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Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group
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Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a
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feature, such that these two features are related in a certain way, then the structure becomes a
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Hegedus, Stephen John; Moreno-Armella, Luis (2011). "The emergence of mathematical structures".
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saw structures as the root of mathematics. They first mentioned them in their "Fascicule" of
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Corry, Leo (September 1992). "Nicolas
Bourbaki and the concept of mathematical structure".
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Logical foundations of mathematics and computational complexity a gentle introduction
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An order: each number is either less than or greater than any other number.
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and expanded it into
Chapter IV of the 1957 edition. They identified three
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Its order and, independently, its metric structure induce its topology.
208:) are of special interest in many fields of mathematics. Examples are
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between sets which preserve structures (i.e., structures in the
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Discrete mathematical structures : theory and applications
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Kolman, Bernard; Busby, Robert C.; Ross, Sharon Cutler (2000).
40:
495:
Biological signal processing and computational neuroscience
115:
provided with some additional features on the set (e.g. an
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Fundamental
Structures of Algebra and Discrete Mathematics
580:(4th ed.). Upper Saddle River, NJ: Prentice Hall.
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Its algebraic structure and topology make it into a
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322:Its order and algebraic structure make it into an
614:PudlĂĄk, Pavel (2013). "Mathematical structures".
358:Equivalent definitions of mathematical structures
263:and the pair of which together make it into a
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228:In 1939, the French group with the pseudonym
216:, which preserve topological structures; and
8:
689:Mathematical structures in computer science
204:are mapped to equivalent structures in the
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636:(21 May 1993). "Mathematical Structures".
220:, which preserve differential structures.
134:A partial list of possible structures are
91:Learn how and when to remove this message
684:(provides a model theoretic definition.)
599:. Australia: Thomson/Course Technology.
54:This article includes a list of general
444:
442:
384:
212:, which preserve algebraic structures;
240:: algebraic, topological, and order.
7:
418:Christiansen, Jacob Stordal (2015).
549:Educational Studies in Mathematics
532:. Hoboken: John Wiley & Sons.
315:There are interfaces among these:
296:A geometry: it is equipped with a
60:it lacks sufficient corresponding
25:
618:. Cham: Springer. pp. 2â24.
307:A topology: there is a notion of
274:of the real line have a specific
252:has several standard structures:
30:For the notion of "structure" in
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578:Discrete mathematical structures
45:
595:Malik, D.S.; Sen, M.K. (2004).
289:A metric: there is a notion of
278:, which can be extended to the
36:Structure (mathematical logic)
27:Additional mathematical object
1:
2397:History of mathematical logic
650:10.1126/science.260.5111.1170
2322:Primitive recursive function
392:Saunders, Mac Lane (1996).
2489:
1386:SchröderâBernstein theorem
1113:Monadic predicate calculus
772:Foundations of mathematics
492:Wells, Richard B. (2010).
394:"Structure in Mathematics"
363:Intuitionistic type theory
29:
2432:
2419:Philosophy of mathematics
2368:Automated theorem proving
1539:
1493:Von NeumannâBernaysâGödel
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561:10.1007/s10649-010-9297-7
420:"Mathematical structures"
244:Example: the real numbers
524:Foldes, Stephan (1994).
2463:Mathematical structures
2069:Self-verifying theories
1890:Tarski's axiomatization
841:Tarski's undefinability
836:incompleteness theorems
401:Philosoph1A Mathemat1Ca
180:differential structures
75:more precise citations.
18:Mathematical structures
2443:Mathematics portal
2054:Proof of impossibility
1702:propositional variable
1012:Propositional calculus
2312:Kolmogorov complexity
2265:Computably enumerable
2165:Model complete theory
1957:Principia Mathematica
1017:Propositional formula
846:BanachâTarski paradox
176:equivalence relations
2260:ChurchâTuring thesis
2247:Computability theory
1456:continuum hypothesis
974:Square of opposition
832:Gödel's completeness
140:algebraic structures
2414:Mathematical object
2305:P versus NP problem
2270:Computable function
2064:Reverse mathematics
1990:Logical consequence
1867:primitive recursive
1862:elementary function
1635:Free/bound variable
1488:TarskiâGrothendieck
1007:Logical connectives
937:Logical equivalence
787:Logical consequence
644:(5111): 1170â1173.
373:Space (mathematics)
368:Mathematical object
2212:Transfer principle
2175:Semantics of logic
2160:Categorical theory
2136:Non-standard model
1650:Logical connective
777:Information theory
726:Mathematical logic
501:. pp. 296â335
463:10.1007/bf00414286
348:Abstract structure
32:mathematical logic
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2382:Abstract category
2185:Theories of truth
1995:Rule of inference
1985:Natural deduction
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1027:Many-valued logic
1002:Boolean functions
885:Russell's paradox
860:diagonal argument
757:First-order logic
606:978-0-619-21558-3
587:978-0-13-083143-9
335:topological group
238:mother structures
192:topological group
156:metric structures
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100:
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16:(Redirected from
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2392:History of logic
2387:Category of sets
2280:Decision problem
2059:Ordinal analysis
2000:Sequent calculus
1898:Boolean algebras
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1783:logical/constant
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230:Nicolas Bourbaki
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71:this article by
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997:Boolean algebra
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947:Equiconsistency
912:Classical logic
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858: and
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816:Theorems (
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2434:
2232:Ultraproduct
2079:Model theory
2044:Independence
1980:Formal proof
1972:Proof theory
1955:
1928:
1885:real numbers
1857:second-order
1768:Substitution
1645:Metalanguage
1586:conservative
1559:Axiom schema
1503:Constructive
1473:MorseâKelley
1439:Set theories
1418:Aleph number
1411:inaccessible
1317:Grothendieck
1201:intersection
1088:Higher-order
1076:Second-order
1022:Truth tables
979:Venn diagram
762:Formal proof
683:
676:
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637:
634:Senechal, M.
615:
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494:
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430:. Retrieved
427:maths.lth.se
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333:, a type of
314:
250:real numbers
247:
237:
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102:
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78:
59:
2468:Type theory
2342:Type theory
2290:undecidable
2222:Truth value
2109:equivalence
1788:non-logical
1401:Enumeration
1391:Isomorphism
1338:cardinality
1322:Von Neumann
1287:Ultrafilter
1252:Uncountable
1186:equivalence
1103:Quantifiers
1093:Fixed-point
1062:First-order
942:Consistency
927:Proposition
904:Traditional
875:Lindström's
865:Compactness
807:Type theory
752:Cardinality
673:"Structure"
353:Isomorphism
270:A measure:
248:The set of
105:mathematics
73:introducing
2473:Set theory
2457:Categories
2153:elementary
1846:arithmetic
1714:Quantifier
1692:functional
1564:Expression
1282:Transitive
1226:identities
1211:complement
1144:hereditary
1127:Set theory
678:PlanetMath
432:2019-12-09
379:References
184:categories
160:geometries
152:topologies
81:April 2016
56:references
2424:Supertask
2327:Recursion
2285:decidable
2119:saturated
2097:of models
2020:deductive
2015:axiomatic
1935:Hilbert's
1922:Euclidean
1903:canonical
1826:axiomatic
1758:Signature
1687:Predicate
1576:Extension
1498:Ackermann
1423:Operation
1302:Universal
1292:Recursive
1267:Singleton
1262:Inhabited
1247:Countable
1237:Types of
1221:power set
1191:partition
1108:Predicate
1054:Predicate
969:Syllogism
959:Soundness
932:Inference
922:Tautology
824:paradoxes
691:(journal)
569:119981368
407:(3): 176.
331:Lie group
309:open sets
272:intervals
150:, etc.),
117:operation
109:structure
2409:Logicism
2402:timeline
2378:Concrete
2237:Validity
2207:T-schema
2200:Kripke's
2195:Tarski's
2190:semantic
2180:Strength
2129:submodel
2124:spectrum
2092:function
1940:Tarski's
1929:Elements
1916:geometry
1872:Robinson
1793:variable
1778:function
1751:spectrum
1741:Sentence
1697:variable
1640:Language
1593:Relation
1554:Automata
1544:Alphabet
1528:language
1382:-jection
1360:codomain
1346:Function
1307:Universe
1277:Infinite
1181:Relation
964:Validity
954:Argument
852:theorem,
658:17806355
479:16981077
471:20117057
451:Synthese
342:See also
291:distance
206:codomain
198:Mappings
136:measures
129:topology
121:relation
2351:Related
2148:Diagram
2046: (
2025:Hilbert
2010:Systems
2005:Theorem
1883:of the
1828:systems
1608:Formula
1603:Grammar
1519: (
1463:General
1176:Forcing
1161:Element
1081:Monadic
856:paradox
797:Theorem
733:General
638:Science
505:7 April
300:and is
284:subsets
224:History
69:improve
2114:finite
1877:Skolem
1830:
1805:Theory
1773:Symbol
1763:String
1746:atomic
1623:ground
1618:closed
1613:atomic
1569:ground
1532:syntax
1428:binary
1355:domain
1272:Finite
1037:finite
895:Logics
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802:Theory
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298:metric
276:length
202:domain
182:, and
172:events
168:graphs
164:orders
148:fields
144:groups
125:metric
58:, but
34:, see
2104:Model
1852:Peano
1709:Proof
1549:Arity
1478:Naive
1365:image
1297:Fuzzy
1257:Empty
1206:union
1151:Class
792:Model
782:Lemma
740:Axiom
565:S2CID
499:(PDF)
475:S2CID
467:JSTOR
423:(PDF)
397:(PDF)
265:field
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127:, or
111:is a
2227:Type
2030:list
1834:list
1811:list
1800:Term
1734:rank
1628:open
1522:list
1334:Maps
1239:sets
1098:Free
1068:list
818:list
745:list
654:PMID
620:ISBN
601:ISBN
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534:ISBN
507:2016
302:flat
107:, a
1914:of
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