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Validity and other concepts are then introduced similarly to modal frames, with a few changes necessary to accommodate for the weaker closure properties of the set of admissible valuations. In particular, an intuitionistic frame
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Descriptive frames are the most important class of frames because of the duality theory (see below). Refined frames are useful as a common generalization of descriptive and Kripke frames.
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are a pair of contravariant functors, which make the category of
Heyting algebras dually equivalent to the category of descriptive intuitionistic frames.
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Kripke frames are refined and atomic. However, infinite Kripke frames are never compact. Every finite differentiated or atomic frame is a Kripke frame.
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A frame and its dual validate the same formulas; hence the general frame semantics and algebraic semantics are in a sense equivalent. The double dual
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It is also possible to define duals of p-morphisms on one hand, and modal algebra homomorphisms on the other hand. In this way the operators
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The frame semantics for intuitionistic and intermediate logics can be developed in parallel to the semantics for modal logics. An
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itself. This is not true in general for double duals of frames, as the dual of every algebra is descriptive. In fact, a frame
4301:) between the categories of descriptive frames, and modal algebras. This is a special case of a more general duality between
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It is possible to construct intuitionistic general frames from transitive reflexive modal frames and vice versa, see
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is complete with respect to its canonical model, and the general frame induced by the canonical model (called the
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3071:{\displaystyle \langle x/{\sim },y/{\sim }\rangle \in S\iff \forall A\in V\,(x\in \Box A\Rightarrow y\in A),}
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Its dual
Heyting algebra, the Rieger–Nishimura lattice. It is the free Heyting algebra over 1 generator.
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279:: it shares the transparent geometrical insight of the former, and robust completeness of the latter.
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5231:, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.
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is complete with respect to a class of general frames. This is a consequence of the fact that
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1796:{\displaystyle V={\big \{}\{x\in F\mid x\Vdash A\}\mid A{\hbox{ is a formula}}{\big \}}.}
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may be identified with a general frame in which all valuations are admissible: i.e.,
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1810:, and disjoint unions of Kripke frames have analogues on general frames. A frame
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of general frames, and the category of modal algebras. These functors provide a
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In full generality, general frames are hardly more than a fancy name for Kripke
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4815:{\displaystyle \mathbf {F} ^{+}=\langle V,\cap ,\cup ,\to ,\emptyset \rangle }
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2721:{\displaystyle V=\{A\subseteq F\mid \forall i\in I\,(A\cap F_{i}\in V_{i})\}.}
4705:
Tight intuitionistic frames are automatically differentiated, hence refined.
4420:
1268:
4027:{\displaystyle x\,R\,y\iff \forall a\in A\,(\Box a\in x\Rightarrow a\in y)}
2900:{\displaystyle x\sim y\iff \forall A\in V\,(x\in A\Leftrightarrow y\in A),}
571:{\displaystyle \Box A=\{x\in F\mid \forall y\in F\,(x\,R\,y\to y\in A)\}}
3380:
The Rieger–Nishimura ladder: a 1-universal intuitionistic Kripke frame.
3761:{\displaystyle \mathbf {A} =\langle A,\wedge ,\vee ,-,\Box \rangle }
1806:
The fundamental truth-preserving operations of generated subframes,
4873:{\displaystyle \mathbf {A} =\langle A,\wedge ,\vee ,\to ,0\rangle }
4701:
with the finite intersection property has a non-empty intersection.
4166:
is descriptive if and only if it is isomorphic to its double dual
3383:
3375:
2424:{\displaystyle \mathbf {F} _{i}=\langle F_{i},R_{i},V_{i}\rangle }
248:
271:
logics. The general frame semantics combines the main virtues of
5220:, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
3519:{\displaystyle \langle {\mathcal {P}}(F),\cap ,\cup ,-\rangle }
1501:{\displaystyle \forall A\in V\,(x\in \Box A\Rightarrow y\in A)}
4612:{\displaystyle \forall A\in V\,(x\in A\Leftrightarrow y\in A)}
1409:{\displaystyle \forall A\in V\,(x\in A\Leftrightarrow y\in A)}
158:
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complex algebras and fields of sets on relational structures
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is to restrict the allowed valuations in the frame: a model
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In the opposite direction, it is possible to construct the
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is closed under substitution, the general frame induced by
4926:{\displaystyle \mathbf {A} _{+}=\langle F,\leq ,V\rangle }
4747:{\displaystyle \mathbf {F} =\langle F,\leq ,V\rangle }
4548:{\displaystyle \mathbf {F} =\langle F,\leq ,V\rangle }
3703:{\displaystyle \mathbf {A} _{+}=\langle F,R,V\rangle }
3462:
is closed under
Boolean operations, therefore it is a
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with an additional structure, which are used to model
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is complete with respect to a class of Kripke models
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2164:{\displaystyle f\colon \mathbf {F} \to \mathbf {G} }
87:. Unsourced material may be challenged and removed.
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3435:{\displaystyle \mathbf {F} =\langle F,R,V\rangle }
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330:{\displaystyle \mathbf {F} =\langle F,R,V\rangle }
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4447:that contains the empty set, and is closed under
3805:{\displaystyle \langle A,\wedge ,\vee ,-\rangle }
3526:. It also carries an additional unary operation,
186:but its sources remain unclear because it lacks
3140:Unlike Kripke frames, every normal modal logic
5216:Alexander Chagrov and Michael Zakharyaschev,
3273:{\displaystyle \langle F,R,{\Vdash }\rangle }
3213:{\displaystyle \langle F,R,{\Vdash }\rangle }
1785:
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1652:{\displaystyle \langle F,R,{\Vdash }\rangle }
748:{\displaystyle \{x\in F\mid x\Vdash p\}\in V}
8:
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637:{\displaystyle \langle F,R,\Vdash \rangle }
50:Learn how and when to remove these messages
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3125:{\displaystyle A/{\sim }\in W\iff A\in V.}
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2211:that is a p-morphism of the Kripke frames
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4348:{\displaystyle \langle F,\leq ,V\rangle }
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2117:{\displaystyle W=\{A\cap G\mid A\in V\}.}
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235:Learn how and when to remove this message
217:Learn how and when to remove this message
147:Learn how and when to remove this message
4694:{\displaystyle V\cup \{F-A\mid A\in V\}}
3596:is a modal algebra, which is called the
3392:General frames bear close connection to
839:{\displaystyle \{x\in F\mid x\Vdash A\}}
584:fields of sets with additional structure
5086:{\displaystyle \{x\in F\mid a\in x\},}
4497:{\displaystyle A\to B=\Box (-A\cup B)}
4201:{\displaystyle (\mathbf {F} ^{+})_{+}}
4122:of any modal algebra is isomorphic to
4115:{\displaystyle (\mathbf {A} _{+})^{+}}
1056:, if all axioms (or equivalently, all
3320:is complete with respect to a single
2946:be the set of equivalence classes of
1693:{\displaystyle \langle F,R,V\rangle }
7:
4708:The dual of an intuitionistic frame
2827:defined as follows. We consider the
2626:{\displaystyle \{R_{i}\mid i\in I\}}
2561:{\displaystyle \{F_{i}\mid i\in I\}}
1598:, if it is differentiated and tight,
453:that is closed under the following:
85:adding citations to reliable sources
2268:{\displaystyle \langle G,S\rangle }
2236:{\displaystyle \langle F,R\rangle }
1963:{\displaystyle \langle F,R\rangle }
1931:{\displaystyle \langle G,S\rangle }
1176:{\displaystyle \langle F,R\rangle }
669:{\displaystyle \langle F,R\rangle }
457:the Boolean operations of (binary)
362:{\displaystyle \langle F,R\rangle }
4806:
4569:
3981:
3022:
2854:
2664:
1615:Operations and morphisms on frames
1455:
1366:
523:
14:
3935:, and the accessibility relation
2061:{\displaystyle S=R\cap G\times G}
1260:{\displaystyle {\mathcal {P}}(F)}
31:This article has multiple issues.
4965:
4889:
4830:
4822:. The dual of a Heyting algebra
4766:
4716:
4517:
4178:
4152:
4130:
4092:
3904:{\displaystyle \mathbf {A} _{+}}
3891:
3848:
3718:
3666:
3644:{\displaystyle \mathbf {F} ^{+}}
3631:
3608:
3404:
2789:
2743:
2465:
2366:
2157:
2149:
1868:
1818:
1314:
1110:
1088:
1042:
925:
688:
582:They are thus a special case of
299:
163:
61:
20:
1604:, if it is refined and compact.
72:needs additional citations for
39:or discuss these issues on the
5169:
5162:
5136:
5129:
4858:
4800:
4606:
4594:
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4189:
4173:
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3486:
3300:-frame. Moreover, every logic
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1212:
965:for all admissible valuations
562:
550:
536:
1:
1566:has a non-empty intersection,
5181:{\displaystyle (\cdot )_{+}}
5148:{\displaystyle (\cdot )^{+}}
4972:{\displaystyle \mathbf {A} }
4880:is the intuitionistic frame
4315:intuitionistic general frame
4270:{\displaystyle (\cdot )_{+}}
4237:{\displaystyle (\cdot )^{+}}
4159:{\displaystyle \mathbf {F} }
4137:{\displaystyle \mathbf {A} }
3882:of admissible valuations in
3855:{\displaystyle \mathbf {A} }
3615:{\displaystyle \mathbf {F} }
3442:be a general frame. The set
2357:of an indexed set of frames
1564:finite intersection property
1117:{\displaystyle \mathbf {F} }
1095:{\displaystyle \mathbf {F} }
1049:{\displaystyle \mathbf {F} }
932:{\displaystyle \mathbf {F} }
695:{\displaystyle \mathbf {F} }
5023:consists of all subsets of
2939:{\displaystyle G=F/{\sim }}
2316:{\displaystyle f^{-1}\in V}
5276:
782:The closure conditions on
644:based on the Kripke frame
3546:. The combined structure
2523:is the disjoint union of
958:{\displaystyle x\Vdash A}
369:is a Kripke frame (i.e.,
5122:. As in the modal case,
1592:contains all singletons,
172:This article includes a
4638:{\displaystyle x\leq y}
4451:intersection and union,
3816:, whose underlying set
1529:{\displaystyle x\,R\,y}
1102:. In this case we call
978:{\displaystyle \Vdash }
890:(not only a variable).
433:is a set of subsets of
201:more precise citations.
5205:Neighborhood semantics
5182:
5149:
5116:
5115:{\displaystyle a\in A}
5087:
5037:
5017:
4993:
4973:
4947:
4927:
4874:
4816:
4748:
4695:
4639:
4613:
4549:
4498:
4441:
4413:
4393:
4369:
4349:
4291:Jónsson–Tarski duality
4279:contravariant functors
4271:
4238:
4202:
4160:
4138:
4116:
4071:
4051:
4028:
3949:
3929:
3905:
3876:
3856:
3830:
3806:
3768:. The Boolean algebra
3762:
3704:
3645:
3616:
3590:
3540:
3520:
3456:
3436:
3389:
3381:
3372:Jónsson–Tarski duality
3362:
3338:
3314:
3294:
3274:
3234:
3214:
3174:
3154:
3126:
3072:
2960:
2940:
2901:
2821:
2775:
2722:
2627:
2582:
2562:
2517:
2497:
2451:
2450:{\displaystyle i\in I}
2425:
2343:
2342:{\displaystyle A\in W}
2317:
2269:
2237:
2205:
2185:
2165:
2118:
2062:
2024:
2004:
1984:
1964:
1932:
1906:, if the Kripke frame
1900:
1850:
1797:
1714:
1694:
1653:
1586:
1556:
1530:
1502:
1436:
1410:
1346:
1285:
1261:
1228:
1177:
1138:
1118:
1096:
1074:
1050:
1034:is valid in the frame
1028:
1005:
1004:{\displaystyle x\in F}
979:
959:
933:
907:
884:
860:
840:
796:
772:
757:propositional variable
749:
696:
670:
638:
600:
572:
486:
447:
427:
407:
383:
363:
331:
5183:
5150:
5117:
5088:
5038:
5018:
4994:
4992:{\displaystyle \leq }
4974:
4948:
4928:
4875:
4817:
4749:
4696:
4651:, if every subset of
4640:
4614:
4550:
4499:
4442:
4414:
4394:
4370:
4368:{\displaystyle \leq }
4350:
4309:Intuitionistic frames
4272:
4239:
4203:
4161:
4139:
4117:
4072:
4052:
4037:for all ultrafilters
4029:
3950:
3930:
3906:
3877:
3857:
3831:
3807:
3763:
3710:to any modal algebra
3705:
3646:
3617:
3591:
3541:
3539:{\displaystyle \Box }
3521:
3457:
3437:
3387:
3379:
3363:
3339:
3315:
3295:
3275:
3235:
3215:
3175:
3155:
3127:
3073:
2961:
2959:{\displaystyle \sim }
2941:
2902:
2822:
2776:
2723:
2628:
2583:
2563:
2518:
2498:
2452:
2426:
2344:
2318:
2270:
2238:
2206:
2186:
2166:
2119:
2063:
2025:
2010:closed upwards under
2005:
1985:
1965:
1933:
1901:
1851:
1798:
1715:
1695:
1654:
1587:
1557:
1542:, if every subset of
1531:
1503:
1437:
1411:
1347:
1286:
1262:
1229:
1178:
1139:
1119:
1097:
1075:
1051:
1029:
1006:
980:
960:
934:
908:
885:
861:
841:
797:
773:
750:
697:
680:in the general frame
671:
639:
601:
573:
487:
485:{\displaystyle \Box }
448:
428:
408:
384:
364:
332:
5159:
5126:
5100:
5050:
5027:
5007:
4983:
4961:
4937:
4884:
4826:
4761:
4712:
4655:
4623:
4566:
4513:
4458:
4431:
4403:
4383:
4359:
4321:
4248:
4215:
4170:
4148:
4126:
4084:
4061:
4041:
3962:
3939:
3919:
3886:
3866:
3844:
3820:
3772:
3714:
3661:
3626:
3604:
3550:
3530:
3473:
3446:
3400:
3352:
3328:
3304:
3284:
3244:
3224:
3184:
3164:
3144:
3083:
2973:
2950:
2914:
2837:
2829:equivalence relation
2785:
2739:
2640:
2592:
2572:
2527:
2507:
2461:
2435:
2361:
2327:
2282:
2247:
2215:
2195:
2175:
2139:
2075:
2034:
2014:
1994:
1974:
1942:
1910:
1864:
1814:
1727:
1704:
1666:
1623:
1576:
1546:
1512:
1452:
1420:
1363:
1310:
1275:
1238:
1187:
1155:
1128:
1106:
1084:
1064:
1038:
1018:
989:
969:
943:
921:
897:
874:
850:
806:
786:
762:
709:
684:
648:
610:
590:
496:
476:
437:
417:
397:
373:
341:
295:
81:improve this article
5223:Patrick Blackburn,
2781:is a refined frame
2171:is a function from
1619:Every Kripke model
1435:{\displaystyle x=y}
289:modal general frame
277:algebraic semantics
5227:, and Yde Venema,
5178:
5145:
5112:
5083:
5033:
5013:
4989:
4969:
4953:is the set of all
4943:
4923:
4870:
4812:
4744:
4691:
4635:
4609:
4545:
4494:
4437:
4409:
4389:
4365:
4345:
4267:
4234:
4198:
4156:
4134:
4112:
4067:
4047:
4024:
3945:
3925:
3901:
3872:
3852:
3836:is the set of all
3826:
3802:
3758:
3700:
3641:
3612:
3586:
3536:
3516:
3452:
3432:
3390:
3382:
3368:) is descriptive.
3358:
3334:
3310:
3290:
3270:
3230:
3210:
3170:
3150:
3122:
3068:
2956:
2936:
2897:
2817:
2771:
2718:
2623:
2578:
2558:
2513:
2493:
2447:
2421:
2339:
2313:
2265:
2233:
2201:
2181:
2161:
2114:
2058:
2020:
2000:
1980:
1960:
1928:
1896:
1858:generated subframe
1846:
1793:
1781:
1779: is a formula
1710:
1690:
1662:the general frame
1649:
1582:
1552:
1526:
1498:
1432:
1406:
1342:
1281:
1257:
1224:
1173:
1134:
1114:
1092:
1070:
1046:
1024:
1013:normal modal logic
1001:
975:
955:
929:
903:
880:
856:
836:
792:
768:
745:
692:
666:
634:
596:
568:
482:
443:
423:
403:
379:
359:
327:
174:list of references
5260:Concepts in logic
5036:{\displaystyle F}
5016:{\displaystyle V}
4946:{\displaystyle F}
4440:{\displaystyle F}
4412:{\displaystyle V}
4392:{\displaystyle F}
4277:become a pair of
4070:{\displaystyle y}
4050:{\displaystyle x}
3948:{\displaystyle R}
3928:{\displaystyle F}
3875:{\displaystyle V}
3829:{\displaystyle F}
3622:, and denoted by
3466:of the power set
3455:{\displaystyle V}
3361:{\displaystyle L}
3337:{\displaystyle L}
3313:{\displaystyle L}
3293:{\displaystyle L}
3233:{\displaystyle L}
3173:{\displaystyle L}
3153:{\displaystyle L}
2581:{\displaystyle R}
2516:{\displaystyle F}
2204:{\displaystyle G}
2184:{\displaystyle F}
2023:{\displaystyle R}
2003:{\displaystyle F}
1983:{\displaystyle G}
1780:
1713:{\displaystyle V}
1585:{\displaystyle V}
1555:{\displaystyle V}
1284:{\displaystyle F}
1137:{\displaystyle L}
1073:{\displaystyle L}
1027:{\displaystyle L}
985:, and all points
906:{\displaystyle A}
883:{\displaystyle A}
859:{\displaystyle V}
802:then ensure that
795:{\displaystyle V}
771:{\displaystyle p}
599:{\displaystyle V}
586:. The purpose of
446:{\displaystyle F}
426:{\displaystyle V}
406:{\displaystyle F}
382:{\displaystyle R}
245:
244:
237:
227:
226:
219:
157:
156:
149:
131:
54:
5267:
5255:Duality theories
5225:Maarten de Rijke
5187:
5185:
5184:
5179:
5177:
5176:
5154:
5152:
5151:
5146:
5144:
5143:
5121:
5119:
5118:
5113:
5092:
5090:
5089:
5084:
5042:
5040:
5039:
5034:
5022:
5020:
5019:
5014:
4998:
4996:
4995:
4990:
4978:
4976:
4975:
4970:
4968:
4952:
4950:
4949:
4944:
4932:
4930:
4929:
4924:
4898:
4897:
4892:
4879:
4877:
4876:
4871:
4833:
4821:
4819:
4818:
4813:
4775:
4774:
4769:
4753:
4751:
4750:
4745:
4719:
4700:
4698:
4697:
4692:
4644:
4642:
4641:
4636:
4618:
4616:
4615:
4610:
4554:
4552:
4551:
4546:
4520:
4503:
4501:
4500:
4495:
4446:
4444:
4443:
4438:
4418:
4416:
4415:
4410:
4398:
4396:
4395:
4390:
4374:
4372:
4371:
4366:
4354:
4352:
4351:
4346:
4276:
4274:
4273:
4268:
4266:
4265:
4243:
4241:
4240:
4235:
4233:
4232:
4207:
4205:
4204:
4199:
4197:
4196:
4187:
4186:
4181:
4165:
4163:
4162:
4157:
4155:
4143:
4141:
4140:
4135:
4133:
4121:
4119:
4118:
4113:
4111:
4110:
4101:
4100:
4095:
4076:
4074:
4073:
4068:
4056:
4054:
4053:
4048:
4033:
4031:
4030:
4025:
3954:
3952:
3951:
3946:
3934:
3932:
3931:
3926:
3911:consists of the
3910:
3908:
3907:
3902:
3900:
3899:
3894:
3881:
3879:
3878:
3873:
3861:
3859:
3858:
3853:
3851:
3835:
3833:
3832:
3827:
3811:
3809:
3808:
3803:
3767:
3765:
3764:
3759:
3721:
3709:
3707:
3706:
3701:
3675:
3674:
3669:
3650:
3648:
3647:
3642:
3640:
3639:
3634:
3621:
3619:
3618:
3613:
3611:
3595:
3593:
3592:
3587:
3545:
3543:
3542:
3537:
3525:
3523:
3522:
3517:
3485:
3484:
3461:
3459:
3458:
3453:
3441:
3439:
3438:
3433:
3407:
3367:
3365:
3364:
3359:
3343:
3341:
3340:
3335:
3319:
3317:
3316:
3311:
3299:
3297:
3296:
3291:
3279:
3277:
3276:
3271:
3266:
3239:
3237:
3236:
3231:
3219:
3217:
3216:
3211:
3206:
3179:
3177:
3176:
3171:
3159:
3157:
3156:
3151:
3131:
3129:
3128:
3123:
3098:
3093:
3077:
3075:
3074:
3069:
3007:
3002:
2991:
2986:
2965:
2963:
2962:
2957:
2945:
2943:
2942:
2937:
2935:
2930:
2906:
2904:
2903:
2898:
2826:
2824:
2823:
2818:
2792:
2780:
2778:
2777:
2772:
2746:
2727:
2725:
2724:
2719:
2708:
2707:
2695:
2694:
2632:
2630:
2629:
2624:
2607:
2606:
2588:is the union of
2587:
2585:
2584:
2579:
2567:
2565:
2564:
2559:
2542:
2541:
2522:
2520:
2519:
2514:
2502:
2500:
2499:
2494:
2468:
2456:
2454:
2453:
2448:
2430:
2428:
2427:
2422:
2417:
2416:
2404:
2403:
2391:
2390:
2375:
2374:
2369:
2348:
2346:
2345:
2340:
2322:
2320:
2319:
2314:
2297:
2296:
2274:
2272:
2271:
2266:
2242:
2240:
2239:
2234:
2210:
2208:
2207:
2202:
2190:
2188:
2187:
2182:
2170:
2168:
2167:
2162:
2160:
2152:
2133:bounded morphism
2123:
2121:
2120:
2115:
2067:
2065:
2064:
2059:
2029:
2027:
2026:
2021:
2009:
2007:
2006:
2001:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1937:
1935:
1934:
1929:
1905:
1903:
1902:
1897:
1871:
1855:
1853:
1852:
1847:
1821:
1808:p-morphic images
1802:
1800:
1799:
1794:
1789:
1788:
1782:
1778:
1742:
1741:
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1658:
1656:
1655:
1650:
1645:
1591:
1589:
1588:
1583:
1561:
1559:
1558:
1553:
1535:
1533:
1532:
1527:
1507:
1505:
1504:
1499:
1441:
1439:
1438:
1433:
1415:
1413:
1412:
1407:
1351:
1349:
1348:
1343:
1317:
1290:
1288:
1287:
1282:
1266:
1264:
1263:
1258:
1247:
1246:
1233:
1231:
1230:
1225:
1211:
1210:
1182:
1180:
1179:
1174:
1143:
1141:
1140:
1135:
1123:
1121:
1120:
1115:
1113:
1101:
1099:
1098:
1093:
1091:
1079:
1077:
1076:
1071:
1055:
1053:
1052:
1047:
1045:
1033:
1031:
1030:
1025:
1010:
1008:
1007:
1002:
984:
982:
981:
976:
964:
962:
961:
956:
938:
936:
935:
930:
928:
912:
910:
909:
904:
889:
887:
886:
881:
865:
863:
862:
857:
845:
843:
842:
837:
801:
799:
798:
793:
777:
775:
774:
769:
754:
752:
751:
746:
701:
699:
698:
693:
691:
675:
673:
672:
667:
643:
641:
640:
635:
605:
603:
602:
597:
577:
575:
574:
569:
491:
489:
488:
483:
452:
450:
449:
444:
432:
430:
429:
424:
412:
410:
409:
404:
388:
386:
385:
380:
368:
366:
365:
360:
336:
334:
333:
328:
302:
273:Kripke semantics
240:
233:
222:
215:
211:
208:
202:
197:this article by
188:inline citations
167:
166:
159:
152:
145:
141:
138:
132:
130:
89:
65:
57:
46:
24:
23:
16:
5275:
5274:
5270:
5269:
5268:
5266:
5265:
5264:
5235:
5234:
5213:
5201:
5193:modal companion
5168:
5157:
5156:
5135:
5124:
5123:
5098:
5097:
5048:
5047:
5025:
5024:
5005:
5004:
4981:
4980:
4979:, the ordering
4959:
4958:
4935:
4934:
4887:
4882:
4881:
4824:
4823:
4764:
4759:
4758:
4756:Heyting algebra
4710:
4709:
4653:
4652:
4621:
4620:
4564:
4563:
4511:
4510:
4456:
4455:
4429:
4428:
4401:
4400:
4381:
4380:
4357:
4356:
4319:
4318:
4311:
4257:
4246:
4245:
4224:
4213:
4212:
4188:
4176:
4168:
4167:
4146:
4145:
4124:
4123:
4102:
4090:
4082:
4081:
4059:
4058:
4039:
4038:
3960:
3959:
3937:
3936:
3917:
3916:
3889:
3884:
3883:
3864:
3863:
3842:
3841:
3818:
3817:
3770:
3769:
3712:
3711:
3664:
3659:
3658:
3629:
3624:
3623:
3602:
3601:
3548:
3547:
3528:
3527:
3471:
3470:
3468:Boolean algebra
3444:
3443:
3398:
3397:
3374:
3350:
3349:
3346:canonical frame
3326:
3325:
3324:frame. Indeed,
3302:
3301:
3282:
3281:
3242:
3241:
3222:
3221:
3182:
3181:
3162:
3161:
3142:
3141:
3138:
3081:
3080:
2971:
2970:
2948:
2947:
2912:
2911:
2835:
2834:
2783:
2782:
2737:
2736:
2699:
2686:
2638:
2637:
2598:
2590:
2589:
2570:
2569:
2533:
2525:
2524:
2505:
2504:
2459:
2458:
2457:, is the frame
2433:
2432:
2408:
2395:
2382:
2364:
2359:
2358:
2325:
2324:
2285:
2280:
2279:
2245:
2244:
2213:
2212:
2193:
2192:
2173:
2172:
2137:
2136:
2073:
2072:
2032:
2031:
2012:
2011:
1992:
1991:
1990:is a subset of
1972:
1971:
1940:
1939:
1908:
1907:
1862:
1861:
1812:
1811:
1725:
1724:
1702:
1701:
1664:
1663:
1621:
1620:
1617:
1574:
1573:
1544:
1543:
1510:
1509:
1450:
1449:
1418:
1417:
1361:
1360:
1308:
1307:
1297:
1295:Types of frames
1273:
1272:
1236:
1235:
1185:
1184:
1153:
1152:
1151:A Kripke frame
1126:
1125:
1104:
1103:
1082:
1081:
1062:
1061:
1036:
1035:
1016:
1015:
987:
986:
967:
966:
941:
940:
919:
918:
895:
894:
872:
871:
848:
847:
804:
803:
784:
783:
760:
759:
707:
706:
682:
681:
646:
645:
608:
607:
588:
587:
494:
493:
474:
473:
435:
434:
415:
414:
395:
394:
391:binary relation
371:
370:
339:
338:
293:
292:
285:
241:
230:
229:
228:
223:
212:
206:
203:
192:
178:related reading
168:
164:
153:
142:
136:
133:
96:"General frame"
90:
88:
78:
66:
25:
21:
12:
11:
5:
5273:
5271:
5263:
5262:
5257:
5252:
5247:
5237:
5236:
5233:
5232:
5221:
5212:
5209:
5208:
5207:
5200:
5197:
5175:
5171:
5167:
5164:
5142:
5138:
5134:
5131:
5111:
5108:
5105:
5094:
5093:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5032:
5012:
4988:
4967:
4942:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4896:
4891:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4832:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4773:
4768:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4718:
4703:
4702:
4690:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4646:
4634:
4631:
4628:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4580:
4577:
4574:
4571:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4519:
4506:
4505:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4454:the operation
4452:
4436:
4408:
4388:
4364:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4310:
4307:
4295:Bjarni JĂłnsson
4264:
4260:
4256:
4253:
4231:
4227:
4223:
4220:
4195:
4191:
4185:
4180:
4175:
4154:
4132:
4109:
4105:
4099:
4094:
4089:
4066:
4046:
4035:
4034:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3992:
3989:
3986:
3983:
3979:
3975:
3971:
3967:
3955:is defined by
3944:
3924:
3898:
3893:
3871:
3850:
3825:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3720:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3673:
3668:
3638:
3633:
3610:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3535:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3483:
3478:
3451:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3406:
3394:modal algebras
3373:
3370:
3357:
3333:
3309:
3289:
3269:
3265:
3261:
3258:
3255:
3252:
3249:
3229:
3209:
3205:
3201:
3198:
3195:
3192:
3189:
3169:
3149:
3137:
3134:
3133:
3132:
3121:
3118:
3115:
3112:
3108:
3104:
3101:
3097:
3092:
3088:
3078:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3033:
3030:
3027:
3024:
3020:
3016:
3013:
3010:
3006:
3001:
2997:
2994:
2990:
2985:
2981:
2978:
2966:. Then we put
2955:
2934:
2929:
2925:
2922:
2919:
2908:
2907:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2865:
2862:
2859:
2856:
2852:
2848:
2845:
2842:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2791:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2745:
2729:
2728:
2717:
2714:
2711:
2706:
2702:
2698:
2693:
2689:
2685:
2682:
2679:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2622:
2619:
2616:
2613:
2610:
2605:
2601:
2597:
2577:
2557:
2554:
2551:
2548:
2545:
2540:
2536:
2532:
2512:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2467:
2446:
2443:
2440:
2420:
2415:
2411:
2407:
2402:
2398:
2394:
2389:
2385:
2381:
2378:
2373:
2368:
2355:disjoint union
2351:
2350:
2338:
2335:
2332:
2312:
2309:
2306:
2303:
2300:
2295:
2292:
2288:
2264:
2261:
2258:
2255:
2252:
2232:
2229:
2226:
2223:
2220:
2200:
2180:
2159:
2155:
2151:
2147:
2144:
2125:
2124:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2019:
1999:
1979:
1959:
1956:
1953:
1950:
1947:
1927:
1924:
1921:
1918:
1915:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1870:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1820:
1804:
1803:
1792:
1787:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1740:
1735:
1732:
1720:is defined as
1709:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1648:
1644:
1640:
1637:
1634:
1631:
1628:
1616:
1613:
1606:
1605:
1599:
1593:
1581:
1567:
1551:
1537:
1525:
1521:
1517:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1466:
1463:
1460:
1457:
1443:
1431:
1428:
1425:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1377:
1374:
1371:
1368:
1357:differentiated
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1316:
1296:
1293:
1280:
1256:
1253:
1250:
1245:
1223:
1220:
1217:
1214:
1209:
1204:
1201:
1198:
1195:
1192:
1172:
1169:
1166:
1163:
1160:
1133:
1112:
1090:
1069:
1044:
1023:
1000:
997:
994:
974:
954:
951:
948:
927:
902:
879:
855:
835:
832:
829:
826:
823:
820:
817:
814:
811:
791:
780:
779:
767:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
690:
665:
662:
659:
656:
653:
633:
630:
627:
624:
621:
618:
615:
595:
580:
579:
567:
564:
561:
558:
555:
552:
549:
545:
541:
538:
534:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
481:
472:the operation
470:
442:
422:
402:
378:
358:
355:
352:
349:
346:
326:
323:
320:
317:
314:
311:
308:
305:
301:
284:
281:
253:general frames
243:
242:
225:
224:
182:external links
171:
169:
162:
155:
154:
69:
67:
60:
55:
29:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
5272:
5261:
5258:
5256:
5253:
5251:
5248:
5246:
5243:
5242:
5240:
5230:
5226:
5222:
5219:
5215:
5214:
5210:
5206:
5203:
5202:
5198:
5196:
5194:
5189:
5173:
5165:
5140:
5132:
5109:
5106:
5103:
5080:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5046:
5045:
5044:
5030:
5010:
5002:
4986:
4956:
4955:prime filters
4940:
4917:
4914:
4911:
4908:
4905:
4899:
4894:
4864:
4861:
4855:
4852:
4849:
4846:
4843:
4840:
4834:
4803:
4797:
4794:
4791:
4788:
4785:
4782:
4776:
4771:
4757:
4738:
4735:
4732:
4729:
4726:
4720:
4706:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4661:
4658:
4650:
4647:
4632:
4629:
4626:
4603:
4600:
4597:
4591:
4588:
4585:
4578:
4575:
4572:
4561:
4558:
4557:
4556:
4539:
4536:
4533:
4530:
4527:
4521:
4488:
4485:
4482:
4479:
4473:
4470:
4467:
4461:
4453:
4450:
4449:
4448:
4434:
4426:
4422:
4421:upper subsets
4406:
4386:
4378:
4377:partial order
4362:
4339:
4336:
4333:
4330:
4327:
4316:
4308:
4306:
4304:
4300:
4299:Alfred Tarski
4296:
4292:
4288:
4284:
4280:
4262:
4254:
4229:
4221:
4209:
4193:
4183:
4107:
4097:
4078:
4064:
4044:
4018:
4015:
4012:
4006:
4003:
4000:
3997:
3990:
3987:
3984:
3973:
3969:
3965:
3958:
3957:
3956:
3942:
3922:
3914:
3896:
3869:
3839:
3823:
3815:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3722:
3694:
3691:
3688:
3685:
3682:
3676:
3671:
3657:
3652:
3636:
3599:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3533:
3510:
3507:
3504:
3501:
3498:
3495:
3489:
3469:
3465:
3449:
3426:
3423:
3420:
3417:
3414:
3408:
3395:
3386:
3378:
3371:
3369:
3355:
3347:
3331:
3323:
3307:
3287:
3263:
3259:
3256:
3253:
3250:
3227:
3203:
3199:
3196:
3193:
3190:
3167:
3147:
3135:
3119:
3116:
3113:
3110:
3102:
3099:
3095:
3090:
3086:
3079:
3065:
3059:
3056:
3053:
3047:
3044:
3041:
3038:
3031:
3028:
3025:
3014:
3011:
3004:
2999:
2995:
2992:
2988:
2983:
2979:
2969:
2968:
2967:
2953:
2932:
2927:
2923:
2920:
2917:
2894:
2888:
2885:
2882:
2876:
2873:
2870:
2863:
2860:
2857:
2846:
2843:
2840:
2833:
2832:
2831:
2830:
2811:
2808:
2805:
2802:
2799:
2793:
2765:
2762:
2759:
2756:
2753:
2747:
2734:
2715:
2704:
2700:
2696:
2691:
2687:
2683:
2680:
2673:
2670:
2667:
2661:
2658:
2655:
2652:
2646:
2643:
2636:
2635:
2634:
2617:
2614:
2611:
2608:
2603:
2599:
2575:
2552:
2549:
2546:
2543:
2538:
2534:
2510:
2487:
2484:
2481:
2478:
2475:
2469:
2444:
2441:
2438:
2413:
2409:
2405:
2400:
2396:
2392:
2387:
2383:
2376:
2371:
2356:
2336:
2333:
2330:
2310:
2307:
2301:
2293:
2290:
2286:
2278:
2277:
2276:
2259:
2256:
2253:
2227:
2224:
2221:
2198:
2178:
2145:
2142:
2134:
2130:
2111:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2081:
2078:
2071:
2070:
2069:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2017:
1997:
1977:
1954:
1951:
1948:
1922:
1919:
1916:
1890:
1887:
1884:
1881:
1878:
1872:
1859:
1840:
1837:
1834:
1831:
1828:
1822:
1809:
1790:
1773:
1770:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1733:
1730:
1723:
1722:
1721:
1707:
1684:
1681:
1678:
1675:
1672:
1661:
1642:
1638:
1635:
1632:
1629:
1614:
1612:
1609:
1603:
1600:
1597:
1594:
1579:
1571:
1568:
1565:
1549:
1541:
1538:
1523:
1519:
1515:
1492:
1489:
1486:
1480:
1477:
1474:
1471:
1464:
1461:
1458:
1447:
1444:
1429:
1426:
1423:
1400:
1397:
1394:
1388:
1385:
1382:
1375:
1372:
1369:
1358:
1355:
1354:
1353:
1336:
1333:
1330:
1327:
1324:
1318:
1304:
1302:
1294:
1292:
1278:
1270:
1251:
1215:
1202:
1199:
1196:
1193:
1167:
1164:
1161:
1149:
1147:
1131:
1080:are valid in
1067:
1059:
1021:
1014:
998:
995:
992:
972:
952:
949:
946:
916:
900:
891:
877:
869:
853:
830:
827:
824:
821:
818:
815:
812:
789:
765:
758:
742:
739:
733:
730:
727:
724:
721:
718:
715:
705:
704:
703:
679:
660:
657:
654:
628:
625:
622:
619:
616:
593:
585:
559:
556:
553:
547:
543:
539:
532:
529:
526:
520:
517:
514:
511:
505:
502:
499:
492:, defined by
479:
471:
468:
464:
460:
456:
455:
454:
440:
420:
400:
392:
376:
353:
350:
347:
321:
318:
315:
312:
309:
303:
290:
282:
280:
278:
274:
270:
266:
262:
261:Kripke frames
258:
254:
250:
239:
236:
221:
218:
210:
207:November 2020
200:
196:
190:
189:
183:
179:
175:
170:
161:
160:
151:
148:
140:
137:November 2020
129:
126:
122:
119:
115:
112:
108:
105:
101:
98: –
97:
93:
92:Find sources:
86:
82:
76:
75:
70:This article
68:
64:
59:
58:
53:
51:
44:
43:
38:
37:
32:
27:
18:
17:
5250:Model theory
5228:
5217:
5190:
5095:
5043:of the form
4707:
4704:
4648:
4559:
4507:
4424:
4419:is a set of
4317:is a triple
4314:
4312:
4290:
4281:between the
4210:
4079:
4036:
3838:ultrafilters
3655:
3653:
3598:dual algebra
3597:
3391:
3345:
3321:
3139:
3136:Completeness
2909:
2732:
2730:
2354:
2352:
2132:
2128:
2126:
1857:
1805:
1659:
1618:
1610:
1607:
1601:
1595:
1569:
1539:
1445:
1356:
1305:
1300:
1298:
1267:denotes the
1150:
1145:
914:
892:
867:
781:
677:
581:
459:intersection
291:is a triple
288:
286:
269:intermediate
256:
252:
246:
231:
213:
204:
193:Please help
185:
143:
134:
124:
117:
110:
103:
91:
79:Please help
74:verification
71:
47:
40:
34:
33:Please help
30:
5245:Modal logic
5229:Modal Logic
5218:Modal Logic
3915:subsets of
3814:Stone space
3322:descriptive
2735:of a frame
1860:of a frame
1602:descriptive
846:belongs to
393:on the set
255:(or simply
199:introducing
5239:Categories
5211:References
4555:is called
3862:. The set
3656:dual frame
3464:subalgebra
2733:refinement
2323:for every
2129:p-morphism
1352:is called
893:A formula
755:for every
678:admissible
467:complement
283:Definition
107:newspapers
36:improve it
5166:⋅
5133:⋅
5107:∈
5072:∈
5066:∣
5060:∈
5001:inclusion
4987:≤
4921:⟩
4912:≤
4903:⟨
4868:⟩
4859:→
4853:∨
4847:∧
4838:⟨
4810:⟩
4807:∅
4801:→
4795:∪
4789:∩
4780:⟨
4742:⟩
4733:≤
4724:⟨
4683:∈
4677:∣
4671:−
4662:∪
4630:≤
4601:∈
4595:⇔
4589:∈
4576:∈
4570:∀
4543:⟩
4534:≤
4525:⟨
4486:∪
4480:−
4474:◻
4465:→
4363:≤
4343:⟩
4334:≤
4325:⟨
4255:⋅
4222:⋅
4016:∈
4010:⇒
4004:∈
3998:◻
3988:∈
3982:∀
3978:⟺
3800:⟩
3797:−
3791:∨
3785:∧
3776:⟨
3756:⟩
3753:◻
3747:−
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3735:∧
3726:⟨
3698:⟩
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3575:−
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3563:∩
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3534:◻
3514:⟩
3511:−
3505:∪
3499:∩
3477:⟨
3430:⟩
3412:⟨
3268:⟩
3264:⊩
3248:⟨
3208:⟩
3204:⊩
3188:⟨
3114:∈
3107:⟺
3100:∈
3096:∼
3057:∈
3051:⇒
3045:◻
3042:∈
3029:∈
3023:∀
3019:⟺
3012:∈
3009:⟩
3005:∼
2989:∼
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2954:∼
2933:∼
2886:∈
2880:⇔
2874:∈
2861:∈
2855:∀
2851:⟺
2844:∼
2815:⟩
2797:⟨
2769:⟩
2751:⟨
2697:∈
2684:∩
2671:∈
2665:∀
2662:∣
2656:⊆
2615:∈
2609:∣
2550:∈
2544:∣
2491:⟩
2473:⟨
2442:∈
2419:⟩
2380:⟨
2334:∈
2308:∈
2291:−
2263:⟩
2251:⟨
2231:⟩
2219:⟨
2154:→
2146::
2103:∈
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2091:∩
2053:×
2047:∩
1958:⟩
1946:⟨
1926:⟩
1914:⟨
1894:⟩
1876:⟨
1844:⟩
1826:⟨
1771:∣
1762:⊩
1756:∣
1750:∈
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1670:⟨
1647:⟩
1643:⊩
1627:⟨
1562:with the
1490:∈
1484:⇒
1478:◻
1475:∈
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1456:∀
1398:∈
1392:⇔
1386:∈
1373:∈
1367:∀
1340:⟩
1322:⟨
1269:power set
1222:⟩
1191:⟨
1171:⟩
1159:⟨
996:∈
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950:⊩
828:⊩
822:∣
816:∈
740:∈
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725:∣
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652:⟨
632:⟩
629:⊩
614:⟨
557:∈
551:→
530:∈
524:∀
521:∣
515:∈
500:◻
480:◻
357:⟩
345:⟨
325:⟩
307:⟨
42:talk page
5199:See also
4933:, where
4619:implies
4355:, where
4289:(called
4283:category
2910:and let
2503:, where
1700:, where
1508:implies
1416:implies
1306:A frame
1234:, where
1058:theorems
870:formula
337:, where
4754:is the
4649:compact
4287:duality
2068:), and
1970:(i.e.,
1660:induces
1596:refined
1540:compact
413:), and
195:improve
121:scholar
5096:where
5003:, and
4399:, and
4293:after
3913:clopen
3812:has a
3396:. Let
3280:is an
2633:, and
2030:, and
1570:atomic
1301:models
465:, and
259:) are
257:frames
123:
116:
109:
102:
94:
4562:, if
4560:tight
4427:) of
4425:cones
4375:is a
3220:: as
1856:is a
1572:, if
1448:, if
1446:tight
1359:, if
1146:frame
1060:) of
939:, if
915:valid
868:every
702:, if
463:union
389:is a
265:modal
249:logic
180:, or
128:JSTOR
114:books
5155:and
4297:and
4244:and
4057:and
2731:The
2353:The
2243:and
2131:(or
1011:. A
866:for
275:and
267:and
100:news
4999:is
4957:of
4379:on
3840:of
3600:of
3348:of
2191:to
1271:of
1124:an
917:in
913:is
676:is
247:In
83:by
5241::
5195:.
4305:.
4208:.
4077:.
3651:.
2568:,
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2127:A
1291:.
1148:.
461:,
287:A
251:,
184:,
176:,
45:.
5174:+
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5163:(
5141:+
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5130:(
5110:A
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5081:,
5078:}
5075:x
5069:a
5063:F
5057:x
5054:{
5031:F
5011:V
4966:A
4941:F
4918:V
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4909:,
4906:F
4900:=
4895:+
4890:A
4865:0
4862:,
4856:,
4850:,
4844:,
4841:A
4835:=
4831:A
4804:,
4798:,
4792:,
4786:,
4783:V
4777:=
4772:+
4767:F
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4730:,
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4721:=
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4689:}
4686:V
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4522:=
4518:F
4504:.
4492:)
4489:B
4483:A
4477:(
4471:=
4468:B
4462:A
4435:F
4423:(
4407:V
4387:F
4340:V
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4331:,
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4263:+
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4252:(
4230:+
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4219:(
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4098:+
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4045:x
4022:)
4019:y
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4007:x
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3995:(
3991:A
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3824:F
3794:,
3788:,
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3750:,
3744:,
3738:,
3732:,
3729:A
3723:=
3719:A
3695:V
3692:,
3689:R
3686:,
3683:F
3677:=
3672:+
3667:A
3637:+
3632:F
3609:F
3578:,
3572:,
3566:,
3560:,
3557:V
3508:,
3502:,
3496:,
3493:)
3490:F
3487:(
3482:P
3450:V
3427:V
3424:,
3421:R
3418:,
3415:F
3409:=
3405:F
3356:L
3332:L
3308:L
3288:L
3260:,
3257:R
3254:,
3251:F
3228:L
3200:,
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3194:,
3191:F
3168:L
3148:L
3120:.
3117:V
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3103:W
3091:/
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3063:)
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3054:y
3048:A
3039:x
3036:(
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3000:/
2996:y
2993:,
2984:/
2980:x
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2924:F
2921:=
2918:G
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2892:)
2889:A
2883:y
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2868:(
2864:V
2858:A
2847:y
2841:x
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2809:,
2806:S
2803:,
2800:G
2794:=
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2757:,
2754:F
2748:=
2744:F
2716:.
2713:}
2710:)
2705:i
2701:V
2692:i
2688:F
2681:A
2678:(
2674:I
2668:i
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2653:A
2650:{
2647:=
2644:V
2621:}
2618:I
2612:i
2604:i
2600:R
2596:{
2576:R
2556:}
2553:I
2547:i
2539:i
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2531:{
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2476:F
2470:=
2466:F
2445:I
2439:i
2414:i
2410:V
2406:,
2401:i
2397:R
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2388:i
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2377:=
2372:i
2367:F
2349:.
2337:W
2331:A
2311:V
2305:]
2302:A
2299:[
2294:1
2287:f
2260:S
2257:,
2254:G
2228:R
2225:,
2222:F
2199:G
2179:F
2158:G
2150:F
2143:f
2112:.
2109:}
2106:V
2100:A
2094:G
2088:A
2085:{
2082:=
2079:W
2056:G
2050:G
2044:R
2041:=
2038:S
2018:R
1998:F
1978:G
1955:R
1952:,
1949:F
1923:S
1920:,
1917:G
1891:V
1888:,
1885:R
1882:,
1879:F
1873:=
1869:F
1841:W
1838:,
1835:S
1832:,
1829:G
1823:=
1819:G
1791:.
1786:}
1774:A
1768:}
1765:A
1759:x
1753:F
1747:x
1744:{
1739:{
1734:=
1731:V
1708:V
1685:V
1682:,
1679:R
1676:,
1673:F
1639:,
1636:R
1633:,
1630:F
1580:V
1550:V
1536:,
1524:y
1520:R
1516:x
1496:)
1493:A
1487:y
1481:A
1472:x
1469:(
1465:V
1459:A
1442:,
1430:y
1427:=
1424:x
1404:)
1401:A
1395:y
1389:A
1383:x
1380:(
1376:V
1370:A
1337:V
1334:,
1331:R
1328:,
1325:F
1319:=
1315:F
1279:F
1255:)
1252:F
1249:(
1244:P
1219:)
1216:F
1213:(
1208:P
1203:,
1200:R
1197:,
1194:F
1168:R
1165:,
1162:F
1144:-
1132:L
1111:F
1089:F
1068:L
1043:F
1022:L
999:F
993:x
953:A
947:x
926:F
901:A
878:A
854:V
834:}
831:A
825:x
819:F
813:x
810:{
790:V
778:.
766:p
743:V
737:}
734:p
728:x
722:F
716:x
713:{
689:F
661:R
658:,
655:F
626:,
623:R
620:,
617:F
594:V
578:.
566:}
563:)
560:A
554:y
548:y
544:R
540:x
537:(
533:F
527:y
518:F
512:x
509:{
506:=
503:A
469:,
441:F
421:V
401:F
377:R
354:R
351:,
348:F
322:V
319:,
316:R
313:,
310:F
304:=
300:F
238:)
232:(
220:)
214:(
209:)
205:(
191:.
150:)
144:(
139:)
135:(
125:·
118:·
111:·
104:·
77:.
52:)
48:(
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