{ localUrl: '../page/operator_mathematics.html', arbitalUrl: 'https://arbital.com/p/operator_mathematics', rawJsonUrl: '../raw/3h7.json', likeableId: '2522', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'OakLoom' ], pageId: 'operator_mathematics', edit: '8', editSummary: '', prevEdit: '7', currentEdit: '8', wasPublished: 'true', type: 'wiki', title: 'Operator', clickbait: '', textLength: '1523', alias: 'operator_mathematics', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-14 12:29:12', pageCreatorId: 'NateSoares', pageCreatedAt: '2016-05-09 07:38:21', 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: '57', text: 'An operation $f$ on a [3jz set] $S$ is a function that takes some values from $S$ and produces a new value. An operation can take any number of values from $S$, including zero (in which case $f$ is simply a constant) or infinitely many (in which case we call $f$ an "infinitary operation"). Common operations take a finite non-zero number of parameters. Operations often produce a value that is also in $S$ (in which case we say $S$ is [3gy closed] under $f$), but that is not always the case.\n\nFor example, the function $+$ is a binary operation on [45h $\\mathbb N$], meaning it takes two values from $\\mathbb N$ and produces another. Because $+$ produces a value that is also in $\\mathbb N$, we say that $\\mathbb N$ is closed under $+$.\n\nThe function $\\operatorname{neg}$ that maps $x$ to $-x$ is a unary operation on [48l $\\mathbb Z$]: It takes one value from $\\mathbb Z$ as input, and produces an output in $\\mathbb Z$ (namely, the negation of the input). $\\operatorname{neg}$ is also a unary operation on $\\mathbb N$, but $\\mathbb N$ is not closed under $\\operatorname{neg}$ (because $\\operatorname{neg}(3)=-3$ is not in $\\mathbb N$).\n\nThe number of values that the operator takes as input is called the [3h8 arity] of the operator. For example, the function $\\operatorname{zero}$ which takes no inputs and returns $0$ is a zero-arity operator; and the operator $f(a, b, c, d) = ac - bd$ is a four-arity operator (which can be used on any [3gq ring], if we interpret multiplication and subtraction as ring operations).', 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 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