4616:. However, this terminology is avoided for Galois connections, since there was a time when posets were transformed into categories in a dual fashion, i.e. with morphisms pointing in the opposite direction. This led to a complementary notation concerning left and right adjoints, which today is ambiguous.
4337:
In this situation, an important feature of Galois connections is that one adjoint uniquely determines the other. Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection. The main property
5036:, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103β125. (Freely available online in various file formats
4565:
is also a Galois connection. When considering categories of complete lattices, this can be simplified to considering just mappings preserving all suprema (or, alternatively, infima). Mapping complete lattices to their duals, these categories display auto
1903:
3483:
antitone Galois connections between power sets arise in this way. This follows from the "Basic
Theorem on Concept Lattices". Theory and applications of Galois connections arising from binary relations are studied in
3109:
242:. Mnemonically, the upper/lower terminology refers to where the function application appears relative to β€. The term "adjoint" refers to the fact that monotone Galois connections are special cases of pairs of
1402:
3274:
1541:
1478:
1346:
1124:
truncating a real number to the greatest integer less than or equal to it. The embedding of integers is customarily done implicitly, but to show the Galois connection we make it explicit. So let
3749:. Thus monotonicity does not have to be included in the definition explicitly. However, mentioning monotonicity helps to avoid confusion about the two alternative notions of Galois connections.
2060:
1991:
1240:
1158:
4255:. In this way, closure operators and Galois connections are seen to be closely related, each specifying an instance of the other. Similar conclusions hold true for kernel operators.
1284:
1204:
1789:
4430:. The existence of a certain Galois connection now implies the existence of the respective least or greatest elements, no matter whether the corresponding posets satisfy any
3495:
in its primitive version incorporates both the monotone and antitone Galois connections to furnish its upper and lower bounds of nodes for the concept lattice, respectively.
1938:
1110:
1081:
4949:
62:; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as
898:
863:
1043:
3537:
is the lower adjoint as introduced above. Some helpful and instructive basic properties can be obtained immediately. By the defining property of Galois connections,
1005:
970:
938:
3488:. That field uses Galois connections for mathematical data analysis. Many algorithms for Galois connections can be found in the respective literature, e.g., in.
1425:
1753:
can be characterized by the existence of suitable adjoints. These considerations give some impression of the ubiquity of Galois connections in order theory.
4938:
4955:
5123:
77:
between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below. The term
4907:
3326:(not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and Zariski closed subsets of the
5037:
3020:
1354:
556:. However the original notion in Galois theory is slightly different. In this alternative definition, a Galois connection is a pair of
4082:(respectively residual mapping). Therefore, the notion of residuated mapping and monotone Galois connection are essentially the same.
3157:
1493:
816:. All of the below statements on Galois connections can thus easily be converted into statements about antitone Galois connections.
5059:
4656:
definition. One can also define Galois connections as a pair of monotone functions that satisfy the laxer condition that for all
4930:"Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints"
5043:, it presents many examples and results, as well as notes on the different notations and definitions that arose in this area.)
4434:. Thus, when one upper adjoint of a Galois connection is given, the other upper adjoint can be defined via this same property.
4327:
1430:
5101:
4979:
4929:
4858:
1289:
791:
The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between
4838:
36:
2073:, one can establish that doubly transitive actions have no blocks other than the trivial ones (singletons or the whole of
5014:
1999:
1962:
5086:
5068:
4759:
1209:
1127:
5170:
1742:. The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function
4311:
3739:
preserves the order of any two elements, i.e. it is monotone. Again, a similar reasoning yields monotonicity of
1898:{\displaystyle {\mathcal {B}}=\{B\subseteq X:x\in B;\forall g\in G,gB=B\ \mathrm {or} \ gB\cap B=\emptyset \},}
4177:. Nuclei induce frame homomorphisms; a subset of a locale is called a sublocale if it is given by a nucleus.
2776:
1245:
1163:
4489:
Galois connections also provide an interesting class of mappings between posets which can be used to obtain
4586:
Every partially ordered set can be viewed as a category in a natural way: there is a unique morphism from
4608:
between two categories that arise from partially ordered sets. In this context, the upper adjoint is the
4431:
2505:
1722:
1652:
4105:
is monotone (being the composite of monotone functions), inflationary, and idempotent. This states that
3348:
3327:
3011:
5175:
3305:
4871:
1725:. Roughly speaking, it turns out that the usual functions β¨ and β§ are lower and upper adjoints to the
641:
in this version erases the distinction between upper and lower, and the two functions are then called
35:(posets). Galois connections find applications in various mathematical theories. They generalize the
4567:
4330:
states that the converse implication is also valid in certain cases: especially, any mapping between
1919:
4575:
1768:
1090:
1061:
261:
An essential property of a Galois connection is that an upper/lower adjoint of a Galois connection
92:(or dual order isomorphism, depending on whether we take monotone or antitone Galois connections).
1949:
5003:
The following books and survey articles include Galois connections using the monotone definition:
4625:
3492:
3485:
5160:
4490:
3588:
2224:, both ordered by inclusion β. There is a further adjoint pair in this situation: for a subset
2109:
1643:
being the lower adjoint. A similar Galois connection whose lower adjoint is given by the meet (
868:
804:
133:
4804:
4493:
of posets. Especially, it is possible to compose Galois connections: given Galois connections
836:
5165:
4574:
that induce adjoint mappings in the other direction are the morphisms usually considered for
4326:. From these properties, one can also conclude monotonicity of the adjoints immediately. The
2817:
1010:
122:
32:
2283:
is the upper adjoint, while in the second Galois connection it serves as the lower adjoint.
975:
4629:
4319:
3115:
943:
911:
411:
742:
are the associated closure operators; they are monotone idempotent maps with the property
8:
4570:, that are quite fundamental for obtaining other duality theorems. More special kinds of
3309:
3144:
2977:
2810:
2291:
2121:
4808:
4090:
The above findings can be summarized as follows: for a Galois connection, the composite
3924:
This shows the desired equality. Furthermore, we can use this property to conclude that
1427:
is restricted to the integers. The well-known properties of the floor function, such as
5140:
4899:
4737:
4147:
4079:
3331:
2959:
2655:
2442:
2345:
2082:
2066:
1702:
1410:
4624:
Galois connections may be used to describe many forms of abstraction in the theory of
3343:
More generally, there is an antitone Galois connection between ideals in the ring and
48:
5097:
5082:
5064:
5054:
4959:
4903:
4891:
4854:
4834:
4765:
4755:
2747:
2740:
2560:
2427:
1556:
901:
514:
130:
89:
74:
5021:
Gerhard Gierz, Karl H. Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove,
1721:
Further interesting examples for Galois connections are described in the article on
5132:
5107:
5009:
4883:
4747:
4605:
4331:
4180:
4121:
3320:
3297:
1750:
1484:
900:
each other's inverse, forms a (trivial) Galois connection, as follows. Because the
428:
386:
243:
59:
4481:, contains a greatest element. Again, this can be dualized for the upper adjoint.
4800:
4733:
3368:
2988:
2623:
2493:
2392:
2295:
1648:
549:
247:
44:
4947:
4751:
4652:. It is only explicit in the definition to distinguish it from the alternative
4446:
3711:. Applying the basic property of Galois connections, one can now conclude that
2970:
2764:
2737:
2341:
2287:
1121:
197:
55:
5118:
5040:
4887:
4142:
is monotone, deflationary, and idempotent. Such mappings are sometimes called
5154:
5022:
4963:
4946:
For a counterexample for the false theorem in Sect.7 (p.243 top right), see:
4895:
4769:
4648:
Monotonicity follows from the following condition. See the discussion of the
4233:
2457:
2167:
1576:
905:
553:
4872:"A general theory of concept lattice with tractable implication exploration"
4807:. The notation is different nowadays; an easier introduction by Peter Smith
2838:. This yields an antitone Galois connection between the set of subspaces of
548:
The above definition is common in many applications today, and prominent in
5092:
Nikolaos
Galatos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono (2007),
2850:
2599:
form a monotone Galois connection, with semantics being the upper adjoint.
2349:
1765:
1726:
24:
3354:
2804:
1483:
The dual orderings give another monotone Galois connection, now with the
1113:
645:
rather than adjoints. Each polarity uniquely determines the other, since
20:
4334:
that preserves all suprema is the lower adjoint of a Galois connection.
4310:
Another important property of Galois connections is that lower adjoints
5144:
4820:
Birkhoff, 1st edition (1940): Β§32, 3rd edition (1967): Ch. V, Β§7 and Β§8
4251:, that maps each closed element to itself, considered as an element of
4208:
4056:
2963:
2876:
529:
446:
2512:
the power set of the set of all mathematical structures. For a theory
3344:
2372:
2070:
1568:
1117:
830:
382:
82:
5136:
1480:
can be derived by elementary reasoning from this Galois connection.
4742:
4571:
4315:
3104:{\displaystyle V(S)=\{x\in K^{n}:f(x)=0{\mbox{ for all }}f\in S\},}
2384:
1941:
40:
5032:
Marcel ErnΓ©, JΓΌrgen
Koslowski, Austin Melton, George E. Strecker,
2746:, there is an antitone Galois connection between subgroups of the
2563:, this is the set of sentences that are true in all structures in
250:
as discussed further below. Other terminology encountered here is
5094:
Residuated
Lattices. An Algebraic Glimpse at Substructural Logics
4987:
Proc. 6th ACM Symp. on
Principles of Programming Languages (POPL)
4948:
Jochen
Burghardt; Florian KammΓΌller; Jeff W. Sanders (Dec 2000).
4323:
4305:
4288:) are mapped to elements within the range of the kernel operator
2388:
1644:
1397:{\displaystyle n\leq x~\Leftrightarrow ~n\leq \lfloor x\rfloor .}
1084:
445:, known as the associated kernel operator. Both are monotone and
2939:
This gives an antitone Galois connection between the subsets of
904:
is reflexive, transitive and antisymmetric, it is, trivially, a
4968:(However the original article only considers complete lattices)
4062:
It can be shown (see Blyth or ErnΓ© for proofs) that a function
3269:{\displaystyle I(U)=\{f\in K:f(x)=0{\mbox{ for all }}x\in U\}.}
2842:
and itself, ordered by inclusion; both polarities are equal to
1580:
1536:{\displaystyle x\leq n~\Leftrightarrow ~\lceil x\rceil \leq n.}
4791:
See
Alperin, Bell, Groups and Representations (GTM 162), p. 32
3752:
Another basic property of Galois connections is the fact that
3468:
yield an antitone Galois connection between the power sets of
4604:. A monotone Galois connection is then nothing but a pair of
3503:
In the following, we consider a (monotone) Galois connection
2531:
4619:
3621:. These properties can be described by saying the composite
16:
Particular correspondence between two partially ordered sets
5048:
Some publications using the original (antitone) definition:
4977:
4927:
4258:
The above considerations also show that closed elements of
4085:
2271:
form a monotone Galois connection between the power set of
2216:
form a monotone Galois connection between the power set of
3856:
is inflationary as shown above. On the other hand, since
2612:
The motivating example comes from Galois theory: suppose
1546:
4811:, which also attribute the concept to the article cited.
1473:{\displaystyle \lfloor x+n\rfloor =\lfloor x\rfloor +n,}
4191:
gives rise to the Galois connection with lower adjoint
4070:
is a lower (respectively upper) adjoint if and only if
3355:
Connections on power sets arising from binary relations
2805:
Linear algebra: annihilators and orthogonal complements
2555:
be the minimum of the axiomatizations that approximate
31:
is a particular correspondence (typically) between two
4338:
to derive this uniqueness is the following: For every
3245:
3080:
2966:
and their zero sets is an antitone Galois connection.
2731:
2077:): this follows from the stabilizers being maximal in
1341:{\displaystyle F(n)\leq x~\Leftrightarrow ~n\leq G(x)}
1116:, each with its usual ordering, is given by the usual
4732:. Lecture Notes in Computer Science. Vol. 2962.
4730:
Semirings for Soft
Constraint Solving and Programming
3160:
3023:
2472:
form a monotone Galois connection between subsets of
2002:
1965:
1922:
1792:
1496:
1433:
1413:
1357:
1292:
1248:
1212:
1166:
1130:
1093:
1064:
1013:
978:
946:
914:
871:
839:
4939:
ACM Symposium on
Principles of Programming Languages
509:
is a Galois connection in which the kernel operator
4831:
Formal
Concept Analysis -- Mathematical Foundations
2508:(axiomatizations) reverse ordered by strength, and
4980:"Systematic Design of Program Analysis Frameworks"
4484:
3587:. By a similar reasoning (or just by applying the
3268:
3103:
2054:
1985:
1932:
1897:
1535:
1472:
1419:
1396:
1340:
1278:
1234:
1198:
1152:
1104:
1075:
1037:
999:
964:
932:
892:
857:
73:A Galois connection is rather weak compared to an
5124:Transactions of the American Mathematical Society
3308:, then the closure on the polynomial ring is the
2055:{\displaystyle B\mapsto H_{B}=\{g\in G:gx\in B\}}
5152:
4803:, Adjointness in foundations, Dialectica, 1969,
4581:
1986:{\displaystyle {\mathcal {B}}\to {\mathcal {G}}}
1120:function of the integers into the reals and the
4322:. Dually, upper adjoints preserve all existing
100:
4870:Liaw, Tsong-Ming; Lin, Simon C. (2020-10-12).
4306:Existence and uniqueness of Galois connections
2694:to be the field consisting of all elements of
2682:, ordered by inclusion β. For such a subgroup
2602:
2537: ; for a set of mathematical structures
2530:be the set of all structures that satisfy the
1235:{\displaystyle G:\mathbb {R} \to \mathbb {Z} }
1153:{\displaystyle F:\mathbb {Z} \to \mathbb {R} }
1048:
258:) for the lower (respectively upper) adjoint.
1655:, where the two mappings can be described by
799:is just a monotone Galois connection between
543:
402:Given a Galois connection with lower adjoint
3260:
3176:
3095:
3039:
2049:
2022:
1889:
1803:
1709:" is the upper adjoint of "conjunction with
1521:
1515:
1458:
1452:
1446:
1434:
1388:
1382:
1270:
1264:
5114:, Amer. Math. Soc. Coll. Pub., Vol 25, 1940
2312:, and the closure operator on subgroups of
1756:
54:A Galois connection can also be defined on
4978:Patrick Cousot; Radhia Cousot (Jan 1979).
4928:Patrick Cousot; Radhia Cousot (Jan 1977).
4727:
4437:On the other hand, some monotone function
2088:
1639:, form a monotone Galois connection, with
5079:Lattices and Ordered Algebraic Structures
4741:
4620:Applications in the theory of programming
1228:
1220:
1189:
1146:
1138:
1095:
1066:
47:, discovered by the French mathematician
5060:Categories for the Working Mathematician
5053:
4328:adjoint functor theorem for order theory
4086:Closure operators and Galois connections
4869:
2698:that are held fixed by all elements of
2487:
1279:{\displaystyle G(x)=\lfloor x\rfloor .}
1199:{\displaystyle F(n)=n\in \mathbb {R} ,}
5153:
1547:Power set; implication and conjunction
2953:
1058:A monotone Galois connection between
129:between these posets consists of two
3287:form an antitone Galois connection.
3010:is a set of polynomials, define the
2875:, consisting of all elements of the
2728:form an antitone Galois connection.
2480:, if both are ordered by inclusion.
1551:For an order-theoretic example, let
1160:denote the embedding function, with
37:fundamental theorem of Galois theory
5117:
5029:, Cambridge University Press, 2003.
5018:, Cambridge University Press, 2002.
4954:(Technical report). Vol. 122.
3689:. Then using the above one obtains
2732:Algebraic topology: covering spaces
2331:
2290:between algebraic objects (such as
1651:. Especially, it is present in any
1407:This is valid because the variable
13:
5015:Introduction to Lattices and Order
4863:
3589:duality principle for order theory
2279:. In the first Galois connection,
1978:
1968:
1925:
1886:
1864:
1861:
1830:
1795:
560:, i.e. order-reversing, functions
14:
5187:
2987:be the set of all subsets of the
2797:, there is a covering space with
2294:), this connection is called the
1053:
373:A consequence of this is that if
39:about the correspondence between
4951:Isomorphism of Galois Embeddings
4199:being just the corestriction of
2996:ordered by inclusion β, and let
2607:
1647:) operation can be found in any
5027:Continuous Lattices and Domains
4971:
4921:
4910:from the original on 2020-05-28
4649:
4612:while the lower adjoint is the
4485:Galois connections as morphisms
3476:, both ordered by inclusion β.
2962:, the relation between sets of
2630:be the set of all subfields of
1242:denotes the floor function, so
5034:A primer on Galois connections
4989:. ACM Press. pp. 269β282.
4843:
4823:
4814:
4794:
4785:
4776:
4721:
4712:
4642:
3235:
3229:
3220:
3188:
3170:
3164:
3070:
3064:
3033:
3027:
2864:we can define its annihilator
2336:Pick some mathematical object
2006:
1973:
1933:{\displaystyle {\mathcal {G}}}
1509:
1370:
1335:
1329:
1314:
1302:
1296:
1258:
1252:
1224:
1176:
1170:
1142:
1029:
1023:
988:
982:
972:partially ordered sets. Since
959:
947:
927:
915:
881:
849:
95:
1:
5121:(1944), "Galois Connexions",
5063:(Second ed.). Springer.
4997:
4582:Connection to category theory
3880:is monotonic, one finds that
3498:
3000:be the set of all subsets of
2777:semi-locally simply connected
2638:, ordered by inclusion β. If
2449:, and take as "subobjects of
1105:{\displaystyle \mathbb {R} ,}
1076:{\displaystyle \mathbb {Z} ,}
1045:we have a Galois connection.
824:
64:(monotone) Galois connections
4876:Theoretical Computer Science
4849:Ganter, B. and Obiedkov, S.
4728:Bistarelli, Stefano (2004).
3147:of polynomials vanishing on
2904:, we define its annihilator
2893:. Similarly, given a subset
101:(Monotone) Galois connection
81:is sometimes used to mean a
7:
4752:10.1007/978-3-540-25925-1_8
4211:mapping the closure system
3731:. But this just shows that
3006:ordered by inclusion β. If
2670:be the set of subgroups of
2642:is such a subfield, write
2603:Antitone Galois connections
1716:
1049:Monotone Galois connections
819:
532:the set of closed elements
525:is an order isomorphism of
68:antitone Galois connections
10:
5192:
3421:Similarly, for any subset
2801:as its fundamental group.
2779:, then for every subgroup
2591:: the "semantics functor"
2492:A very general comment of
544:Antitone Galois connection
427:, known as the associated
127:monotone Galois connection
4888:10.1016/j.tcs.2020.05.014
4829:Ganter, B. and Wille, R.
3382:is given. For any subset
3367:are arbitrary sets and a
2595:and the "syntax functor"
2418:be the underlying set of
1959:Then, the correspondence
1705:terms: "implication from
893:{\displaystyle g:Y\to X,}
4635:
4318:that exist within their
3319:More generally, given a
2567:). We can then say that
2344:, for instance a group,
2302:connect to subgroups of
2085:for further discussion.
2069:Galois connection. As a
1757:Transitive group actions
858:{\displaystyle f:X\to Y}
5171:Abstract interpretation
4626:abstract interpretation
4432:completeness properties
4183:, any closure operator
3870:is deflationary, while
3797:. Clearly we find that
3493:general concept lattice
3486:formal concept analysis
2089:Image and inverse image
2083:Doubly transitive group
1723:completeness properties
1038:{\displaystyle x=g(y),}
833:of a pair of functions
691:is the largest element
659:is the largest element
331:is the largest element
4851:Conceptual Exploration
4809:in these lecture notes
3312:of ideal generated by
3296:is the closure in the
3270:
3118:of the polynomials in
3105:
2484:is the lower adjoint.
2352:, etc. For any subset
2112:, then for any subset
2056:
1987:
1934:
1899:
1537:
1474:
1421:
1398:
1342:
1280:
1236:
1200:
1154:
1106:
1077:
1039:
1001:
1000:{\displaystyle f(x)=y}
966:
934:
894:
859:
410:, we can consider the
265:determines the other:
123:partially ordered sets
33:partially ordered sets
4630:programming languages
4449:each set of the form
4358:is the least element
4232:is then given by the
4222:). The upper adjoint
3347:of the corresponding
3271:
3106:
2818:orthogonal complement
2736:Analogously, given a
2504:to be the set of all
2275:and the power set of
2220:and the power set of
2057:
1988:
1935:
1900:
1538:
1475:
1422:
1399:
1343:
1281:
1237:
1201:
1155:
1107:
1078:
1040:
1002:
967:
965:{\displaystyle (Y,=)}
935:
933:{\displaystyle (X,=)}
895:
860:
279:is the least element
79:Galois correspondence
5077:Thomas Scott Blyth,
4384:. Dually, for every
4146:. In the context of
3306:algebraically closed
3158:
3021:
2771:. In particular, if
2498:syntax and semantics
2488:Syntax and semantics
2422:. (We can even take
2399:. For any subobject
2000:
1963:
1920:
1790:
1775:and pick some point
1749:Going further, even
1494:
1431:
1411:
1355:
1290:
1246:
1210:
1164:
1128:
1091:
1062:
1011:
976:
944:
912:
869:
837:
164:, such that for all
88:; this is simply an
5010:Hilary A. Priestley
5008:Brian A. Davey and
4943:. pp. 238β252.
4853:, Springer (2016),
4833:, Springer (1999),
4445:is a lower adjoint
3438: ) = {
3399: ) = {
3300:, and if the field
3247: for all
3082: for all
2943:and the subsets of
2913: ) = {
2811:inner product space
2763:and path-connected
2656:field automorphisms
2158:and for any subset
1348:then translates to
588:between two posets
389:of the other, i.e.
218:In this situation,
5081:, Springer, 2005,
5057:(September 1998).
5055:Mac Lane, Saunders
4302:, and vice versa.
4148:frames and locales
4080:residuated mapping
3591:), one finds that
3479:Up to isomorphism
3266:
3249:
3114:the set of common
3101:
3084:
2960:algebraic geometry
2954:Algebraic geometry
2816:, we can form the
2587:logically entails
2500:are adjoint: take
2476:and subobjects of
2178: ) =
2132: ) =
2081:in that case. See
2052:
1983:
1930:
1895:
1533:
1470:
1417:
1394:
1338:
1276:
1232:
1196:
1150:
1102:
1073:
1035:
997:
962:
930:
890:
855:
406:and upper adjoint
5176:Closure operators
5102:978-0-444-52141-5
4859:978-3-662-49290-1
4332:complete lattices
3557:is equivalent to
3248:
3083:
2748:fundamental group
2741:topological space
2654:for the group of
2561:first-order logic
2428:topological space
2286:In the case of a
1870:
1859:
1751:complete lattices
1514:
1508:
1420:{\displaystyle n}
1375:
1369:
1319:
1313:
902:equality relation
714:The compositions
385:then each is the
90:order isomorphism
86:Galois connection
75:order isomorphism
29:Galois connection
5183:
5147:
5108:Garrett Birkhoff
5074:
4991:
4990:
4984:
4975:
4969:
4967:
4958:. p. 9-14.
4944:
4934:
4925:
4919:
4918:
4916:
4915:
4867:
4861:
4847:
4841:
4839:978-3-540-627715
4827:
4821:
4818:
4812:
4798:
4792:
4789:
4783:
4780:
4774:
4773:
4745:
4725:
4719:
4716:
4710:
4708:
4690:
4686:
4682:
4663:
4659:
4646:
4606:adjoint functors
4603:
4564:
4539:, the composite
4538:
4534:
4530:
4515:
4511:
4507:
4480:
4476:
4472:
4444:
4429:
4414:
4410:
4407:is the greatest
4406:
4391:
4387:
4383:
4365:
4361:
4357:
4345:
4341:
4301:
4287:
4265:
4261:
4254:
4250:
4246:
4231:
4221:
4206:
4203:to the image of
4202:
4198:
4190:
4186:
4176:
4164:
4150:, the composite
4144:kernel operators
4141:
4127:
4122:closure operator
4119:
4104:
4077:
4069:
4054:
4039:
4022:
3971:
3919:
3879:
3869:
3855:
3836:
3796:
3792:
3788:
3748:
3738:
3730:
3710:
3688:
3678:
3653:
3634:
3620:
3616:
3612:
3586:
3582:
3578:
3556:
3536:
3521:
3475:
3471:
3467:
3463:
3459:
3428:
3424:
3420:
3389:
3385:
3381:
3377:
3373:
3366:
3362:
3339:
3325:
3321:commutative ring
3315:
3303:
3298:Zariski topology
3295:
3282:
3275:
3273:
3272:
3267:
3250:
3246:
3219:
3218:
3200:
3199:
3150:
3142:
3131:
3125:
3121:
3110:
3108:
3107:
3102:
3085:
3081:
3057:
3056:
3009:
3005:
2999:
2995:
2986:
2982:
2975:
2949:
2942:
2938:
2903:
2896:
2892:
2888:
2884:
2874:
2863:
2859:
2855:
2845:
2841:
2837:
2833:
2830:of any subspace
2829:
2815:
2800:
2796:
2782:
2774:
2770:
2762:
2745:
2727:
2716:
2702:. Then the maps
2701:
2697:
2693:
2685:
2681:
2669:
2665:
2661:
2653:
2641:
2637:
2633:
2629:
2621:
2598:
2594:
2590:
2586:
2578:
2570:
2566:
2558:
2554:
2546:
2536:
2529:
2521:
2511:
2506:logical theories
2503:
2483:
2479:
2475:
2471:
2467:
2463:
2455:
2452:
2448:
2440:
2425:
2421:
2417:
2406:
2402:
2398:
2382:
2378:
2371:be the smallest
2370:
2359:
2355:
2339:
2332:Span and closure
2327:
2322:
2315:
2311:
2301:
2282:
2278:
2274:
2270:
2266:
2262:
2231:
2227:
2223:
2219:
2215:
2211:
2207:
2166:we can form the
2165:
2161:
2157:
2120:we can form the
2119:
2115:
2107:
2080:
2076:
2061:
2059:
2058:
2053:
2018:
2017:
1992:
1990:
1989:
1984:
1982:
1981:
1972:
1971:
1955:
1947:
1939:
1937:
1936:
1931:
1929:
1928:
1915:
1904:
1902:
1901:
1896:
1868:
1867:
1857:
1799:
1798:
1782:
1778:
1774:
1764:
1748:
1741:
1712:
1708:
1700:
1673:
1642:
1638:
1615:
1597:
1593:
1590:. Then the maps
1589:
1585:
1574:
1566:
1562:
1554:
1542:
1540:
1539:
1534:
1512:
1506:
1485:ceiling function
1479:
1477:
1476:
1471:
1426:
1424:
1423:
1418:
1403:
1401:
1400:
1395:
1373:
1367:
1347:
1345:
1344:
1339:
1317:
1311:
1286:The equivalence
1285:
1283:
1282:
1277:
1241:
1239:
1238:
1233:
1231:
1223:
1205:
1203:
1202:
1197:
1192:
1159:
1157:
1156:
1151:
1149:
1141:
1111:
1109:
1108:
1103:
1098:
1082:
1080:
1079:
1074:
1069:
1044:
1042:
1041:
1036:
1006:
1004:
1003:
998:
971:
969:
968:
963:
939:
937:
936:
931:
899:
897:
896:
891:
864:
862:
861:
856:
815:
811:
802:
798:
794:
787:
783:
779:
764:
760:
756:
741:
727:
709:
694:
690:
677:
662:
658:
640:
636:
633:The symmetry of
628:
613:
595:
591:
587:
573:
539:
535:
528:
524:
520:
512:
508:
504:
499:Galois insertion
493:
489:
485:
471:
467:
463:
444:
429:closure operator
426:
409:
405:
398:
380:
376:
368:
363:
362:
361:
356:
345:
344:
343:
342:
337:
330:
317:
315:
314:
313:
308:
293:
292:
291:
290:
285:
278:
244:adjoint functors
233:
229:
221:
213:
196:
179:
175:
171:
167:
163:
149:
120:
112:
23:, especially in
5191:
5190:
5186:
5185:
5184:
5182:
5181:
5180:
5151:
5150:
5137:10.2307/1990305
5071:
5000:
4995:
4994:
4982:
4976:
4972:
4945:
4932:
4926:
4922:
4913:
4911:
4868:
4864:
4848:
4844:
4828:
4824:
4819:
4815:
4801:William Lawvere
4799:
4795:
4790:
4786:
4782:Galatos, p. 145
4781:
4777:
4762:
4736:. p. 102.
4734:Springer-Verlag
4726:
4722:
4717:
4713:
4692:
4688:
4684:
4665:
4661:
4657:
4647:
4643:
4638:
4622:
4595:
4594:if and only if
4584:
4562:
4555:
4549: ,
4540:
4536:
4532:
4528:
4517:
4513:
4509:
4508:between posets
4505:
4499: ,
4494:
4487:
4478:
4474:
4450:
4438:
4416:
4412:
4408:
4400:
4393:
4389:
4385:
4377:
4367:
4363:
4359:
4347:
4343:
4339:
4308:
4300:
4289:
4274:
4267:
4263:
4259:
4252:
4248:
4237:
4230:
4223:
4212:
4204:
4200:
4192:
4188:
4184:
4170:
4158:
4151:
4140:
4129:
4125:
4113:
4106:
4098:
4091:
4088:
4071:
4063:
4048:
4041:
4038:
4027:
4012:
3997:
3986:
3979:
3965:
3950:
3939:
3928:
3913:
3902:
3891:
3884:
3878:
3871:
3868:
3857:
3849:
3842:
3830:
3819:
3808:
3801:
3794:
3790:
3782:
3771:
3760:
3753:
3747:
3740:
3732:
3712:
3700:
3690:
3680:
3666:
3647:
3640:
3633:
3622:
3618:
3614:
3603:
3592:
3584:
3580:
3568:
3558:
3538:
3523:
3519:
3513: ,
3504:
3501:
3473:
3469:
3465:
3461:
3430:
3426:
3422:
3391:
3387:
3383:
3379:
3375:
3371:
3369:binary relation
3364:
3360:
3357:
3330:
3323:
3313:
3301:
3291:
3290:The closure on
3280:
3244:
3214:
3210:
3195:
3191:
3159:
3156:
3155:
3148:
3133:
3127:
3126:is a subset of
3123:
3119:
3079:
3052:
3048:
3022:
3019:
3018:
3007:
3001:
2997:
2991:
2989:polynomial ring
2984:
2980:
2973:
2956:
2944:
2940:
2905:
2898:
2894:
2890:
2889:that vanish on
2886:
2879:
2865:
2861:
2857:
2853:
2843:
2839:
2835:
2831:
2820:
2813:
2807:
2798:
2790:
2784:
2780:
2772:
2768:
2765:covering spaces
2756:
2750:
2743:
2734:
2718:
2703:
2699:
2695:
2687:
2683:
2671:
2667:
2663:
2659:
2643:
2639:
2635:
2631:
2627:
2624:field extension
2613:
2610:
2605:
2596:
2592:
2588:
2580:
2579:if and only if
2572:
2571:is a subset of
2568:
2564:
2556:
2548:
2538:
2534:
2523:
2513:
2509:
2501:
2494:William Lawvere
2490:
2481:
2477:
2473:
2469:
2465:
2461:
2453:
2450:
2446:
2431:
2423:
2419:
2408:
2404:
2400:
2396:
2380:
2376:
2361:
2357:
2353:
2337:
2334:
2318:
2317:
2313:
2303:
2299:
2298:: subgroups of
2296:lattice theorem
2280:
2276:
2272:
2268:
2264:
2233:
2229:
2225:
2221:
2217:
2213:
2209:
2170:
2163:
2159:
2124:
2117:
2113:
2094:
2091:
2078:
2074:
2065:is a monotone,
2013:
2009:
2001:
1998:
1997:
1977:
1976:
1967:
1966:
1964:
1961:
1960:
1953:
1948:containing the
1945:
1940:consist of the
1924:
1923:
1921:
1918:
1917:
1916:. Further, let
1913:
1860:
1794:
1793:
1791:
1788:
1787:
1780:
1776:
1772:
1762:
1759:
1743:
1729:
1719:
1710:
1706:
1675:
1656:
1653:Boolean algebra
1649:Heyting algebra
1640:
1633: \
1617:
1599:
1595:
1591:
1587:
1583:
1579:. Pick a fixed
1572:
1564:
1560:
1552:
1549:
1495:
1492:
1491:
1432:
1429:
1428:
1412:
1409:
1408:
1356:
1353:
1352:
1291:
1288:
1287:
1247:
1244:
1243:
1227:
1219:
1211:
1208:
1207:
1188:
1165:
1162:
1161:
1145:
1137:
1129:
1126:
1125:
1094:
1092:
1089:
1088:
1065:
1063:
1060:
1059:
1056:
1051:
1012:
1009:
1008:
1007:if and only if
977:
974:
973:
945:
942:
941:
913:
910:
909:
870:
867:
866:
838:
835:
834:
827:
822:
813:
807:
800:
796:
792:
785:
781:
766:
762:
758:
743:
729:
715:
696:
692:
681:
664:
660:
649:
638:
634:
615:
614:if and only if
600:
593:
589:
575:
561:
546:
537:
533:
526:
522:
518:
510:
506:
502:
491:
487:
473:
469:
465:
450:
432:
414:
407:
403:
390:
378:
374:
357:
354:
353:
352:
347:
338:
335:
334:
333:
332:
321:
309:
306:
305:
304:
295:
286:
283:
282:
281:
280:
269:
248:category theory
231:
227:
219:
200:
184:
177:
173:
169:
165:
151:
137:
114:
106:
103:
98:
56:preordered sets
49:Γvariste Galois
17:
12:
11:
5:
5189:
5179:
5178:
5173:
5168:
5163:
5149:
5148:
5131:(3): 493β513,
5115:
5112:Lattice Theory
5105:
5090:
5075:
5069:
5045:
5044:
5030:
5019:
4999:
4996:
4993:
4992:
4970:
4920:
4862:
4842:
4822:
4813:
4805:available here
4793:
4784:
4775:
4760:
4720:
4711:
4640:
4639:
4637:
4634:
4621:
4618:
4583:
4580:
4578:(or locales).
4560:
4553:
4526:
4503:
4486:
4483:
4447:if and only if
4398:
4375:
4307:
4304:
4298:
4294: β
4272:
4228:
4187:on some poset
4165:is called the
4156:
4138:
4134: β
4111:
4096:
4087:
4084:
4046:
4036:
4032: β
4024:
4023:
4010:
4006:)))) =
3995:
3991: (
3984:
3973:
3972:
3963:
3959: (
3955:)))) =
3948:
3944: (
3937:
3933: (
3922:
3921:
3911:
3900:
3896: (
3889:
3876:
3866:
3862: β
3847:
3839:
3838:
3828:
3817:
3813: (
3806:
3780:
3769:
3765: (
3758:
3745:
3698:
3645:
3631:
3627: β
3601:
3597: (
3566:
3528: :
3517:
3500:
3497:
3356:
3353:
3349:affine variety
3328:affine variety
3277:
3276:
3265:
3262:
3259:
3256:
3253:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3217:
3213:
3209:
3206:
3203:
3198:
3194:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3112:
3111:
3100:
3097:
3094:
3091:
3088:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3055:
3051:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
2971:natural number
2955:
2952:
2806:
2803:
2788:
2754:
2738:path-connected
2733:
2730:
2609:
2606:
2604:
2601:
2489:
2486:
2458:closed subsets
2379:that contains
2342:underlying set
2333:
2330:
2090:
2087:
2063:
2062:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2016:
2012:
2008:
2005:
1980:
1975:
1970:
1927:
1906:
1905:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1866:
1863:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1797:
1758:
1755:
1718:
1715:
1548:
1545:
1544:
1543:
1532:
1529:
1526:
1523:
1520:
1517:
1511:
1505:
1502:
1499:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1416:
1405:
1404:
1393:
1390:
1387:
1384:
1381:
1378:
1372:
1366:
1363:
1360:
1337:
1334:
1331:
1328:
1325:
1322:
1316:
1310:
1307:
1304:
1301:
1298:
1295:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1230:
1226:
1222:
1218:
1215:
1195:
1191:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1148:
1144:
1140:
1136:
1133:
1122:floor function
1101:
1097:
1072:
1068:
1055:
1054:Floor; ceiling
1052:
1050:
1047:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
996:
993:
990:
987:
984:
981:
961:
958:
955:
952:
949:
929:
926:
923:
920:
917:
889:
886:
883:
880:
877:
874:
854:
851:
848:
845:
842:
826:
823:
821:
818:
712:
711:
679:
631:
630:
545:
542:
449:, and we have
371:
370:
319:
254:(respectively
234:is called the
222:is called the
216:
215:
198:if and only if
102:
99:
97:
94:
15:
9:
6:
4:
3:
2:
5188:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5161:Galois theory
5159:
5158:
5156:
5146:
5142:
5138:
5134:
5130:
5126:
5125:
5120:
5116:
5113:
5109:
5106:
5103:
5099:
5095:
5091:
5088:
5087:1-85233-905-5
5084:
5080:
5076:
5072:
5070:0-387-98403-8
5066:
5062:
5061:
5056:
5052:
5051:
5050:
5049:
5042:
5039:
5035:
5031:
5028:
5024:
5023:Dana S. Scott
5020:
5017:
5016:
5011:
5007:
5006:
5005:
5004:
4988:
4981:
4974:
4965:
4961:
4957:
4953:
4952:
4942:
4940:
4931:
4924:
4909:
4905:
4901:
4897:
4893:
4889:
4885:
4881:
4877:
4873:
4866:
4860:
4856:
4852:
4846:
4840:
4836:
4832:
4826:
4817:
4810:
4806:
4802:
4797:
4788:
4779:
4771:
4767:
4763:
4761:3-540-21181-0
4757:
4753:
4749:
4744:
4739:
4735:
4731:
4724:
4715:
4707:
4703:
4699:
4695:
4680:
4676:
4672:
4668:
4655:
4651:
4645:
4641:
4633:
4631:
4627:
4617:
4615:
4611:
4610:right adjoint
4607:
4602:
4598:
4593:
4589:
4579:
4577:
4573:
4569:
4559:
4552:
4548:
4544:
4525:
4521:
4502:
4498:
4492:
4482:
4470:
4466:
4462:
4458:
4454:
4448:
4442:
4435:
4433:
4428:
4424:
4420:
4404:
4397:
4381:
4374:
4370:
4355:
4351:
4335:
4333:
4329:
4325:
4321:
4317:
4313:
4303:
4297:
4293:
4286:
4282:
4278:
4271:
4256:
4244:
4240:
4235:
4227:
4219:
4215:
4210:
4196:
4182:
4178:
4174:
4168:
4162:
4155:
4149:
4145:
4137:
4133:
4123:
4120:is in fact a
4117:
4110:
4102:
4095:
4083:
4081:
4075:
4067:
4060:
4058:
4052:
4045:
4035:
4031:
4020:
4016:
4009:
4005:
4001:
3994:
3990:
3983:
3978:
3977:
3976:
3969:
3962:
3958:
3954:
3947:
3943:
3936:
3932:
3927:
3926:
3925:
3917:
3910:
3907:))) β€
3906:
3899:
3895:
3888:
3883:
3882:
3881:
3875:
3865:
3861:
3853:
3846:
3834:
3827:
3824:))) β₯
3823:
3816:
3812:
3805:
3800:
3799:
3798:
3786:
3779:
3776:))) =
3775:
3768:
3764:
3757:
3750:
3744:
3736:
3728:
3724:
3720:
3716:
3708:
3704:
3697:
3693:
3687:
3683:
3677:
3673:
3669:
3665:Now consider
3663:
3661:
3657:
3651:
3644:
3638:
3630:
3626:
3611:
3607:
3600:
3596:
3590:
3576:
3572:
3565:
3561:
3554:
3550:
3546:
3542:
3535:
3531:
3527:
3516:
3512:
3508:
3496:
3494:
3489:
3487:
3482:
3477:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3370:
3352:
3350:
3346:
3341:
3337:
3333:
3329:
3322:
3317:
3311:
3307:
3299:
3294:
3288:
3286:
3263:
3257:
3254:
3251:
3241:
3238:
3232:
3226:
3223:
3215:
3211:
3207:
3204:
3201:
3196:
3192:
3185:
3182:
3179:
3173:
3167:
3161:
3154:
3153:
3152:
3146:
3140:
3136:
3130:
3117:
3098:
3092:
3089:
3086:
3076:
3073:
3067:
3061:
3058:
3053:
3049:
3045:
3042:
3036:
3030:
3024:
3017:
3016:
3015:
3013:
3004:
2994:
2990:
2979:
2972:
2967:
2965:
2961:
2951:
2947:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2901:
2882:
2878:
2872:
2868:
2856:and a subset
2852:
2847:
2827:
2823:
2819:
2812:
2802:
2794:
2787:
2778:
2766:
2760:
2753:
2749:
2742:
2739:
2729:
2725:
2721:
2714:
2710:
2706:
2691:
2679:
2675:
2657:
2651:
2647:
2634:that contain
2625:
2620:
2616:
2608:Galois theory
2600:
2584:
2576:
2562:
2552:
2545:
2541:
2533:
2527:
2520:
2516:
2507:
2499:
2495:
2485:
2459:
2444:
2438:
2434:
2429:
2415:
2411:
2395:generated by
2394:
2390:
2386:
2374:
2368:
2364:
2351:
2347:
2343:
2329:
2326:
2321:
2310:
2306:
2297:
2293:
2289:
2284:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2205:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2168:inverse image
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2123:
2111:
2106:
2102:
2098:
2086:
2084:
2072:
2068:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2019:
2014:
2010:
2003:
1996:
1995:
1994:
1957:
1951:
1943:
1911:
1892:
1883:
1880:
1877:
1874:
1871:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1800:
1786:
1785:
1784:
1770:
1767:
1754:
1752:
1746:
1740:
1736:
1732:
1728:
1724:
1714:
1704:
1698:
1694:
1690:
1686:
1682:
1678:
1671:
1667:
1663:
1659:
1654:
1650:
1646:
1636:
1632:
1628:
1624:
1620:
1614:
1610:
1606:
1602:
1582:
1578:
1575:, ordered by
1570:
1558:
1530:
1527:
1524:
1518:
1503:
1500:
1497:
1490:
1489:
1488:
1486:
1481:
1467:
1464:
1461:
1455:
1449:
1443:
1440:
1437:
1414:
1391:
1385:
1379:
1376:
1364:
1361:
1358:
1351:
1350:
1349:
1332:
1326:
1323:
1320:
1308:
1305:
1299:
1293:
1273:
1267:
1261:
1255:
1249:
1216:
1213:
1193:
1185:
1182:
1179:
1173:
1167:
1134:
1131:
1123:
1119:
1115:
1099:
1086:
1070:
1046:
1032:
1026:
1020:
1017:
1014:
994:
991:
985:
979:
956:
953:
950:
924:
921:
918:
907:
906:partial order
903:
887:
884:
878:
875:
872:
852:
846:
843:
840:
832:
817:
810:
806:
789:
777:
773:
769:
754:
750:
746:
740:
736:
732:
726:
722:
718:
707:
703:
699:
688:
684:
680:
675:
671:
667:
656:
652:
648:
647:
646:
644:
626:
622:
618:
611:
607:
603:
599:
598:
597:
586:
582:
578:
572:
568:
564:
559:
555:
554:domain theory
551:
541:
531:
516:
500:
495:
484:
480:
476:
461:
457:
453:
448:
443:
439:
435:
430:
425:
421:
417:
413:
400:
397:
393:
388:
384:
367:
360:
350:
341:
328:
324:
320:
312:
302:
298:
289:
276:
272:
268:
267:
266:
264:
259:
257:
256:right adjoint
253:
249:
245:
241:
237:
236:upper adjoint
225:
224:lower adjoint
211:
207:
203:
199:
195:
191:
187:
183:
182:
181:
162:
158:
154:
148:
144:
140:
135:
132:
128:
124:
118:
110:
93:
91:
87:
84:
80:
76:
71:
69:
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
5166:Order theory
5128:
5122:
5119:Ore, Γystein
5111:
5096:, Elsevier,
5093:
5078:
5058:
5047:
5046:
5033:
5026:
5013:
5002:
5001:
4986:
4973:
4950:
4936:
4923:
4912:. Retrieved
4879:
4875:
4865:
4850:
4845:
4830:
4825:
4816:
4796:
4787:
4778:
4729:
4723:
4718:Gierz, p. 23
4714:
4705:
4701:
4697:
4693:
4683:and for all
4678:
4674:
4670:
4666:
4653:
4644:
4623:
4614:left adjoint
4613:
4609:
4600:
4596:
4591:
4587:
4585:
4557:
4550:
4546:
4542:
4523:
4519:
4500:
4496:
4488:
4468:
4464:
4460:
4456:
4452:
4440:
4436:
4426:
4422:
4418:
4402:
4395:
4379:
4372:
4368:
4353:
4349:
4336:
4309:
4295:
4291:
4284:
4280:
4276:
4269:
4257:
4242:
4238:
4225:
4217:
4213:
4194:
4179:
4172:
4166:
4160:
4153:
4143:
4135:
4131:
4115:
4108:
4100:
4093:
4089:
4073:
4065:
4061:
4050:
4043:
4033:
4029:
4025:
4018:
4014:
4007:
4003:
3999:
3992:
3988:
3981:
3974:
3967:
3960:
3956:
3952:
3945:
3941:
3934:
3930:
3923:
3915:
3908:
3904:
3897:
3893:
3886:
3873:
3863:
3859:
3851:
3844:
3840:
3832:
3825:
3821:
3814:
3810:
3803:
3784:
3777:
3773:
3766:
3762:
3755:
3751:
3742:
3734:
3726:
3722:
3718:
3714:
3706:
3702:
3695:
3691:
3685:
3681:
3675:
3671:
3667:
3664:
3659:
3656:inflationary
3655:
3649:
3642:
3637:deflationary
3636:
3628:
3624:
3609:
3605:
3598:
3594:
3574:
3570:
3563:
3559:
3552:
3548:
3544:
3540:
3533:
3529:
3525:
3514:
3510:
3506:
3502:
3490:
3480:
3478:
3455:
3451:
3447:
3443:
3439:
3435:
3431:
3416:
3412:
3408:
3404:
3400:
3396:
3392:
3390:, we define
3358:
3342:
3335:
3318:
3292:
3289:
3284:
3278:
3138:
3134:
3128:
3113:
3014:of zeros as
3002:
2992:
2968:
2957:
2945:
2934:
2930:
2926:
2922:
2918:
2914:
2910:
2906:
2899:
2880:
2870:
2866:
2851:vector space
2848:
2825:
2821:
2808:
2792:
2785:
2758:
2751:
2735:
2723:
2719:
2712:
2708:
2704:
2689:
2677:
2673:
2649:
2645:
2618:
2614:
2611:
2582:
2574:
2550:
2543:
2539:
2525:
2518:
2514:
2497:
2491:
2436:
2432:
2413:
2409:
2366:
2362:
2350:vector space
2340:that has an
2335:
2324:
2319:
2316:is given by
2308:
2304:
2288:quotient map
2285:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2104:
2100:
2096:
2092:
2064:
1958:
1909:
1907:
1769:transitively
1760:
1744:
1738:
1734:
1730:
1727:diagonal map
1720:
1696:
1692:
1688:
1684:
1683:) = (
1680:
1676:
1669:
1665:
1661:
1657:
1634:
1630:
1626:
1622:
1618:
1612:
1608:
1604:
1600:
1567:both be the
1550:
1482:
1406:
1114:real numbers
1057:
828:
808:
790:
775:
771:
767:
752:
748:
744:
738:
734:
730:
724:
720:
716:
713:
705:
701:
697:
686:
682:
673:
669:
665:
654:
650:
642:
632:
624:
620:
616:
609:
605:
601:
596:, such that
584:
580:
576:
570:
566:
562:
557:
547:
521:, and hence
498:
496:
482:
478:
474:
459:
455:
451:
441:
437:
433:
423:
419:
415:
412:compositions
401:
395:
391:
372:
365:
358:
348:
339:
326:
322:
310:
300:
296:
287:
274:
270:
262:
260:
255:
252:left adjoint
251:
239:
235:
223:
217:
209:
205:
201:
193:
189:
185:
160:
156:
152:
146:
142:
138:
126:
116:
108:
104:
85:
78:
72:
67:
63:
53:
28:
25:order theory
18:
4207:(i.e. as a
4169:induced by
3721:) β€
3547:) β€
2964:polynomials
2666:fixed. Let
2383:, i.e. the
1912:containing
1908:the set of
1783:. Consider
1625: ) =
1607: ) =
1112:the set of
1083:the set of
536: of
96:Definitions
21:mathematics
5155:Categories
4998:References
4937:Proc. 4th
4914:2023-07-19
4882:: 84β114.
4743:cs/0208008
4650:properties
4491:categories
4415:such that
4366:such that
4262:(elements
4209:surjective
4181:Conversely
4128:. Dually,
4057:idempotent
3789:, for all
3679:such that
3613:, for all
3579:, for all
3509:= (
3499:Properties
3345:subschemes
3151:, that is
2877:dual space
2662:that hold
2140:= {
2067:one-to-one
1950:stabilizer
1559:, and let
825:Bijections
805:order dual
643:polarities
447:idempotent
180:, we have
4964:1435-2702
4904:219514253
4896:0304-3975
4770:0302-9743
4572:morphisms
4545:β
4471: },
4459:|
4371:β€
4234:inclusion
3694:β€
3660:extensive
3562:β€
3458: }.
3429:, define
3419: }.
3255:∈
3205:…
3183:∈
3132:, define
3090:∈
3046:∈
2937: }.
2809:Given an
2686:, define
2373:subobject
2249:|
2232:, define
2194:|
2071:corollary
2044:∈
2029:∈
2007:↦
1974:→
1942:subgroups
1887:∅
1878:∩
1837:∈
1831:∀
1822:∈
1810:⊆
1577:inclusion
1569:power set
1525:≤
1522:⌉
1516:⌈
1510:⇔
1501:≤
1459:⌋
1453:⌊
1447:⌋
1435:⌊
1389:⌋
1383:⌊
1380:≤
1371:⇔
1362:≤
1324:≤
1315:⇔
1306:≤
1271:⌋
1265:⌊
1225:→
1186:∈
1143:→
1118:embedding
908:, making
882:→
850:→
831:bijection
383:bijective
134:functions
83:bijective
45:subfields
41:subgroups
4908:Archived
4696: (
4677: (
4673:(
4654:antitone
4531:between
4495:(
4451:{
4312:preserve
4275:(
4159:β
4114:β
4099:β
4049:β
4013:(
3998:(
3987:(
3940:(
3892:(
3850:β
3841:because
3809:(
3761:(
3701:(
3648:β
3639:, while
3569:(
3522:, where
3359:Suppose
3141: )
2983:and let
2873: )
2849:Given a
2828: )
2585: )
2577: )
2553: )
2528: )
2496:is that
2439: )
2426:to be a
2416: )
2393:subspace
2385:subgroup
2369: )
2198: (
2144: (
2110:function
2099: :
1717:Lattices
1679:(
1598:, where
1555:be some
1085:integers
820:Examples
803:and the
780:for all
757:for all
733: :
719: :
579: :
565: :
558:antitone
515:identity
486:for all
464:for all
436: :
418: :
263:uniquely
155: :
141: :
131:monotone
5145:1990305
4568:duality
4463: (
4439:
4421: (
4417:
4394:
4352: (
4348:
4316:suprema
4290:
4279: (
4268:
4224:
4193:
4171:
4167:nucleus
4152:
4130:
4107:
4092:
4076:
4072:
4068:
4064:
4042:
4028:
4017: (
4002: (
3980:
3929:
3885:
3872:
3858:
3843:
3802:
3754:
3741:
3733:
3725: (
3717: (
3713:
3705: (
3641:
3623:
3593:
3573: (
3551: (
3543: (
3539:
3524:
3505:
3310:radical
3143:as the
3012:variety
2948:
2929:) = 0 β
2902:
2883:
2464:.) Now
2443:closure
2389:subring
2253: {
2136:
2095:
1703:logical
1645:infimum
550:lattice
513:is the
387:inverse
121:be two
60:classes
5143:
5100:
5085:
5067:
4962:
4941:(POPL)
4902:
4894:
4857:
4837:
4768:
4758:
4576:frames
4443:
4324:infima
4320:domain
4197:
4175:
4163:
4118:
4103:
4053:
4026:i.e.,
3854:
3737:
3652:
2976:and a
2969:Fix a
2722:β¦ Fix(
2707:β¦ Gal(
2626:. Let
2547:, let
2532:axioms
2522:, let
2430:, let
2407:, let
2360:, let
2292:groups
2182:
1910:blocks
1869:
1858:
1747:β {1}.
1616:, and
1581:subset
1513:
1507:
1374:
1368:
1318:
1312:
1206:while
431:, and
5141:JSTOR
5038:PS.GZ
4983:(PDF)
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