{ localUrl: '../page/algebraic_structure.html', arbitalUrl: 'https://arbital.com/p/algebraic_structure', rawJsonUrl: '../raw/3gx.json', likeableId: '2501', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'EricRogstad' ], pageId: 'algebraic_structure', edit: '9', editSummary: '', prevEdit: '8', currentEdit: '9', wasPublished: 'true', type: 'wiki', title: 'Algebraic structure', clickbait: '', textLength: '1103', alias: 'algebraic_structure', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'NateSoares', editCreatedAt: '2016-06-07 05:58:30', pageCreatorId: 'NateSoares', pageCreatedAt: '2016-05-09 05:27:23', 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: '227', text: 'Roughly speaking, an algebraic structure is a set $X$, known as the [3gz underlying set], paired with a collection of [3h7 operations] that obey a given set of laws. For example, a [3gd group] is a set paired with a single binary operation that satisfies the four group axioms, and a [3gq ring] is a set paired with two binary operations that satisfy the ten ring axioms.\n\nIn fact, algebraic structures can have more than one underlying set. Most have only one (including [3h3 monoids], [3gd groups], [3gq rings], [algebraic_field fields], [algebraic_lattice lattices], and [algebraic_arithmetic arithmetics]), and differ in how their associated operations work. More complex algebraic structures (such as [algebraic_algebra algebras], [algebraic_module modules], and [3w0 vector spaces]) have two underlying sets. For example, vector spaces are defined using both an underlying [algebraic_field field] of scalars and an underlying [3h2 commutative group] of vectors.\n\nFor a map of algebraic structures and how they relate to each other, see the [algebraic_structure_tree tree of algebraic structures].', 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 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