interval  nonstandard neutrosophic infimum  nonstandard neutrosophic supremum  intuitionistic fuzzy parameters  uncertainty modeling  neutrosophic goal programming approach  shale gas water management system  emarketing  Internet of Things  neutrsophic set  multicriteria decision making techniques  probabilistic neutrosophic hesitant fuzzy set (PNHFS)  decisionmaking  Choquet integral  neutrosophic extended triplet group  neutrosophic quadruple numbers  neutrosophic set  BNHWA aggregation operator  BNHOWA aggregation operator  BNHHA aggregation operator  score function  accuracy function  certainty function  group decision making  nonstandard analysis  extended nonstandard analysis  monad  binad  left monad closed to the right  right monad closed to the left  pierced binad  unpierced binad  nonstandard neutrosophic mobinad set  neutrosophic topology  nonstandard neutrosophic topology  neutrosophic sets  neutrosophic triangular norms  residuated lattices  representable neutrosophic tnorms  De Morgan neutrosophic triples  neutrosophic residual implications  infinitely ?distributive  visual tracking  neutrosophic weight  objectness  weighted multiple instance learning  supply chain sustainability metrics  plithogeny  aggregation operations  neutrosophic set  quality function deployment  multigranulation neutrosophic rough set  nondual  two universes  multiattribute group decision making  neutrosophic extended triplet group  neutrosophic quadruple numbers  refined neutrosophic numbers  refined neutrosophic quadruple numbers  neutrosophic set  fuzzy numbers  neutrosophic numbers  neutrosophic symmetric scenarios  performance indicators  financial assets  simplified neutrosophic hesitant fuzzy set  multiattribute decisionmaking  maximizing deviation  TOPSIS  single valued neutrosophic set  covering  symmetric relation  graph representation  matrix representation  paper defect diagnosis  ordinary single valued neutrosophic (co)topology  ordinary single valued neutrosophic subspace  ?level  ordinary single valued neutrosophic neighborhood system  ordinary single valued neutrosophic base  ordinary single valued neutrosophic subbase  neutrosophic offset  uninorm  neutrosophic offuninorm  neutrosophic offnorm  neutrosophic offconorm  implicator  prospector  nperson cooperative game  hypergroup  semihypergroup  neutrosophic extended triplet group  neutrosophic extended triplet semihypergroup (NETsemihypergroup)  NEThypergroup  neutrosophic rings  neutrosophic quadruple rings  idempotents  neutrosophic extended triplet group  neutrosophic set
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Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (,
generalized aggregation operators  interval neutrosophic set (INS)  multiattribute decision making (MADM)  Choquet integral  fuzzy measure  clustering algorithm  neutrosophic association rule  data mining  neutrosophic sets  big data  analytic hierarchy process (AHP)  SWOT analysis  multicriteria decisionmaking (MCDM) techniques  neutrosophic set theory  neutrosophic clustering  image segmentation  neutrosophic cmeans clustering  region growing  dermoscopy  skin cancer  neutosophic extended triplet subgroups  neutrosophic triplet cosets  neutrosophic triplet normal subgroups  neutrosophic triplet quotient groups  shopping mall  photovoltaic plan  decisionmaking trial and evaluation laboratory (DEMATEL)  intervalvalued neutrosophic set  extended ELECTRE III  symmetry  single valued neutrosophic set (SVNS)  neutrosophic multiset (NM)  single valued neutrosophic multiset (SVNM)  cosine measure  multiple attribute decisionmaking  LNGPBM operator  LNGWPBM operator  Linguistic neutrosophic sets  generalized partitioned Bonferroni mean operator  multiple attribute group decisionmaking (MAGDM)  pseudoBCI algebra  hesitant fuzzy set  neutrosophic set  filter  action learning  school administrator  SVM  neutrosophic classification  neutrosophic set  soft set  totally dependentneutrosophic set  totally dependentneutrosophic soft set  generalized De Morgan algebra  complex neutrosophic set  complex neutrosophic graph  fuzzy graph  matrix representation  neutrosophic triplet groups  semigroup  semineutrosophic triplets  classical group of neutrosophic triplets  Ssemigroup of neutrosophic triplets  pseudo primitive elements  neutrosophic sets (NSs)  interval neutrosophic numbers (INNs)  exponential operational laws of interval neutrosophic numbers  interval neutrosophic weighted exponential aggregation (INWEA) operator  multiple attribute decision making (MADM)  typhoon disaster evaluation  simplified neutrosophic linguistic numbers  cloud model  Maclaurin symmetric mean  multicriteria decisionmaking  neutrosophy  DSmT  decisionmaking algorithms  robotic dexterous hands  grasping configurations  grasp type  generalized group  neutrosophic triplet set  neutrosophic triplet group  group  neutrosophic cubic set  neutrosophic cubic graphs  applications of neutrosophic cubic graphs  singlevalued neutrosophic multisets  medical diagnosis  probabilistic rough sets over two universes  threeway decisions  similarity measures  neutrosophic cubic set  decisionmaking  soft sets  support soft sets  interval valued neutrosophic support soft sets  sustainable supplier selection problems (SSSPs)  analytic network process  interdependency of criteria  TOPSIS  neutrosophic set  2inglevalued neutrosophic set  Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)  integrated weight  maximizing deviation  multiattribute decisionmaking (MADM)  neutrosophic triplet set (NTS)  partial metric spaces (PMS)  fixed point theory (FPT)  neutrosophic triplet  quasi neutrosophic triplet loop  quasi neutrosophic triplet group  BEalgebra  CIalgebra  fuzzy time series  forecasting  twofactor fuzzy logical relationship  multivalued neutrosophic set  Hamming distance  neutrosophic set  prioritized operator  Muirhead mean  multicriteria decisionmaking  aggregation operators  dual aggregation operators  neutrosophic triplet group (NTG)  NTsubgroup  homomorphism theorem  weak commutative neutrosophic triplet group  neutrosophic rough set  MGNRS  dual domains  inclusion relation  decisionmaking  neutromonomorphism  neutroepimorphism  neutroautomorphism  fundamental neutrohomomorphism theorem  first neutroisomorphism theorem  and second neutroisomorphism theorem  linear and nonlinear neutrosophic number  deneutrosophication methods  neutrosophic set  bipolar fuzzy set  neutrosophic bipolar fuzzy set  neutrosophic bipolar fuzzy weighted averaging operator  similarity measure  algorithm  multiple attribute decision making problem  neutrosophic duplets  semigroup  neutrosophic triplet groups  neutrosophic set  fault diagnosis  normal distribution  defuzzification  simplified neutrosophic weighted averaging operator  (commutative) ideal  generalized neutrosophic set  generalized neutrosophic ideal  commutative generalized neutrosophic ideal  linguistic neutrosophic sets  multicriteria group decisionmaking  power aggregation operator  extended TOPSIS method  probabilistic singlevalued (interval) neutrosophic hesitant fuzzy set  multiattribute decision making  aggregation operator  quasigroup  loop  BCIalgebra  BolMoufang  quasi neutrosophic loops  Fenyves identities  Gmetric  neutrosophic Gmetric  neutrosophic sets  clustering  neutrosophic big data  neutrosophic logic  aggregation operator  complement  intersection  membership  neutrosophic soft set  NC power dual MM operator (NCPDMM) operator  NCPMM operator  MADM  MM operator  Neutrosophic cubic sets  PA operator  interval neutrosophic sets  Bonferroni mean  power operator  multiattribute decision making (MADM)  multiple attribute group decision making (MAGDM)  2tuple linguistic neutrosophic sets (2TLNSs)  TODIM model  2TLNNs TODIM method  construction project  MCGDM problems  triangular fuzzy neutrosophic sets (TFNSs)  VIKOR model  TFNNs VIKOR method  potential evaluation  emerging technology commercialization  Qlinguistic neutrosophic variable set  vector similarity measure  cosine measure  Dice measure  Jaccard measure  decision making  inclusion relation  neutrosophic rough set  multiattribute group decisionmaking (MAGDM)  multigranulation neutrosophic rough set (MNRS)  two universes  single valued trapezoidal neutrosophic number  multicriteria group decision making  possibility degree  power aggregation operators  LAsemihypergroups  neutrosophic triplet set  neutrohomomorphism  algorithm  decision making  expert set  generalized neutrosophic set  neutrosophic sets  Qneutrosophic  soft sets  simplified neutrosophic sets (SNSs)  interval number  dependent degree  multicriteria group decisionmaking (MCGDM)  computability  oracle Turing machines  neutrosophic sets  neutrosophic logic  recursive enumerability  oracle computation  criterion functions  neutrosophic computation  neutrosophic logic  quantum computation  computation  logic
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Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (,
generalized aggregation operators  interval neutrosophic set (INS)  multiattribute decision making (MADM)  Choquet integral  fuzzy measure  clustering algorithm  neutrosophic association rule  data mining  neutrosophic sets  big data  analytic hierarchy process (AHP)  SWOT analysis  multicriteria decisionmaking (MCDM) techniques  neutrosophic set theory  neutrosophic clustering  image segmentation  neutrosophic cmeans clustering  region growing  dermoscopy  skin cancer  neutosophic extended triplet subgroups  neutrosophic triplet cosets  neutrosophic triplet normal subgroups  neutrosophic triplet quotient groups  shopping mall  photovoltaic plan  decisionmaking trial and evaluation laboratory (DEMATEL)  intervalvalued neutrosophic set  extended ELECTRE III  symmetry  single valued neutrosophic set (SVNS)  neutrosophic multiset (NM)  single valued neutrosophic multiset (SVNM)  cosine measure  multiple attribute decisionmaking  LNGPBM operator  LNGWPBM operator  Linguistic neutrosophic sets  generalized partitioned Bonferroni mean operator  multiple attribute group decisionmaking (MAGDM)  pseudoBCI algebra  hesitant fuzzy set  neutrosophic set  filter  action learning  school administrator  SVM  neutrosophic classification  neutrosophic set  soft set  totally dependentneutrosophic set  totally dependentneutrosophic soft set  generalized De Morgan algebra  complex neutrosophic set  complex neutrosophic graph  fuzzy graph  matrix representation  neutrosophic triplet groups  semigroup  semineutrosophic triplets  classical group of neutrosophic triplets  Ssemigroup of neutrosophic triplets  pseudo primitive elements  neutrosophic sets (NSs)  interval neutrosophic numbers (INNs)  exponential operational laws of interval neutrosophic numbers  interval neutrosophic weighted exponential aggregation (INWEA) operator  multiple attribute decision making (MADM)  typhoon disaster evaluation  simplified neutrosophic linguistic numbers  cloud model  Maclaurin symmetric mean  multicriteria decisionmaking  neutrosophy  DSmT  decisionmaking algorithms  robotic dexterous hands  grasping configurations  grasp type  generalized group  neutrosophic triplet set  neutrosophic triplet group  group  neutrosophic cubic set  neutrosophic cubic graphs  applications of neutrosophic cubic graphs  singlevalued neutrosophic multisets  medical diagnosis  probabilistic rough sets over two universes  threeway decisions  similarity measures  neutrosophic cubic set  decisionmaking  soft sets  support soft sets  interval valued neutrosophic support soft sets  sustainable supplier selection problems (SSSPs)  analytic network process  interdependency of criteria  TOPSIS  neutrosophic set  2inglevalued neutrosophic set  Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)  integrated weight  maximizing deviation  multiattribute decisionmaking (MADM)  neutrosophic triplet set (NTS)  partial metric spaces (PMS)  fixed point theory (FPT)  neutrosophic triplet  quasi neutrosophic triplet loop  quasi neutrosophic triplet group  BEalgebra  CIalgebra  fuzzy time series  forecasting  twofactor fuzzy logical relationship  multivalued neutrosophic set  Hamming distance  neutrosophic set  prioritized operator  Muirhead mean  multicriteria decisionmaking  aggregation operators  dual aggregation operators  neutrosophic triplet group (NTG)  NTsubgroup  homomorphism theorem  weak commutative neutrosophic triplet group  neutrosophic rough set  MGNRS  dual domains  inclusion relation  decisionmaking  neutromonomorphism  neutroepimorphism  neutroautomorphism  fundamental neutrohomomorphism theorem  first neutroisomorphism theorem  and second neutroisomorphism theorem  linear and nonlinear neutrosophic number  deneutrosophication methods  neutrosophic set  bipolar fuzzy set  neutrosophic bipolar fuzzy set  neutrosophic bipolar fuzzy weighted averaging operator  similarity measure  algorithm  multiple attribute decision making problem  neutrosophic duplets  semigroup  neutrosophic triplet groups  neutrosophic set  fault diagnosis  normal distribution  defuzzification  simplified neutrosophic weighted averaging operator  (commutative) ideal  generalized neutrosophic set  generalized neutrosophic ideal  commutative generalized neutrosophic ideal  linguistic neutrosophic sets  multicriteria group decisionmaking  power aggregation operator  extended TOPSIS method  probabilistic singlevalued (interval) neutrosophic hesitant fuzzy set  multiattribute decision making  aggregation operator  quasigroup  loop  BCIalgebra  BolMoufang  quasi neutrosophic loops  Fenyves identities  Gmetric  neutrosophic Gmetric  neutrosophic sets  clustering  neutrosophic big data  neutrosophic logic  aggregation operator  complement  intersection  membership  neutrosophic soft set  NC power dual MM operator (NCPDMM) operator  NCPMM operator  MADM  MM operator  Neutrosophic cubic sets  PA operator  interval neutrosophic sets  Bonferroni mean  power operator  multiattribute decision making (MADM)  multiple attribute group decision making (MAGDM)  2tuple linguistic neutrosophic sets (2TLNSs)  TODIM model  2TLNNs TODIM method  construction project  MCGDM problems  triangular fuzzy neutrosophic sets (TFNSs)  VIKOR model  TFNNs VIKOR method  potential evaluation  emerging technology commercialization  Qlinguistic neutrosophic variable set  vector similarity measure  cosine measure  Dice measure  Jaccard measure  decision making  inclusion relation  neutrosophic rough set  multiattribute group decisionmaking (MAGDM)  multigranulation neutrosophic rough set (MNRS)  two universes  single valued trapezoidal neutrosophic number  multicriteria group decision making  possibility degree  power aggregation operators  LAsemihypergroups  neutrosophic triplet set  neutrohomomorphism  algorithm  decision making  expert set  generalized neutrosophic set  neutrosophic sets  Qneutrosophic  soft sets  simplified neutrosophic sets (SNSs)  interval number  dependent degree  multicriteria group decisionmaking (MCGDM)  computability  oracle Turing machines  neutrosophic sets  neutrosophic logic  recursive enumerability  oracle computation  criterion functions  neutrosophic computation  neutrosophic logic  quantum computation  computation  logic  n/a
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