{
  localUrl: '../page/order_relation.html',
  arbitalUrl: 'https://arbital.com/p/order_relation',
  rawJsonUrl: '../raw/549.json',
  likeableId: '2952',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '2',
  dislikeCount: '0',
  likeScore: '2',
  individualLikes: [
    'EricBruylant',
    'JaimeSevillaMolina'
  ],
  pageId: 'order_relation',
  edit: '9',
  editSummary: '',
  prevEdit: '8',
  currentEdit: '9',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Order relation',
  clickbait: 'A way of determining which elements of a set come "before" or "after" other elements.',
  textLength: '2768',
  alias: 'order_relation',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-07-07 16:32:44',
  pageCreatorId: 'JoeZeng',
  pageCreatedAt: '2016-07-05 21:17:15',
  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: '92',
  text: 'An **order relation** (also called an **order** or **ordering**) is a [3nt binary relation] $\\le$ on a [3jz set] $S$ that can be used to order the elements in that set.\n\nAn order relation satisfies the following properties:\n\n1. For all $a \\in S$, $a \\le a$. (the [reflexive_relation reflexive] property)\n2. For all $a, b \\in S$, if $a \\le b$ and $b \\le a$, then $a = b$. (the [antisymmetric_relation antisymmetric] property)\n3. For all $a, b, c \\in S$, if $a \\le b$ and $b \\le c$, then $a \\le c$. (the [transitive_relation transitive] property)\n\nA set that has an order relation is called a [3rb partially ordered set] (or "poset"), and $\\le$ is its *partial order*.\n\n## Totality of an order\n\nThere is also a fourth property that distinguishes between two different types of orders:\n\n4. For all $a, b \\in S$, either $a \\le b$ or $b \\le a$ or both. (the [total_relation total] property)\n\nThe total property implies the reflexive property, by setting $a = b$.\n\nIf the order relation satisfies the total property, then $S$ is called a [-540], and $\\le$ is its *total order*.\n\n## Well-ordering\n\nA fifth property that extends the idea of a "total order" is that of the [55r well-ordering]:\n\n5. For every subset $X$ of $S$, $X$ has a least element: an element $x$ such that for all $y \\in X$, we have $x \\leq y$.\n\nWell-orderings are very useful: they are the orderings we can perform [mathematical_induction induction] over. (For more on this viewpoint, see the page on [structural_induction].)\n\n# Derived relations\n\nThe order relation immediately affords several other relations.\n\n## Reverse order\n\nWe can define a *reverse order* $\\ge$ as follows: $a \\ge b$ when $b \\le a$.  \n\n## Strict order \n\nFrom any poset $(S, \\le)$, we can derive a *strict order* $<$, which disallows equality. For $a, b \\in S$, $a < b$ when $a \\le b$ and $a \\neq b$. This strict order is still antisymmetric and transitive, but it is no longer reflexive.\n\nWe can then also define a reverse strict order $>$ as follows: $a > b$ when $b \\le a$ and $a \\neq b$.\n\n## Incomparability\n\nIn a poset that is not totally ordered, there exist elements $a$ and $b$ where the order relation is undefined. If neither $a \\leq b$ nor $b \\leq a$ then we say that $a$ and $b$ are *incomparable*, and write $a \\parallel b$. \n\n## Cover relation\n\nFrom any poset $(S, \\leq)$, we can derive an underlying *cover relation* $\\prec$, defined such that for $a, b \\in S$, $a \\prec b$ whenever the following two conditions are satisfied:\n\n1. $a < b$.\n2. For all $s \\in S$, $a \\leq s < b$ implies that $a = s$.\n\nSimply put, $a \\prec b$ means that $b$ is the smallest element of $S$ which is strictly greater than $a$.\n$a \\prec b$ is pronounced "$a$ is covered by $b$", or "$b$ covers $a$", and $b$ is said to be a *cover* of $a$.',
  metaText: '',
  isTextLoaded: 'true',
  isSubscribedToDiscussion: 'false',
  isSubscribedToUser: 'false',
  isSubscribedAsMaintainer: 'false',
  discussionSubscriberCount: '1',
  maintainerCount: '1',
  userSubscriberCount: '0',
  lastVisit: '',
  hasDraft: 'false',
  votes: [],
  voteSummary: 'null',
  muVoteSummary: '0',
  voteScaling: '0',
  currentUserVote: '-2',
  voteCount: '0',
  lockedVoteType: '',
  maxEditEver: '0',
  redLinkCount: '0',
  lockedBy: '',
  lockedUntil: '',
  nextPageId: '',
  prevPageId: '',
  usedAsMastery: 'false',
  proposalEditNum: '0',
  permissions: {
    edit: {
      has: 'false',
      reason: 'You don't have domain permission to edit this page'
    },
    proposeEdit: {
      has: 'true',
      reason: ''
    },
    delete: {
      has: 'false',
      reason: 'You don't have domain permission to delete this page'
    },
    comment: {
      has: 'false',
      reason: 'You can't comment in this domain because you are not a member'
    },
    proposeComment: {
      has: 'true',
      reason: ''
    }
  },
  summaries: {},
  creatorIds: [
    'JoeZeng',
    'PatrickStevens'
  ],
  childIds: [],
  parentIds: [
    'relation_mathematics'
  ],
  commentIds: [
    '55t'
  ],
  questionIds: [],
  tagIds: [
    'formal_definition_meta_tag'
  ],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [],
  subjects: [],
  lenses: [],
  lensParentId: '',
  pathPages: [],
  learnMoreTaughtMap: {},
  learnMoreCoveredMap: {},
  learnMoreRequiredMap: {},
  editHistory: {},
  domainSubmissions: {},
  answers: [],
  answerCount: '0',
  commentCount: '0',
  newCommentCount: '0',
  linkedMarkCount: '0',
  changeLogs: [
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15992',
      pageId: 'order_relation',
      userId: 'PatrickStevens',
      edit: '9',
      type: 'newEdit',
      createdAt: '2016-07-07 16:32:44',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15991',
      pageId: 'order_relation',
      userId: 'PatrickStevens',
      edit: '8',
      type: 'newEdit',
      createdAt: '2016-07-07 16:28:19',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15990',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '7',
      type: 'newEdit',
      createdAt: '2016-07-07 16:26:51',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15989',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '6',
      type: 'newEdit',
      createdAt: '2016-07-07 16:25:45',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '2996',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '1',
      dislikeCount: '0',
      likeScore: '1',
      individualLikes: [],
      id: '15988',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '5',
      type: 'newEdit',
      createdAt: '2016-07-07 16:23:49',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15987',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '4',
      type: 'newEdit',
      createdAt: '2016-07-07 16:05:01',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15843',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '3',
      type: 'newEdit',
      createdAt: '2016-07-06 23:00:55',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15703',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-07-06 15:35:11',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15699',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-07-06 15:24:37',
      auxPageId: 'relation_mathematics',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15440',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-07-05 21:17:46',
      auxPageId: 'formal_definition_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15438',
      pageId: 'order_relation',
      userId: 'JoeZeng',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-07-05 21:17:15',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}