36:
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699:
862:
gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward LöwenheimâSkolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
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More generally, any first-order theory with an infinite model has non-isomorphic, elementarily equivalent models, which can be obtained via the
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465:, which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model.
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to be an elementary substructure. It can be useful for constructing an elementary substructure of a large structure.
694:{\displaystyle N\models \varphi (a_{1},\dots ,a_{n}){\text{ if and only if }}M\models \varphi (a_{1},\dots ,a_{n}).}
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with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense
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For example, consider the language with one binary relation symbol '<'. The model
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1973:
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1378:, Graduate Texts in Mathematics, New York âą Heidelberg âą Berlin: Springer Verlag,
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1329:, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier,
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1968:
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is complete if and only if any two of its models are elementarily equivalent.
442:. This is sufficient to ensure elementary equivalence, because the theory of
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1953:
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if every first-order sentence (formula without free variables) over
333:. One can prove that two structures are elementarily equivalent with the
703:
This definition first appears in Tarski, Vaught (1957). It follows that
2701:
1493:
1269:
Elementary embeddings are the most important maps in model theory. In
1391:
1277:(the universe of set theory) play an important role in the theory of
2245:
1591:
1436:
1395:
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together with a new constant symbol for every element of
29:
879:) is a necessary and sufficient condition for a substructure
1217:
1175:
1048:
990:
971:
838:
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155:, one often needs a stronger condition. In this case
1303:
The use of elementary substructures in combinatorics
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2212:
2135:
2029:
1933:
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1371:
1223:
1181:
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996:
977:
844:
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730:can be interpreted as structures in the signature
693:
469:Elementary substructures and elementary extensions
57:but its sources remain unclear because it lacks
1266:, and its image is an elementary substructure.
340:Elementary embeddings are used in the study of
1407:
914:if and only if for every first-order formula
8:
1309:, vol. 136, issues 1--3, 1994, pp.243--252.
289:is elementary if and only if it passes the
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1828:
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1174:
1047:
989:
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837:
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611:
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88:Learn how and when to remove this message
1273:, elementary embeddings whose domain is
404:are elementarily equivalent, one writes
274:) is an elementary substructure of
1294:
7:
446:is complete, as can be shown by the
430:with its usual order and the model
991:
352:Elementarily equivalent structures
215:if and only if it is true in
25:
910:is an elementary substructure of
787:is an elementary substructure of
751:is an elementary substructure of
223:is an elementary substructure of
3133:
1262:Every elementary embedding is a
1121:such that for every first-order
444:unbounded dense linear orderings
34:
771:are elementarily equivalent as
457:. Thus, for example, there are
685:
653:
636:
604:
504:such that for all first-order
1:
3094:History of mathematical logic
380:if and only if it is true in
3019:Primitive recursive function
1031:), then there is an element
894:be a structure of signature
325:also has a solution in
293:: every first-order formula
584:if and only if it holds in
496:are structures of the same
364:of the same signature
3186:
2083:SchröderâBernstein theorem
1810:Monadic predicate calculus
1469:Foundations of mathematics
1353:Cambridge University Press
641: if and only if
3160:Equivalence (mathematics)
3129:
3116:Philosophy of mathematics
3065:Automated theorem proving
2236:
2190:Von NeumannâBernaysâGödel
1831:
335:EhrenfeuchtâFraĂŻssĂ© games
130:if they satisfy the same
1370:Monk, J. Donald (1976),
1224:{\displaystyle \models }
1182:{\displaystyle \models }
1055:{\displaystyle \models }
997:{\displaystyle \exists }
978:{\displaystyle \models }
860:LöwenheimâSkolem theorem
845:{\displaystyle \succeq }
807:{\displaystyle \preceq }
455:LöwenheimâSkolem theorem
43:This article includes a
2766:Self-verifying theories
2587:Tarski's axiomatization
1538:Tarski's undefinability
1533:incompleteness theorems
877:TarskiâVaught criterion
478:elementary substructure
396:first-order theory. If
370:elementarily equivalent
329:when evaluated in
321:that has a solution in
161:elementary substructure
128:elementarily equivalent
72:more precise citations.
27:Concept in model theory
18:Elementarily equivalent
3140:Mathematics portal
2751:Proof of impossibility
2399:propositional variable
1709:Propositional calculus
1349:A shorter model theory
1225:
1183:
1105:of the same signature
1056:
998:
979:
846:
808:
695:
528:) with free variables
3009:Kolmogorov complexity
2962:Computably enumerable
2862:Model complete theory
2654:Principia Mathematica
1714:Propositional formula
1543:BanachâTarski paradox
1305:(1993). Appearing in
1226:
1184:
1089:Elementary embeddings
1057:
999:
980:
847:
809:
759:is a substructure of
718:is a substructure of
707:is a substructure of
696:
317:) with parameters in
167:if every first-order
2957:ChurchâTuring thesis
2944:Computability theory
2153:continuum hypothesis
1671:Square of opposition
1529:Gödel's completeness
1307:Discrete Mathematics
1215:
1173:
1095:elementary embedding
1046:
988:
969:
836:
823:elementary extension
798:
592:
256:elementary embedding
233:elementary extension
3111:Mathematical object
3002:P versus NP problem
2967:Computable function
2761:Reverse mathematics
2687:Logical consequence
2564:primitive recursive
2559:elementary function
2332:Free/bound variable
2185:TarskiâGrothendieck
1704:Logical connectives
1634:Logical equivalence
1484:Logical consequence
1264:strong homomorphism
1145:) and all elements
544:, and all elements
482:elementary submodel
459:non-standard models
3165:Mathematical logic
2909:Transfer principle
2872:Semantics of logic
2857:Categorical theory
2833:Non-standard model
2347:Logical connective
1474:Information theory
1423:Mathematical logic
1374:Mathematical Logic
1323:Keisler, H. Jerome
1221:
1179:
1052:
994:
975:
902:a substructure of
873:TarskiâVaught test
867:TarskiâVaught test
842:
804:
691:
291:TarskiâVaught test
191:) with parameters
106:mathematical logic
45:list of references
3147:
3146:
3079:Abstract category
2882:Theories of truth
2692:Rule of inference
2682:Natural deduction
2663:
2662:
2208:
2207:
1913:Cartesian product
1818:
1817:
1724:Many-valued logic
1699:Boolean functions
1582:Russell's paradox
1557:diagonal argument
1454:First-order logic
1362:978-0-521-58713-6
1336:978-0-444-88054-3
1319:Chang, Chen Chung
1208:) if and only if
1101:into a structure
942:and all elements
642:
98:
97:
90:
16:(Redirected from
3177:
3138:
3137:
3089:History of logic
3084:Category of sets
2977:Decision problem
2756:Ordinal analysis
2697:Sequent calculus
2595:Boolean algebras
2535:
2534:
2509:
2480:logical/constant
2234:
2220:
2143:ZermeloâFraenkel
1894:Set operations:
1829:
1766:
1597:
1577:LöwenheimâSkolem
1464:Formal semantics
1416:
1409:
1402:
1393:
1388:
1377:
1365:
1339:
1310:
1299:
1245:), âŠ,
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851:
849:
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843:
813:
811:
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805:
700:
698:
697:
692:
684:
683:
665:
664:
643:
640:
635:
634:
616:
615:
463:Peano arithmetic
436:rational numbers
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
3185:
3184:
3180:
3179:
3178:
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3150:
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3143:
3132:
3125:
3070:Category theory
3060:Algebraic logic
3043:
3014:Lambda calculus
2952:Church encoding
2938:
2914:Truth predicate
2770:
2736:Complete theory
2659:
2528:
2524:
2520:
2515:
2507:
2227: and
2223:
2218:
2204:
2180:New Foundations
2148:axiom of choice
2131:
2093:Gödel numbering
2033: and
2025:
1929:
1814:
1764:
1745:
1694:Boolean algebra
1680:
1644:Equiconsistency
1609:Classical logic
1586:
1567:Halting problem
1555: and
1531: and
1519: and
1518:
1513:Theorems (
1508:
1425:
1420:
1386:
1369:
1363:
1345:Hodges, Wilfrid
1343:
1337:
1317:
1314:
1313:
1300:
1296:
1291:
1279:large cardinals
1257:
1244:
1213:
1212:
1207:
1199:, âŠ,
1198:
1171:
1170:
1158:
1152:, âŠ,
1151:
1144:
1136:, âŠ,
1135:
1097:of a structure
1091:
1084:
1076:, âŠ,
1075:
1044:
1043:
1030:
1022:, âŠ,
1021:
986:
985:
967:
966:
957:
949:, âŠ,
948:
937:
929:, âŠ,
928:
883:of a structure
869:
834:
833:
796:
795:
779:
755:if and only if
738:
675:
656:
626:
607:
590:
589:
579:
573:, âŠ,
572:
557:
551:, âŠ,
550:
543:
535:, âŠ,
534:
527:
519:, âŠ,
518:
471:
448:ĆoĆâVaught test
440:linear ordering
356:Two structures
354:
342:large cardinals
316:
308:, âŠ,
307:
281:A substructure
206:
198:, âŠ,
197:
190:
182:, âŠ,
181:
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
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3130:
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3123:
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3113:
3108:
3103:
3102:
3101:
3091:
3086:
3081:
3072:
3067:
3062:
3057:
3055:Abstract logic
3051:
3049:
3045:
3044:
3042:
3041:
3036:
3034:Turing machine
3031:
3026:
3021:
3016:
3011:
3006:
3005:
3004:
2999:
2994:
2989:
2984:
2974:
2972:Computable set
2969:
2964:
2959:
2954:
2948:
2946:
2940:
2939:
2937:
2936:
2931:
2926:
2921:
2916:
2911:
2906:
2901:
2900:
2899:
2894:
2889:
2879:
2874:
2869:
2867:Satisfiability
2864:
2859:
2854:
2853:
2852:
2842:
2841:
2840:
2830:
2829:
2828:
2823:
2818:
2813:
2808:
2798:
2797:
2796:
2791:
2784:Interpretation
2780:
2778:
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2768:
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2753:
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2738:
2733:
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2699:
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2664:
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2658:
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2649:
2648:
2647:
2646:
2641:
2640:
2639:
2634:
2629:
2609:
2608:
2607:
2605:minimal axioms
2602:
2591:
2590:
2589:
2578:
2577:
2576:
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2566:
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2556:
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2334:
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2297:
2295:Formation rule
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2214:Formal systems
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2137:
2133:
2132:
2130:
2129:
2128:
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2117:
2112:
2111:
2110:
2103:Large cardinal
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2095:
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2011:
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1996:
1991:
1986:
1981:
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1868:Extensionality
1865:
1863:Ordinal number
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1547:Cantor's
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1501:
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1404:
1396:
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1335:
1312:
1311:
1301:E. C. Milner,
1293:
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1290:
1287:
1283:Critical point
1260:
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1203:
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1133:
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1073:
1051:
1026:
1019:
993:
974:
953:
946:
933:
926:
868:
865:
841:
817:and says that
803:
775:
739:consisting of
734:
690:
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555:
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470:
467:
415:A first-order
392:have the same
353:
350:
346:rank-into-rank
312:
305:
202:
195:
186:
179:
104:, a branch of
96:
95:
53:external links
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3029:Recursive set
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2644:non-Euclidean
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2165:KripkeâPlatek
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2017:
2015:
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2009:constructible
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1686:Propositional
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858:The downward
856:
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820:
816:
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794:
791:, one writes
790:
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780:-structures.
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254:is called an
253:
250: â
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231:is called an
230:
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222:
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170:
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78:February 2023
71:
67:
61:
60:
54:
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41:
32:
31:
19:
3170:Model theory
3131:
2929:Ultraproduct
2805:
2776:Model theory
2741:Independence
2677:Formal proof
2669:Proof theory
2652:
2625:
2582:real numbers
2554:second-order
2465:Substitution
2342:Metalanguage
2283:conservative
2256:Axiom schema
2200:Constructive
2170:MorseâKelley
2136:Set theories
2115:Aleph number
2108:inaccessible
2014:Grothendieck
1898:intersection
1785:Higher-order
1773:Second-order
1719:Truth tables
1676:Venn diagram
1459:Formal proof
1373:
1348:
1327:Model Theory
1326:
1306:
1297:
1274:
1268:
1261:
1254:
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719:
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489:
485:
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477:
473:
472:
452:
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428:real numbers
423:
421:
414:
409:
405:
401:
397:
389:
385:
381:
377:
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369:
365:
361:
357:
355:
344:, including
339:
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176:
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156:
152:
149:substructure
144:
142:
135:
127:
123:
119:of the same
116:
112:
102:model theory
99:
84:
75:
64:Please help
56:
3039:Type theory
2987:undecidable
2919:Truth value
2806:equivalence
2485:non-logical
2098:Enumeration
2088:Isomorphism
2035:cardinality
2019:Von Neumann
1984:Ultrafilter
1949:Uncountable
1883:equivalence
1800:Quantifiers
1790:Fixed-point
1759:First-order
1639:Consistency
1624:Proposition
1601:Traditional
1572:Lindström's
1562:Compactness
1504:Type theory
1449:Cardinality
580:) holds in
376:is true in
211:is true in
132:first-order
126:are called
70:introducing
3154:Categories
2850:elementary
2543:arithmetic
2411:Quantifier
2389:functional
2261:Expression
1979:Transitive
1923:identities
1908:complement
1841:hereditary
1824:Set theory
1289:References
1281:(see also
1271:set theory
1039:such that
508:-formulas
384:, i.e. if
138:-sentences
110:structures
3121:Supertask
3024:Recursion
2982:decidable
2816:saturated
2794:of models
2717:deductive
2712:axiomatic
2632:Hilbert's
2619:Euclidean
2600:canonical
2523:axiomatic
2455:Signature
2384:Predicate
2273:Extension
2195:Ackermann
2120:Operation
1999:Universal
1989:Recursive
1964:Singleton
1959:Inhabited
1944:Countable
1934:Types of
1918:power set
1888:partition
1805:Predicate
1751:Predicate
1666:Syllogism
1656:Soundness
1629:Inference
1619:Tautology
1521:paradoxes
1325:(1990) ,
1219:⊨
1177:⊨
1125:-formula
1109:is a map
1050:⊨
992:∃
973:⊨
840:⪰
802:⪯
670:…
651:φ
648:⊨
621:…
602:φ
599:⊨
498:signature
241:embedding
171:-formula
121:signature
3106:Logicism
3099:timeline
3075:Concrete
2934:Validity
2904:T-schema
2897:Kripke's
2892:Tarski's
2887:semantic
2877:Strength
2826:submodel
2821:spectrum
2789:function
2637:Tarski's
2626:Elements
2613:geometry
2569:Robinson
2490:variable
2475:function
2448:spectrum
2438:Sentence
2394:variable
2337:Language
2290:Relation
2251:Automata
2241:Alphabet
2225:language
2079:-jection
2057:codomain
2043:Function
2004:Universe
1974:Infinite
1878:Relation
1661:Validity
1651:Argument
1549:theorem,
1347:(1997),
1159:of
558:of
394:complete
235:of
3048:Related
2845:Diagram
2743: (
2722:Hilbert
2707:Systems
2702:Theorem
2580:of the
2525:systems
2305:Formula
2300:Grammar
2216: (
2160:General
1873:Forcing
1858:Element
1778:Monadic
1553:paradox
1494:Theorem
1430:General
1113::
1069:,
1015:,
938:) over
922:,
906:. Then
747:. Then
301:,
246::
227:, then
66:improve
2811:finite
2574:Skolem
2527:
2502:Theory
2470:Symbol
2460:String
2443:atomic
2320:ground
2315:closed
2310:atomic
2266:ground
2229:syntax
2125:binary
2052:domain
1969:Finite
1734:finite
1592:Logics
1551:
1499:Theory
1382:
1359:
1333:
1007:
821:is an
500:
476:is an
417:theory
108:, two
2801:Model
2549:Peano
2406:Proof
2246:Arity
2175:Naive
2062:image
1994:Fuzzy
1954:Empty
1903:union
1848:Class
1489:Model
1479:Lemma
1437:Axiom
962:, if
958:from
262:into
239:. An
219:. If
207:from
147:is a
51:, or
2924:Type
2727:list
2531:list
2508:list
2497:Term
2431:rank
2325:open
2219:list
2031:Maps
1936:sets
1795:Free
1765:list
1515:list
1442:list
1380:ISBN
1357:ISBN
1331:ISBN
898:and
890:Let
875:(or
871:The
767:and
763:and
726:and
492:and
400:and
388:and
368:are
360:and
115:and
2611:of
2593:of
2541:of
2073:Sur
2047:Map
1854:Ur-
1836:Set
1285:).
1258:)).
1163:,
1093:An
1085:).
1035:in
825:of
783:If
714:If
488:if
484:of
480:or
461:of
434:of
426:of
285:of
266:if
258:of
163:of
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143:If
100:In
3156::
2997:NP
2621::
2615::
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2077:Bi
2069:In
1355:,
1321:;
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3077:/
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