2923:
Kripke semantics in which sentences of the theory hold. Moving from the field of sets to a
Boolean space somewhat obfuscates this connection. By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and Kripke frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics. The pre-order can be obtained as the specialization pre-order of the McKinsey–Tarski topology. The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates. Via a functor that instead replaces the pre-order with its corresponding Alexandrov topology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico-reflection of the McKinsey–Tarski topology. The approach of formulating a topological duality for interior algebras using both the Stone topology of the Jónsson–Tarski approach and the Alexandrov topology of the pre-order to form a bi-topological space has been investigated by G. Bezhanishvili, R.Mines, and P.J. Morandi. The McKinsey–Tarski topology of an interior algebra is the intersection of the former two topologies.
679:
applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C. Naturman returned to
Sikorski's approach while dropping σ-completeness to produce topomorphisms as defined above. In this terminology, Sikorski's original "continuous homomorphisms"
2890:
Whereas the Jónsson–Tarski generalization of Stone duality applies to
Boolean algebras with operators in general, the connection between interior algebras and topology allows for another method of generalizing Stone duality that is unique to interior algebras. An intermediate step in the development
2922:
The pre-order obtained in the Jónsson–Tarski approach corresponds to the accessibility relation in the Kripke semantics for an S4 theory, while the intermediate field of sets corresponds to a representation of the
Lindenbaum–Tarski algebra for the theory using the sets of possible worlds in the
638:
producing a dual of a continuous map rather than a generalization. On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ-complete homomorphisms but
2583:
can be represented as the open elements of an interior algebra and the latter may be chosen to be an interior algebra generated by its open elements—such interior algebras correspond one-to-one with
Heyting algebras (up to isomorphism) being the free Boolean extensions of the latter.
2919:, C. Naturman showed that this approach can be formalized as a category theoretic Stone duality in which the usual Stone duality for Boolean algebras corresponds to the case of interior algebras having redundant interior operator (Boolean interior algebras).
2578:
pseudo-complemented elements of these algebras respectively and thus form
Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every
276:
pairs, and are paradigmatic instances of "Boolean algebras with operators." The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm following the work of
344:
if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary
Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of
2871:. In the case of interior algebras the interior (or closure) operator corresponds to a pre-order on the Boolean space. Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as
557:
Early research often considered mappings between interior algebras that were homomorphisms of the underlying
Boolean algebras but that did not necessarily preserve the interior or closure operator. Such mappings were called
2498:
can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic
Boolean algebras are then precisely the
2511:. In other words, they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the
1715:
of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying
Boolean algebra of the interior algebra. Moreover, given a neighbourhood function
1145:
Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties:
1534:
Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added relations, so that standard results from
2911:
and Tarski showed that by generating a topology equivalent to using only the complexes that correspond to open elements as a basis, a representation of an interior algebra is obtained as a
2623:. The one-to-one correspondence between Heyting algebras and interior algebras generated by their open elements reflects the correspondence between extensions of intuitionistic logic and
1328:
2864:
2915:—a field of sets on a topological space that is closed with respect to taking interiors or closures. By equipping topological fields of sets with appropriate morphisms known as
610:. This is a Boolean homomorphism, preserves unions of sequences and includes the closure of an inverse image in the inverse image of the closure. Sikorski thus defined a
2879:
for short. This generalization of Stone duality to interior algebras based on the Jónsson–Tarski representation was investigated by Leo Esakia and is also known as the
2940:
2554:) can be interpreted in the Kripke semantics using monadic universal and existential quantification, respectively, without reference to an accessibility relation.
1775:
2904:
1846:
1230:
2892:
956:) are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called
260:′))′. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers
671:). This generalizes the forward image map of a continuous map—the image of a closure is contained in the closure of the image. This construction is
272:, ·, +, ′, 0, 1⟩ is again a Boolean algebra and satisfies the above identities for the closure operator. Closure and interior algebras form
2868:
570:
were used in the case where these were preserved, but this terminology is now redundant as the standard definition of a homomorphism in
321:. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called
2674:
2646:
1783:
in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.
574:
requires that it preserves all operations.) Applications involving countably complete interior algebras (in which countable
2768:, respectively. Thus every derivative algebra can be regarded as an interior algebra. Moreover, given an interior algebra
1484:
Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space
549:.) Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism.
2607:. The relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and
2526:
In the relationship between preordered sets and interior algebras they correspond to the case where the preorder is an
2071:, and the connection between interior algebras and preorders is deeply related to their connection with modal logic.
3014:
2302:
and Kripke frames. In this regard, interior algebras are particularly interesting because of their connection to
1081:
3092:
2690:
2588:
2539:
1874:
1076:. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If
2847:
provides a category theoretic duality between Boolean algebras and a class of topological spaces known as
3097:
2678:
2600:
2500:
2124:
2024:
1547:
996:
903:
110:
59:
47:
940:
3107:
2912:
2899:. The Stone topology of the corresponding Boolean space is then generated using the field of sets as a
1246:
349:
interior algebras, which are the single element interior algebras characterized by the identity 0 = 1.
3009:
2698:
304:
3102:
2982:
3063:
2967:
2495:
1355:
1301:
2575:
2567:
1101:
672:
635:
273:
205:
175:
2298:
This construction and representation theorem is a special case of the more general result for
1276:
i.e. its interior and closure operators distribute over arbitrary meets and joins respectively
3028:
2981:
According to footnote 19 in McKinsey and Tarski, 1944, the result had been proved earlier by
2004:
906:
881:. The open, closed, regular open, regular closed and clopen elements of the interior algebra
3112:
2944:
2592:
2527:
1585:
1312:
1751:
8:
3049:
2962:
2604:
2570:
Heyting algebra. The regular open elements and regular closed elements correspond to the
2320:
2001:
1927:
1375:
1257:
579:
575:
363:
79:
55:
32:
28:
2515:
interior algebras. They are also the interior algebras corresponding to the modal logic
1853:
i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.
2932:
2624:
2512:
676:
252:
If the closure operator is taken as primitive, the interior operator can be defined as
2900:
2616:
2324:
2316:
1535:
1367:
926:
694:
571:
83:
2856:
1981:
true, while the closed elements correspond to those that are only false if they are
1786:
In terms of neighbourhood functions, the open elements are precisely those elements
2947:
of elementary classes of interior algebras with hereditarily undecidable theories.
2936:
2650:
2571:
2064:
1179:
323:
180:
20:
2867:
by equipping Boolean spaces with relations that correspond to the operators via a
2908:
2580:
2563:
2530:, reflecting the fact that such preordered sets provide the Kripke semantics for
2044:
2028:
1285:
1089:
1948:) is an interior algebra. The interior operator in this case corresponds to the
938:
in such an interior algebra giving a representation of an interior algebra as a
893:) are just the open, closed, regular open, regular closed and clopen subsets of
2884:
1862:
1206:
1195:
968:
607:
599:
394:
114:
1370:
of open subsets, motivates an alternative formulation of interior algebras: A
3086:
3023:
2896:
2860:
2848:
2844:
2620:
2535:
2299:
2288:
2040:
2036:
2032:
1961:
1339:
603:
586:) typically made use of countably complete Boolean homomorphisms also called
63:
2048:
1531:. Thus generalized topological spaces are equivalent to interior algebras.
471:
367:
1425:
is closed under arbitrary joins (i.e. if a join of an arbitrary subset of
2852:
2851:. Building on nascent ideas of relational semantics (later formalized by
2839:
2705:
as interior/closure algebras stand to topological interiors/closures and
2051:. The Kripke frames corresponding to interior algebras are precisely the
1949:
1842:
910:
40:
1527:
thereby producing an interior algebra whose open elements are precisely
333:
309:
51:
2263:
2121:
1541:
1155:
1092:
of complete atomic interior algebras and complete topomorphisms then
935:
711:
2035:
on appropriate relational structures. In particular, since they are
1960:). This construction is a special case of a more general result for
680:
are σ-complete topomorphisms between σ-complete interior algebras.)
3045:
Topo-canonical completions of closure algebras and Heyting algebras
2943:(all its subtheories are undecidable) and demonstrated an infinite
2307:
2303:
2279:
2075:
2056:
2052:
1139:
634:). This definition had several difficulties: The construction acts
598:
The earliest generalization of continuity to interior algebras was
455:
299:
278:
36:
3040:, Ph.D. thesis, University of Cape Town Department of Mathematics.
2611:, in which one can interpret theories of intuitionistic logic as
2538:
of quantification (for which monadic Boolean algebras provide an
1121:
1938:/ ~ is the set of equivalence classes under this relation. Then
454:
Topomorphisms are another important, and more general, class of
683:
400:
is a homomorphism between the underlying Boolean algebras of
2448:), and the closed sets are the ones for which every outside
2718:
with derivative operator , we can form an interior algebra
774:
where is the usual topological interior operator. For all
1588:. This leads to another formulation of interior algebras:
478:
is a homomorphism between the Boolean algebras underlying
1977:) correspond to sentences that are only true if they are
1841:
Neighbourhood functions may be defined more generally on
1366:
The modern formulation of topological spaces in terms of
289:
Elements of an interior algebra satisfying the condition
2670:
by using the closure operator as a derivative operator.
1849:. Interior algebras may thus be viewed as precisely the
2965:(1951), "Undecidability of some topological theories,"
1779:
will then be precisely the filter of neighbourhoods of
3043:
Bezhanishvili, G., Mines, R. and Morandi, P.J., 2008,
2907:
introduced by Tang Tsao-Chen for Lewis's modal logic,
2840:
Stone duality and representation for interior algebras
486:, that also preserves the open and closed elements of
2985:
in 1939, but remained unpublished and not accessible
1754:
408:, that also preserves interiors and closures. Hence:
1906:
where ~ is the equivalence relation on sentences in
1550:
can be generalized to interior algebras: An element
1398:, ·, +, ′, 0, 1⟩ is a Boolean algebra as usual, and
331:. Elements that are both open and closed are called
618:between two σ-complete interior algebras such that
2881:Esakia duality for S4-algebras (interior algebras)
2534:. This also reflects the relationship between the
1769:
1542:Neighbourhood functions and neighbourhood lattices
2739:, with interior and closure operators defined by
2697:. Hence derivative algebras stand to topological
2440:In other words, the open sets are the ones whose
352:
208:, the closure operator satisfies the identities:
3084:
2677:. From this perspective, they are precisely the
2595:that interior algebras play for the modal logic
2562:The open elements of an interior algebra form a
1956:), while the closure operator corresponds to ◊ (
651:for interior algebras as a Boolean homomorphism
2681:of derivative algebras satisfying the identity
2645:, the closure operator obeys the axioms of the
2287:giving the above-mentioned representation as a
2185:The corresponding closure operator is given by
825:the corresponding closure operator is given by
3010:Intuitionistic logic and modality via topology
2830:necessarily hold for every derivative algebra
2689:. Derivative algebras provide the appropriate
2664:) with the same underlying Boolean algebra as
2503:of interior algebras satisfying the identity
899:respectively in the usual topological sense.
2733:with the same underlying Boolean algebra as
3073:Closure homomorphisms and interior mappings
3059:On the compactification of closure algebras
2939:. Naturman demonstrated that the theory is
2489:
934:. Moreover, every interior algebra can be
684:Relationships to other areas of mathematics
590:—these preserve countable meets and joins.
284:
2895:, which represents a Boolean algebra as a
2087:, «⟩ we can construct an interior algebra
1124:that is a dual isomorphism of categories.
639:included σ-completeness in his axioms for
2863:and G. Hansoul extended Stone duality to
2131:where the interior operator is given by
1992:, interior algebras are sometimes called
1719:on a Boolean algebra with underlying set
655:between two interior algebras satisfying
327:and closures of open elements are called
31:that encodes the idea of the topological
1820:if and only if there is an open element
1723:, we can define an interior operator by
552:
1746:thereby obtaining an interior algebra.
1554:of an interior algebra is said to be a
1361:
593:
313:and are characterized by the condition
3085:
3005:Ph.D. thesis, University of Amsterdam.
2987:in view of the present war conditions
2634:
2007:, who first proposed the modal logics
1500:we can define an interior operator on
541:(Such morphisms have also been called
362:Interior algebras, by virtue of being
2023:Since interior algebras are (normal)
740:and extend it to an interior algebra
35:of a set. Interior algebras are to
3026:, 1944, "The Algebra of Topology,"
2557:
2444:are inaccessible from outside (the
2389:The corresponding closed sets are:
2266:in an interior algebra of the form
1134:) is a homomorphism if and only if
913:to an interior algebra of the form
871:is the smallest closed superset of
13:
2926:
1845:producing the structures known as
1603:to its set of filters, such that:
1595:on a Boolean algebra is a mapping
944:. The properties of the structure
547:closure algebra semi-homomorphisms
14:
3124:
2452:is inaccessible from inside (the
458:between interior algebras. A map
3015:Annals of Pure and Applied Logic
2855:) and a result of R. S. Pierce,
2306:. The construction provides the
2262:. Every interior algebra can be
1329:finitely subdirectly irreducible
449:
357:
178:of the interior operator is the
3003:Varieties of interior algebras,
2865:Boolean algebras with operators
2566:and the closed elements form a
2519:, and so have also been called
1572:. The set of neighbourhoods of
1342:ultra-connected if and only if
971:between two topological spaces
3038:Interior Algebras and Topology
2975:
2956:
2893:Stone's representation theorem
2883:and is closely related to the
2627:extensions of the modal logic
1856:
1851:Boolean neighbourhood lattices
1764:
1758:
1082:category of topological spaces
863:is the largest open subset of
370:. Given two interior algebras
353:Morphisms of interior algebras
340:An interior algebra is called
1:
2995:
2546:where the modal operators □ (
2039:, they can be represented as
2031:, they can be represented by
1988:Because of their relation to
1372:generalized topological space
643:.) Later J. Schmid defined a
392:interior algebra homomorphism
121:, satisfying the identities:
69:
2018:
1847:neighbourhood (semi)lattices
1805:. In terms of open elements
1439:is closed under finite meets
1260:(Alexandrov) if and only if
962:topological Boolean algebras
614:as a Boolean σ-homomorphism
307:of open elements are called
7:
2712:Given a derivative algebra
2673:Thus interior algebras are
1546:The topological concept of
688:
268:, ·, +, ′, 0, 1, ⟩, where ⟨
58:. Interior algebras form a
10:
3129:
3077:Fundamenta Mathematicae 41
2639:Given an interior algebra
1865:(set of formal sentences)
1668:if and only if there is a
1429:exists then it will be in
582:always exist, also called
113:and postfix designates a
2913:topological field of sets
1875:Lindenbaum–Tarski algebra
941:topological field of sets
2950:
2941:hereditarily undecidable
2649:, . Hence we can form a
2490:Monadic Boolean algebras
1599:from its underlying set
1481:in the Boolean algebra.
1084:and continuous maps and
568:topological homomorphism
285:Open and closed elements
3064:Fundamenta Mathematicae
2968:Fundamenta Mathematicae
2931:Grzegorczyk proved the
2496:monadic Boolean algebra
2338:) whose open sets are:
2043:on a set with a single
1402:is a unary relation on
1356:subdirectly irreducible
1302:directly indecomposable
645:continuous homomorphism
612:continuous homomorphism
588:Boolean σ-homomorphisms
3036:Naturman, C.A., 1991,
3022:McKinsey, J.C.C. and
2887:for Heyting algebras.
2869:power set construction
1771:
1593:neighbourhood function
337:. 0 and 1 are clopen.
3029:Annals of Mathematics
2905:topological semantics
2540:algebraic description
2254:accessible from some
1967:The open elements of
1772:
1122:contravariant functor
958:topo-Boolean algebras
675:but not suitable for
560:Boolean homomorphisms
553:Boolean homomorphisms
27:is a certain type of
3093:Algebraic structures
2935:of closure algebras
2891:of Stone duality is
2693:for the modal logic
2593:intuitionistic logic
2528:equivalence relation
1928:logically equivalent
1770:{\displaystyle N(x)}
1752:
1479:generalized topology
1362:Generalized topology
909:interior algebra is
594:Continuous morphisms
564:closure homomorphism
543:stable homomorphisms
364:algebraic structures
206:principle of duality
3071:Sikorski R., 1955,
3050:Algebra Universalis
3008:Esakia, L., 2004, "
2983:Stanisław Jaśkowski
2963:Andrzej Grzegorczyk
2873:pseudo-epimorphisms
2691:algebraic semantics
2675:derivative algebras
2647:derivative operator
2635:Derivative algebras
2605:propositional logic
2572:pseudo-complemented
2321:Alexandrov topology
2067:of the modal logic
1869:in the modal logic
1843:(meet)-semilattices
1376:algebraic structure
714:Boolean algebra of
710:⟩ one can form the
649:continuous morphism
80:algebraic structure
56:propositional logic
29:algebraic structure
16:Algebraic structure
3098:Mathematical logic
3057:Schmid, J., 1973,
3001:Blok, W.A., 1976,
2933:first-order theory
2903:. Building on the
2651:derivative algebra
2589:play the same role
2250:is the set of all
2238:inaccessible from
2234:is the set of all
1873:, we can form its
1767:
1442:For every element
1258:finitely generated
677:category theoretic
93:, ·, +, ′, 0, 1, ⟩
3108:Closure operators
2901:topological basis
2587:Heyting algebras
2325:topological space
2201:| there exists a
1964:and modal logic.
1536:universal algebra
1492:, ·, +, ′, 0, 1,
1386:, ·, +, ′, 0, 1,
1274:operator complete
1102:dually isomorphic
927:topological space
782:it is defined by
695:topological space
602:'s, based on the
572:universal algebra
119:interior operator
3120:
2990:
2979:
2973:
2960:
2825:
2799:
2767:
2753:
2732:
2601:Boolean algebras
2558:Heyting algebras
2435:
2384:
2218:
2172:
2116:
2065:Kripke semantics
2025:Boolean algebras
1837:
1819:
1804:
1778:
1776:
1774:
1773:
1768:
1745:
1706:
1691:
1677:
1667:
1637:
1571:
1526:
1477:is said to be a
1469:
1419:
1138:is a continuous
1119:
1066:for all subsets
1038:
995:we can define a
991:
856:
813:
769:
736:
641:closure algebras
262:closure algebras
181:closure operator
105:, ·, +, ′, 0, 1⟩
76:interior algebra
48:Boolean algebras
25:interior algebra
21:abstract algebra
3128:
3127:
3123:
3122:
3121:
3119:
3118:
3117:
3103:Boolean algebra
3083:
3082:
2998:
2993:
2980:
2976:
2961:
2957:
2953:
2929:
2927:Metamathematics
2842:
2801:
2775:
2755:
2740:
2719:
2637:
2581:Heyting algebra
2564:Heyting algebra
2560:
2492:
2393:
2342:
2189:
2135:
2125:Boolean algebra
2111:), ∩, ∪, ′, ø,
2091:
2057:Preordered sets
2053:preordered sets
2045:binary relation
2021:
1918:if and only if
1859:
1825:
1806:
1791:
1753:
1750:
1749:
1747:
1724:
1693:
1679:
1669:
1642:
1608:
1563:
1544:
1505:
1451:
1414:
1364:
1315:if and only if
1288:if and only if
1233:if and only if
1231:almost discrete
1209:if and only if
1182:if and only if
1158:if and only if
1105:
1003:
975:
829:
786:
764:), ∩, ∪, ′, ø,
744:
731:), ∩, ∪, ′, ø,
722:
691:
686:
636:contravariantly
596:
555:
533:) is closed in
474:if and only if
452:
360:
355:
287:
111:Boolean algebra
72:
17:
12:
11:
5:
3126:
3116:
3115:
3110:
3105:
3100:
3095:
3081:
3080:
3069:
3055:
3041:
3034:
3020:
3006:
2997:
2994:
2992:
2991:
2974:
2954:
2952:
2949:
2928:
2925:
2885:Esakia duality
2849:Boolean spaces
2841:
2838:
2636:
2633:
2559:
2556:
2491:
2488:
2438:
2437:
2387:
2386:
2323:, producing a
2308:preordered set
2300:modal algebras
2293:preorder field
2280:preordered set
2229:
2228:
2183:
2182:
2118:
2117:
2076:preordered set
2063:) provide the
2041:fields of sets
2037:modal algebras
2033:fields of sets
2020:
2017:
1998:Lewis algebras
1962:modal algebras
1950:modal operator
1904:
1903:
1894:/ ~, ∧, ∨, ¬,
1858:
1855:
1766:
1763:
1760:
1757:
1709:
1708:
1639:
1584:) and forms a
1576:is denoted by
1558:of an element
1548:neighbourhoods
1543:
1540:
1498:
1497:
1472:
1471:
1440:
1434:
1420:
1392:
1391:
1363:
1360:
1359:
1358:
1331:
1313:ultraconnected
1304:
1277:
1249:
1222:
1198:
1171:
1064:
1063:
1040:
1039:
1025:) →
1013:) :
993:
992:
969:continuous map
858:
857:
815:
814:
772:
771:
738:
737:
690:
687:
685:
682:
608:continuous map
595:
592:
554:
551:
539:
538:
515:
451:
448:
447:
446:
428:
395:if and only if
359:
356:
354:
351:
329:regular closed
286:
283:
250:
249:
246:
228:
219:
196:is called the
163:is called the
158:
157:
154:
141:
132:
115:unary operator
107:
106:
95:
94:
71:
68:
64:modal algebras
15:
9:
6:
4:
3:
2:
3125:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3090:
3088:
3078:
3074:
3070:
3067:
3065:
3060:
3056:
3053:
3051:
3046:
3042:
3039:
3035:
3032:
3030:
3025:
3024:Alfred Tarski
3021:
3018:
3016:
3011:
3007:
3004:
3000:
2999:
2988:
2984:
2978:
2971:
2969:
2964:
2959:
2955:
2948:
2946:
2942:
2938:
2934:
2924:
2920:
2918:
2914:
2910:
2906:
2902:
2898:
2897:field of sets
2894:
2888:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2845:Stone duality
2837:
2835:
2834:
2829:
2824:
2823:
2818:
2817:
2812:
2811:
2806:
2805:
2798:
2797:
2792:
2791:
2786:
2785:
2780:
2779:
2773:
2772:
2766:
2762:
2758:
2751:
2747:
2743:
2738:
2737:
2730:
2729:
2724:
2723:
2717:
2716:
2710:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2671:
2669:
2668:
2663:
2662:
2657:
2656:
2652:
2648:
2644:
2643:
2632:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2598:
2594:
2590:
2585:
2582:
2577:
2574:elements and
2573:
2569:
2565:
2555:
2553:
2549:
2545:
2541:
2537:
2536:monadic logic
2533:
2529:
2524:
2522:
2518:
2514:
2510:
2506:
2502:
2497:
2487:
2485:
2484:
2479:
2478:
2473:
2472:
2467:
2466:
2461:
2460:
2456:). Moreover,
2455:
2451:
2447:
2443:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2397:
2392:
2391:
2390:
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2341:
2340:
2339:
2337:
2336:
2331:
2330:
2326:
2322:
2318:
2314:
2313:
2309:
2305:
2301:
2296:
2294:
2290:
2289:field of sets
2286:
2285:
2281:
2277:
2276:
2271:
2270:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2226:
2222:
2216:
2212:
2208:
2204:
2200:
2196:
2192:
2188:
2187:
2186:
2180:
2176:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2133:
2132:
2130:
2126:
2123:
2114:
2110:
2106:
2102:
2101:
2096:
2095:
2090:
2089:
2088:
2086:
2082:
2081:
2077:
2072:
2070:
2066:
2062:
2059:(also called
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2016:
2014:
2010:
2006:
2003:
1999:
1995:
1991:
1986:
1984:
1980:
1976:
1972:
1971:
1965:
1963:
1959:
1955:
1951:
1947:
1943:
1942:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1901:
1897:
1893:
1889:
1885:
1884:
1880:
1879:
1878:
1876:
1872:
1868:
1864:
1854:
1852:
1848:
1844:
1839:
1836:
1832:
1828:
1823:
1817:
1813:
1809:
1802:
1798:
1794:
1789:
1784:
1782:
1761:
1755:
1743:
1739:
1735:
1731:
1727:
1722:
1718:
1714:
1704:
1700:
1696:
1690:
1686:
1682:
1676:
1672:
1665:
1661:
1657:
1653:
1649:
1645:
1640:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1606:
1605:
1604:
1602:
1598:
1594:
1589:
1587:
1583:
1579:
1575:
1570:
1566:
1561:
1557:
1556:neighbourhood
1553:
1549:
1539:
1537:
1532:
1530:
1524:
1520:
1516:
1512:
1508:
1503:
1495:
1491:
1487:
1486:
1485:
1482:
1480:
1476:
1467:
1463:
1459:
1455:
1449:
1445:
1441:
1438:
1435:
1432:
1428:
1424:
1421:
1418:
1413:
1412:
1411:
1410:) such that:
1409:
1405:
1401:
1397:
1389:
1385:
1381:
1380:
1379:
1377:
1373:
1369:
1357:
1353:
1352:
1347:
1346:
1341:
1337:
1336:
1332:
1330:
1326:
1325:
1320:
1319:
1314:
1310:
1309:
1305:
1303:
1299:
1298:
1293:
1292:
1287:
1283:
1282:
1278:
1275:
1271:
1270:
1265:
1264:
1259:
1255:
1254:
1250:
1248:
1244:
1243:
1238:
1237:
1232:
1228:
1227:
1223:
1220:
1219:
1214:
1213:
1208:
1204:
1203:
1199:
1197:
1193:
1192:
1187:
1186:
1181:
1177:
1176:
1172:
1169:
1168:
1163:
1162:
1157:
1153:
1152:
1148:
1147:
1146:
1143:
1141:
1137:
1133:
1129:
1128:
1123:
1118:
1115: →
1114:
1111: :
1110:
1109:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1074:
1069:
1062:
1058:
1054:
1050:
1049:
1045:
1044:
1043:
1036:
1035:
1030:
1029:
1024:
1023:
1018:
1017:
1012:
1008:
1007:
1002:
1001:
1000:
999:topomorphism
998:
990:
989:
985: →
984:
983:
979: :
978:
974:
973:
972:
970:
965:
963:
959:
955:
954:
949:
948:
943:
942:
937:
933:
932:
928:
924:
923:
918:
917:
912:
908:
905:
900:
898:
897:
892:
891:
886:
885:
880:
879:
874:
870:
866:
862:
854:
853:
849:is closed in
848:
844:
840:
836:
832:
828:
827:
826:
824:
820:
811:
810:
805:
801:
797:
793:
789:
785:
784:
783:
781:
777:
767:
763:
759:
755:
754:
749:
748:
743:
742:
741:
734:
730:
726:
721:
720:
719:
717:
713:
709:
705:
701:
700:
696:
681:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
637:
633:
629:
625:
621:
617:
613:
609:
605:
604:inverse image
601:
591:
589:
585:
581:
577:
573:
569:
565:
562:. (The terms
561:
550:
548:
544:
536:
532:
528:
524:
521:is closed in
520:
516:
513:
510:) is open in
509:
505:
501:
497:
493:
492:
491:
489:
485:
481:
477:
473:
469:
465:
461:
457:
450:Topomorphisms
444:
440:
436:
432:
429:
426:
422:
418:
414:
411:
410:
409:
407:
403:
399:
396:
393:
389:
385:
381:
377:
373:
369:
368:homomorphisms
365:
358:Homomorphisms
350:
348:
343:
338:
336:
335:
330:
326:
325:
320:
316:
312:
311:
306:
302:
301:
296:
292:
282:
280:
275:
271:
267:
264:of the form ⟨
263:
259:
255:
247:
245:
241:
237:
233:
229:
227:
223:
220:
218:
214:
211:
210:
209:
207:
203:
199:
195:
191:
187:
183:
182:
177:
172:
170:
166:
162:
155:
153:
150:
146:
142:
140:
136:
133:
131:
127:
124:
123:
122:
120:
116:
112:
104:
100:
99:
98:
92:
88:
87:
86:
85:
81:
77:
67:
65:
61:
57:
54:and ordinary
53:
49:
45:
42:
38:
34:
30:
26:
22:
3076:
3072:
3062:
3058:
3048:
3044:
3037:
3027:
3013:
3002:
2986:
2977:
2966:
2958:
2930:
2921:
2916:
2889:
2880:
2876:
2872:
2843:
2832:
2831:
2827:
2821:
2820:
2815:
2814:
2809:
2808:
2803:
2802:
2795:
2794:
2789:
2788:
2783:
2782:
2777:
2776:
2770:
2769:
2764:
2760:
2756:
2749:
2745:
2741:
2735:
2734:
2727:
2726:
2721:
2720:
2714:
2713:
2711:
2706:
2702:
2699:derived sets
2694:
2686:
2682:
2672:
2666:
2665:
2660:
2659:
2654:
2653:
2641:
2640:
2638:
2628:
2612:
2608:
2596:
2586:
2561:
2551:
2547:
2543:
2531:
2525:
2520:
2516:
2508:
2504:
2493:
2482:
2481:
2476:
2475:
2470:
2469:
2464:
2463:
2458:
2457:
2453:
2449:
2445:
2441:
2439:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2388:
2380:
2376:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2334:
2333:
2328:
2327:
2311:
2310:
2297:
2292:
2283:
2282:
2274:
2273:
2268:
2267:
2259:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2230:
2224:
2220:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2184:
2178:
2174:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2136:
2128:
2119:
2112:
2108:
2104:
2099:
2098:
2093:
2092:
2084:
2079:
2078:
2073:
2068:
2060:
2049:Kripke frame
2022:
2012:
2008:
2000:, after the
1997:
1993:
1989:
1987:
1982:
1978:
1974:
1969:
1968:
1966:
1957:
1953:
1945:
1940:
1939:
1935:
1931:
1923:
1919:
1915:
1911:
1907:
1905:
1899:
1895:
1891:
1887:
1882:
1881:
1870:
1866:
1860:
1850:
1840:
1834:
1830:
1826:
1821:
1815:
1811:
1807:
1800:
1796:
1792:
1787:
1785:
1780:
1741:
1737:
1733:
1729:
1725:
1720:
1716:
1712:
1711:The mapping
1710:
1702:
1698:
1694:
1688:
1684:
1680:
1674:
1670:
1663:
1659:
1655:
1651:
1647:
1643:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1600:
1596:
1592:
1590:
1581:
1577:
1573:
1568:
1564:
1559:
1555:
1551:
1545:
1533:
1528:
1522:
1518:
1514:
1510:
1506:
1501:
1499:
1493:
1489:
1483:
1478:
1474:
1473:
1465:
1461:
1457:
1453:
1447:
1443:
1436:
1430:
1426:
1422:
1416:
1407:
1403:
1399:
1395:
1393:
1387:
1383:
1378:of the form
1371:
1365:
1350:
1349:
1344:
1343:
1334:
1333:
1323:
1322:
1317:
1316:
1307:
1306:
1296:
1295:
1290:
1289:
1280:
1279:
1273:
1268:
1267:
1262:
1261:
1252:
1251:
1241:
1240:
1235:
1234:
1225:
1224:
1221:) is Boolean
1217:
1216:
1211:
1210:
1201:
1200:
1190:
1189:
1184:
1183:
1174:
1173:
1170:) is trivial
1166:
1165:
1160:
1159:
1150:
1149:
1144:
1135:
1131:
1126:
1125:
1116:
1112:
1107:
1106:
1097:
1093:
1085:
1077:
1072:
1071:
1067:
1065:
1060:
1056:
1052:
1047:
1046:
1041:
1033:
1032:
1027:
1026:
1021:
1020:
1015:
1014:
1010:
1005:
1004:
994:
987:
986:
981:
980:
976:
966:
961:
957:
952:
951:
946:
945:
939:
930:
929:
921:
920:
915:
914:
901:
895:
894:
889:
888:
883:
882:
877:
876:
872:
868:
864:
860:
859:
851:
850:
846:
842:
838:
834:
830:
822:
818:
816:
808:
807:
803:
799:
795:
791:
787:
779:
775:
773:
765:
761:
757:
752:
751:
746:
745:
739:
732:
728:
724:
715:
707:
703:
698:
697:
692:
668:
664:
660:
656:
652:
648:
644:
640:
631:
627:
623:
619:
615:
611:
597:
587:
583:
567:
563:
559:
556:
546:
542:
540:
534:
530:
526:
522:
518:
511:
507:
503:
499:
495:
487:
483:
479:
475:
472:topomorphism
467:
463:
459:
453:
442:
438:
434:
430:
424:
420:
416:
412:
405:
401:
397:
391:
387:
383:
379:
375:
371:
361:
346:
341:
339:
332:
328:
324:regular open
322:
318:
314:
308:
298:
294:
290:
288:
269:
265:
261:
257:
253:
251:
243:
239:
235:
231:
225:
221:
216:
212:
201:
197:
193:
189:
185:
179:
173:
168:
164:
160:
159:
151:
148:
144:
138:
134:
129:
125:
118:
108:
102:
96:
90:
75:
73:
43:
24:
18:
3113:Modal logic
2937:undecidable
2877:p-morphisms
2800:. However,
2548:necessarily
2521:S5 algebras
2278:) for some
2047:, called a
2005:C. I. Lewis
1994:S4 algebras
1983:necessarily
1979:necessarily
1954:necessarily
1857:Modal logic
1450:, the join
1406:(subset of
925:) for some
806:is open in
498:is open in
305:complements
297:are called
184:defined by
41:modal logic
3087:Categories
2996:References
2917:field maps
2774:, we have
2513:semisimple
2402:| for all
2351:| for all
2147:| for all
1824:such that
1790:such that
1728:= max{y ∈
1678:such that
1368:topologies
1247:semisimple
1180:indiscrete
911:isomorphic
584:σ-complete
70:Definition
52:set theory
3033:: 141-91.
3019:: 155-70.
2972:: 137–52.
2621:necessity
2615:theories
2603:play for
2550:) and ◊ (
2454:down-sets
2122:power set
2120:from the
2061:S4-frames
2029:operators
2019:Preorders
1910:given by
1286:connected
712:power set
673:covariant
606:map of a
490:. Hence:
456:morphisms
204:. By the
84:signature
82:with the
2909:McKinsey
2552:possibly
2426:implies
2410:and all
2375:implies
2359:and all
2317:topology
2304:topology
2264:embedded
2242:outside
2219:for all
2173:for all
2163:implies
2074:Given a
2002:logician
1958:possibly
1861:Given a
1641:For all
1607:For all
1207:discrete
1140:open map
1090:category
997:complete
967:Given a
936:embedded
904:complete
817:For all
693:Given a
689:Topology
600:Sikorski
462: :
382: :
378:, a map
279:Wim Blok
165:interior
39:and the
37:topology
33:interior
3079:: 12-20
3068:: 33-48
3054:: 1-34.
2857:Jónsson
2679:variety
2501:variety
2446:up-sets
2422:«
2371:«
2315:with a
2213:«
2159:«
1985:false.
1777:
1748:
1538:apply.
1394:where ⟨
1340:compact
1088:is the
1080:is the
525:, then
502:, then
366:, have
347:trivial
342:Boolean
198:closure
60:variety
50:are to
2861:Tarski
2853:Kripke
2629:S4.Grz
2625:normal
2619:under
2617:closed
2542:) and
2442:worlds
2319:, the
2252:worlds
2246:, and
2240:worlds
2236:worlds
1934:, and
1863:theory
1638:exists
1616:, max{
1586:filter
1470:exists
1415:0,1 ∈
1374:is an
1196:simple
907:atomic
902:Every
390:is an
334:clopen
310:closed
303:. The
192:′))′.
117:, the
97:where
78:is an
2951:Notes
2945:chain
2826:does
2819:)) =
2793:)) =
2450:world
2256:world
2209:with
2103:) = ⟨
2027:with
1890:) = ⟨
1354:) is
1327:) is
1300:) is
1272:) is
1245:) is
1194:) is
1156:empty
1120:is a
833:= ∩ {
790:= ∪ {
756:) = ⟨
580:joins
576:meets
470:is a
248:0 = 0
156:1 = 1
109:is a
46:what
23:, an
2754:and
2752:′ ′
2701:and
2599:and
2591:for
2576:dual
2568:dual
2494:Any
2486:)).
2468:) =
2011:and
1926:are
1922:and
1902:, □⟩
1692:and
1509:= Σ{
1104:and
1100:are
1096:and
1059:) =
867:and
845:and
802:and
663:) ≤
626:) ≤
578:and
545:and
482:and
437:) =
419:) =
404:and
374:and
300:open
274:dual
256:= ((
238:) =
188:= ((
176:dual
174:The
147:) =
3017:127
3012:,"
2875:or
2828:not
2703:wK4
2695:wK4
2295:).
2291:(a
2258:in
2193:= {
2139:= {
2127:of
2115:, ⟩
2083:= ⟨
1996:or
1952:□ (
1930:in
1562:if
1504:by
1446:of
1338:is
1311:is
1284:is
1256:is
1229:is
1205:is
1178:is
1154:is
1117:Cit
1113:Top
1098:Cit
1094:Top
1086:Cit
1078:Top
1070:of
1042:by
960:or
875:in
768:, ⟩
702:= ⟨
647:or
566:or
517:If
494:If
200:of
167:of
74:An
62:of
19:In
3089::
3075:,
3066:79
3061:,
3052:58
3047:,
3031:45
2970:38
2859:,
2836:.
2763:+
2759:=
2744:=
2709:.
2707:S4
2685:≥
2631:.
2613:S4
2609:S4
2597:S4
2544:S5
2532:S5
2523:.
2517:S5
2507:=
2430:∈
2418:,
2414:∈
2406:∈
2398:⊆
2379:∈
2367:,
2363:∈
2355:∈
2347:⊆
2223:⊆
2205:∈
2197:∈
2177:⊆
2167:∈
2155:,
2151:∈
2143:∈
2069:S4
2055:.
2015:.
2013:S5
2009:S4
1990:S4
1914:~
1898:,
1877::
1871:S4
1838:.
1833:≤
1829:≤
1810:∈
1795:∈
1744:)}
1736:∈
1732:|
1697:∈
1687:≤
1683:≤
1673:∈
1658:∈
1654:,
1650:∈
1636:)}
1628:∈
1624:|
1620:∈
1612:∈
1591:A
1567:≤
1521:≤
1517:|
1464:≤
1460:|
1452:Σ{
1142:.
1055:)(
964:.
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794:|
778:⊆
718::
706:,
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427:);
386:→
317:=
293:=
281:.
242:+
234:+
224:=
215:≥
171:.
145:xy
137:=
128:≤
66:.
44:S4
2989:.
2833:V
2822:V
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2807:(
2804:D
2796:A
2790:A
2787:(
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2781:(
2778:I
2771:A
2765:x
2761:x
2757:x
2750:x
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2742:x
2736:V
2731:)
2728:V
2725:(
2722:I
2715:V
2687:x
2683:x
2667:A
2661:A
2658:(
2655:D
2642:A
2509:x
2505:x
2483:X
2480:(
2477:T
2474:(
2471:A
2465:X
2462:(
2459:B
2436:.
2434:}
2432:C
2428:y
2424:x
2420:y
2416:X
2412:y
2408:C
2404:x
2400:X
2396:C
2394:{
2385:.
2383:}
2381:O
2377:y
2373:y
2369:x
2365:X
2361:y
2357:O
2353:x
2349:X
2345:O
2343:{
2335:X
2332:(
2329:T
2312:X
2284:X
2275:X
2272:(
2269:B
2260:S
2248:S
2244:S
2232:S
2227:.
2225:X
2221:S
2217:}
2215:x
2211:y
2207:S
2203:y
2199:X
2195:x
2191:S
2181:.
2179:X
2175:S
2171:}
2169:S
2165:y
2161:y
2157:x
2153:X
2149:y
2145:X
2141:x
2137:S
2129:X
2113:X
2109:X
2107:(
2105:P
2100:X
2097:(
2094:B
2085:X
2080:X
1975:M
1973:(
1970:L
1946:M
1944:(
1941:L
1936:M
1932:M
1924:q
1920:p
1916:q
1912:p
1908:M
1900:T
1896:F
1892:M
1888:M
1886:(
1883:L
1867:M
1835:x
1831:z
1827:y
1822:z
1818:)
1816:y
1814:(
1812:N
1808:x
1803:)
1801:x
1799:(
1797:N
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1765:)
1762:x
1759:(
1756:N
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1738:N
1734:x
1730:B
1726:x
1721:B
1717:N
1713:N
1707:.
1705:)
1703:z
1701:(
1699:N
1695:z
1689:x
1685:z
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1675:B
1671:z
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1664:y
1662:(
1660:N
1656:x
1652:B
1648:y
1646:,
1644:x
1634:y
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1630:N
1626:x
1622:B
1618:y
1614:B
1610:x
1601:B
1597:N
1582:x
1580:(
1578:N
1574:x
1569:y
1565:x
1560:x
1552:y
1529:T
1525:}
1523:b
1519:a
1515:T
1513:∈
1511:a
1507:b
1502:B
1496:⟩
1494:T
1490:B
1488:⟨
1475:T
1468:}
1466:b
1462:a
1458:T
1456:∈
1454:a
1448:B
1444:b
1437:T
1433:)
1431:T
1427:T
1423:T
1417:T
1408:B
1404:B
1400:T
1396:B
1390:⟩
1388:T
1384:B
1382:⟨
1351:X
1348:(
1345:A
1335:X
1324:X
1321:(
1318:A
1308:X
1297:X
1294:(
1291:A
1281:X
1269:X
1266:(
1263:A
1253:X
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1239:(
1236:A
1226:X
1218:X
1215:(
1212:A
1202:X
1191:X
1188:(
1185:A
1175:X
1167:X
1164:(
1161:A
1151:X
1136:f
1132:f
1130:(
1127:A
1108:A
1073:Y
1068:S
1061:f
1057:S
1053:f
1051:(
1048:A
1037:)
1034:X
1031:(
1028:A
1022:Y
1019:(
1016:A
1011:f
1009:(
1006:A
988:Y
982:X
977:f
953:X
950:(
947:A
931:X
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919:(
916:A
896:X
890:X
887:(
884:A
878:X
873:S
869:S
865:S
861:S
855:}
852:X
847:C
843:C
839:S
835:C
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823:X
819:S
812:}
809:X
804:O
800:S
796:O
792:O
788:S
780:X
776:S
770:,
766:X
762:X
760:(
758:P
753:X
750:(
747:A
735:⟩
733:X
729:X
727:(
725:P
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716:X
708:T
704:X
699:X
669:x
667:(
665:f
661:x
659:(
657:f
653:f
632:x
630:(
628:f
624:x
622:(
620:f
616:f
537:.
535:B
531:x
529:(
527:f
523:A
519:x
514:;
512:B
508:x
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504:f
500:A
496:x
488:A
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476:f
468:B
464:A
460:f
443:x
441:(
439:f
435:x
433:(
431:f
425:x
423:(
421:f
417:x
415:(
413:f
406:B
402:A
398:f
388:B
384:A
380:f
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319:x
315:x
295:x
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270:S
266:S
258:x
254:x
244:y
240:x
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232:x
230:(
226:x
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143:(
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135:x
130:x
126:x
103:S
101:⟨
91:S
89:⟨
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