{
localUrl: '../page/closure_mathematics.html',
arbitalUrl: 'https://arbital.com/p/closure_mathematics',
rawJsonUrl: '../raw/3gy.json',
likeableId: '2504',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
],
pageId: 'closure_mathematics',
edit: '4',
editSummary: '',
prevEdit: '3',
currentEdit: '4',
wasPublished: 'true',
type: 'wiki',
title: 'Closure',
clickbait: '',
textLength: '519',
alias: 'closure_mathematics',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
editCreatorId: 'NateSoares',
editCreatedAt: '2016-06-07 05:57:29',
pageCreatorId: 'NateSoares',
pageCreatedAt: '2016-05-09 06:18:07',
seeDomainId: '0',
editDomainId: 'AlexeiAndreev',
submitToDomainId: '0',
isAutosave: 'false',
isSnapshot: 'false',
isLiveEdit: 'true',
isMinorEdit: 'false',
indirectTeacher: 'false',
todoCount: '0',
isEditorComment: 'false',
isApprovedComment: 'true',
isResolved: 'false',
snapshotText: '',
anchorContext: '',
anchorText: '',
anchorOffset: '0',
mergedInto: '',
isDeleted: 'false',
viewCount: '32',
text: 'A [3jz set] $S$ is _closed_ under an [3h7 operation] $f$ if, whenever $f$ is fed elements of $S$, it produces another element of $S$. For example, if $f$ is a trinary operation (i.e., a function of three arguments) then "$S$ is closed under $f$" means "if $x, y, z \\in S$ then $f(x, y, z) \\in S$".\n\nFor example, the set [integer $\\mathbb Z$] is closed under addition (because adding two integers yields another integer), but the set $\\mathbb Z_5 = \\{0, 1, 2, 3, 4, 5\\}$ is not (because $1 + 5$ is not in $\\mathbb Z_5$).',
metaText: '',
isSubscribedToDiscussion: 'false',
isSubscribedToUser: 'false',
isSubscribedAsMaintainer: 'false',
discussionSubscriberCount: '1',
maintainerCount: '1',
userSubscriberCount: '0',
lastVisit: '',
hasDraft: 'false',
currentUserVote: '-2',
voteCount: '0',
lockedVoteType: '',
maxEditEver: '0',
lockedBy: '',
lockedUntil: '',
nextPageId: '',
prevPageId: '',
usedAsMastery: 'false',
proposalEditNum: '0',
permissions: {
edit: {
has: 'false',
},
proposeEdit: {
has: 'true',
reason: ''
},
delete: {
has: 'false',
},
comment: {
has: 'false',
reason: 'You can't comment in this domain because you are not a member'
},
proposeComment: {
has: 'true',
reason: ''
}
},
summaries: {},
creatorIds: [
'NateSoares'
],
childIds: [],
parentIds: [
'math'
],
commentIds: [],
questionIds: [],
tagIds: [
'needs_clickbait_meta_tag',
'definition_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [],
subjects: [],
lenses: [],
lensParentId: '',
pathPages: [],
learnMoreTaughtMap: {},
learnMoreCoveredMap: {},
learnMoreRequiredMap: {},
editHistory: {},
domainSubmissions: {},
commentCount: '0',
newCommentCount: '0',
changeLogs: [
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '17085',
pageId: 'closure_mathematics',
userId: 'EricBruylant',
edit: '0',
type: 'newTag',
createdAt: '2016-07-19 00:15:35',
auxPageId: 'needs_clickbait_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '11917',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-07 05:57:29',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10096',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '3',
type: 'newEdit',
createdAt: '2016-05-12 07:09:22',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9951',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '2',
type: 'newParent',
createdAt: '2016-05-10 23:57:12',
auxPageId: 'math',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9949',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '2',
type: 'newTag',
createdAt: '2016-05-10 23:57:07',
auxPageId: 'definition_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9946',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '0',
type: 'newAlias',
createdAt: '2016-05-10 23:56:49',
auxPageId: '',
oldSettingsValue: 'closure_of_a_set',
newSettingsValue: 'closure_mathematics'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9947',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '2',
type: 'newEdit',
createdAt: '2016-05-10 23:56:49',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9704',
pageId: 'closure_mathematics',
userId: 'NateSoares',
edit: '1',
type: 'newEdit',
createdAt: '2016-05-09 06:18:07',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}