{
localUrl: '../page/algebraic_monoid.html',
arbitalUrl: 'https://arbital.com/p/algebraic_monoid',
rawJsonUrl: '../raw/3h3.json',
likeableId: '2652',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'JaimeSevillaMolina'
],
pageId: 'algebraic_monoid',
edit: '7',
editSummary: '',
prevEdit: '6',
currentEdit: '7',
wasPublished: 'true',
type: 'wiki',
title: 'Monoid',
clickbait: '',
textLength: '3466',
alias: 'algebraic_monoid',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'PatrickStevens',
editCreatedAt: '2016-06-15 11:21:48',
pageCreatorId: 'NateSoares',
pageCreatedAt: '2016-05-09 06:55:47',
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: '95',
text: 'A monoid $M$ is a pair $(X, \\diamond)$ where $X$ is a [set_theory_set set] and $\\diamond$ is an [associative_function associative] binary [3h7 operator] with an identity. $\\diamond$ is often interpreted as concatenation; data structures that support concatenation and have an "empty element" (such as lists, strings, and the natural numbers under addition) are examples of monoids.\n\n Monoids are [3gx algebraic structures]. We write $x \\diamond y$ for the application of $\\diamond$ to $x, y \\in X$, which must be defined. $x \\diamond y$ is commonly abbreviated $xy$ when $\\diamond$ can be inferred from context. The monoid axioms (which govern the behavior of $\\diamond$) are as follows.\n\n1. (Closure) For all $x, y$ in $X$, $xy$ is also in $X$.\n1. (Associativity) For all $x, y, z$ in $X$, $x(yz) = (xy)z$.\n2. (Identity) There is an $e$ in $X$ such that, for all $x$ in $X$, $xe = ex = x.$\n\nThe axiom of closure says that $x \\diamond y \\in X$, i.e. that combining two elements of $X$ using $\\diamond$ yields another element of $X$. In other words, $X$ is [3gy closed] under $\\diamond$.\n\nThe axiom of associativity says that $\\diamond$ is an [3h4 associative] operation, which justifies omitting parenthesis when describing the application of $\\diamond$ to many elements in sequence..\n\nThe axiom of identity says that there is some element $e$ in $X$ that $\\diamond$ treats as "empty": If you apply $\\diamond$ to $e$ and $x$, then $\\diamond$ simply returns $x$. The identity is unique: Given two elements $e$ and $z$ that satisfy the axiom of identity, we have $ze = e = ez = z.$ Thus, we can speak of "the identity" $e$ of $M$. $e$ is often written $1$ or $1_M$.\n\n%%%knows-requisite([4c7]):\nEquivalently, a monoid is a category with exactly one object.\n%%%\n\nMonoids are [algebraic_semigroup semigroups] equipped with an identity. [3gd Groups] are monoids with inverses. For more on how monoids relate to other [algerbraic_structure algebraic structures], refer to the [algebraic_structure_tree tree of algebraic structures].\n\n# Notation\n\nGiven a monoid $M = (X, \\diamond)$, we say "$X$ forms a monoid under $\\diamond$." For example, the set of finite bitstrings forms a monoid under concatenation: The set of finite bitstrings is closed under concatenation; concatenation is an associative operation; and the empty bitstring is a finite bitstring that acts like an identity under concatenation. \n\n$X$ is called the [3gz underlying set] of $M$, and $\\diamond$ is called the _monoid operation_. $x \\diamond y$ is usually abbreviated $xy$. $M$ is generally allowed to substitute for $X$ when discussing the monoid. For example, we say that the elements $x, y \\in X$ are "in $M$," and sometimes write "$x, y \\in M$" or talk about the "elements of $M$."\n\n# Examples\n\nBitstrings form a monoid under concatenation, with the empty string as the identity.\n\nThe set of finite lists of elements drawn from $Y$ (for any set $Y$) form a monoid under concatenation, with the empty list as the identity.\n\nThe natural numbers [45h $\\mathbb N$] form a monid under addition, with $0$ as the identity.\n\nMonoids have found some use in functional programming languages such as [https://en.wikipedia.org/wiki/Haskell_(programming_language) Haskell] and [https://en.wikipedia.org/wiki/Scala_(programming_language) Scala], where they are used to generalize over data types in which values can be "combined" (by some operation $\\diamond$) and which include an "empty" value (the identity).\n',
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: '8',
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: [
'NateSoares',
'PatrickStevens',
'LouisPaquin'
],
childIds: [],
parentIds: [
'algebraic_structure'
],
commentIds: [],
questionIds: [],
tagIds: [
'needs_clickbait_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: '23229',
pageId: 'algebraic_monoid',
userId: 'LouisPaquin',
edit: '8',
type: 'newEditProposal',
createdAt: '2019-11-12 19:30:41',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'Typo: monid > monoid'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '17129',
pageId: 'algebraic_monoid',
userId: 'EricBruylant',
edit: '0',
type: 'newTag',
createdAt: '2016-07-19 02:08:00',
auxPageId: 'needs_clickbait_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13022',
pageId: 'algebraic_monoid',
userId: 'PatrickStevens',
edit: '7',
type: 'newEdit',
createdAt: '2016-06-15 11:21:48',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12627',
pageId: 'algebraic_monoid',
userId: 'PatrickStevens',
edit: '6',
type: 'newEdit',
createdAt: '2016-06-14 12:29:01',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12336',
pageId: 'algebraic_monoid',
userId: 'NateSoares',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-10 16:43:41',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9936',
pageId: 'algebraic_monoid',
userId: 'NateSoares',
edit: '3',
type: 'newEdit',
createdAt: '2016-05-10 23:32:55',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9934',
pageId: 'algebraic_monoid',
userId: 'NateSoares',
edit: '2',
type: 'newEdit',
createdAt: '2016-05-10 23:28:45',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9710',
pageId: 'algebraic_monoid',
userId: 'NateSoares',
edit: '1',
type: 'newEdit',
createdAt: '2016-05-09 06:55:47',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9707',
pageId: 'algebraic_monoid',
userId: 'NateSoares',
edit: '0',
type: 'newParent',
createdAt: '2016-05-09 06:25:58',
auxPageId: 'algebraic_structure',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {
lessTechnical: {
likeableId: '3410',
likeableType: 'contentRequest',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [],
id: '54',
pageId: 'algebraic_monoid',
requestType: 'lessTechnical',
createdAt: '2016-08-19 01:16:05'
}
}
}