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| In [[abstract algebra]], a branch of [[mathematics]], a '''monoid''' is an [[algebraic structure]] with a single [[associative]] [[binary operation]] and an [[identity element]]. Monoids are studied in [[semigroup]] theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as [[category (mathematics)|categories]] with a single [[object (category theory)|object]]. Thus, they capture the idea of [[function composition]] within a set. Monoids are also commonly used in [[computer science]], both in its foundational aspects and in practical programming. The set of [[string (computer science)|strings]] built from a given set of [[Character (computing)|characters]] is a [[free monoid]]. The [[transition monoid]] and [[syntactic monoid]] are used in describing [[finite state machines]], whereas [[trace monoid]]s and [[history monoid]]s provide a foundation for [[process calculi]] and [[concurrent computing]]. Some of the more important results in the study of monoids are the [[Krohn–Rhodes theorem]] and the [[star height problem]]. The history of monoids, as well as a discussion of additional general properties, are found in the article on [[semigroup]]s.
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| {{Algebraic structures |Group}}
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| == Definition ==
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| Suppose that ''S'' is a [[set (mathematics)|set]] and • is a [[binary operation]] {{nowrap|1=''S'' × ''S'' → ''S''}}. (If instead • is a function {{nowrap|1=''S'' × ''S'' → ''T''}} ''a priori'' taking values in a ''larger'' set ''T'', then one should first check '''closure''', ''i.e.'', that {{nowrap|1=''a'' • ''b''}} is in ''S'' for all ''a'' and ''b'' in ''S''.) Then ''S'' with • is a '''monoid''' if it satisfies the following two axioms:
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| ;Associativity: For all ''a'', ''b'' and ''c'' in ''S'', the equation {{nowrap|1=(''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'')}} holds.
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| ;Identity element: There exists an element ''e'' in ''S'' such that for every element ''a'' in ''S'', the equation {{nowrap|1=''e'' • ''a'' = ''a'' • ''e'' = ''a''}} holds.
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| In other words, a monoid is a [[semigroup]] with an [[identity element]]. It can also be thought of as a [[magma (algebra)|magma]] with associativity and identity. A [[group (mathematics)|group]] is the same thing as a monoid in which each element has an [[inverse element|inverse]].
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| Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written <math>(ab)c = a(bc)</math> and <math>ea=ae=a</math>. This notation does not imply that it is numbers being multiplied.
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| == Monoid structures ==
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| ===Submonoids===
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| A '''submonoid''' of a monoid {{nowrap|(''M'', •)}} is a [[subset]] ''N'' of ''M'' that is closed under the monoid operation and contains the identity element ''e'' of ''M''.<ref>Jacobson (2009)</ref><ref>Some authors omit the requirement that a submonoid must contain the identity element from its definition, requiring only that it have ''an'' identity element, which can be distinct from that of ''M''.</ref> Symbolically, ''N'' is a submonoid of ''M'' if {{nowrap|''N'' ⊆ ''M''}}, {{nowrap|''x'' • ''y'' ∈ ''N''}} whenever {{nowrap|''x'', ''y'' ∈ ''N''}}, and {{nowrap|''e'' ∈ ''N''}}. ''N'' is thus a monoid under the binary operation inherited from ''M''. | |
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| ===Generators===
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| A subset ''S'' of ''M'' is said to be a '''generator''' of ''M'' if ''M'' is the smallest set containing ''S'' that is closed under the monoid operation, or equivalently ''M'' is the result of applying the [[finitary closure operator]] to ''S''. If there is a generator of ''M'' that has finite cardinality, then ''M'' is said to be '''finitely generated'''. Not every set ''S'' will generate a monoid, as the generated structure may lack an identity element.
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| ===Commutative monoid===
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| A monoid whose operation is [[commutative]] is called a '''commutative monoid''' (or, less commonly, an '''abelian monoid'''). Commutative monoids are often written additively. Any commutative monoid is endowed with its '''algebraic''' [[preorder]]ing ≤, defined by {{nowrap|''x'' ≤ ''y''}} if there exists ''z'' such that {{nowrap|1=''x'' + ''z'' = ''y''}}. An '''order-unit''' of a commutative monoid ''M'' is an element ''u'' of ''M'' such that for any element ''x'' of ''M'', there exists a positive integer ''n'' such that {{nowrap|''x'' ≤ ''nu''}}. This is often used in case ''M'' is the [[Ordered group|positive cone]] of a [[Partially ordered set|partially ordered]] [[abelian group]] ''G'', in which case we say that ''u'' is an order-unit of ''G''.
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| ===Partially commutative monoid===
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| A monoid for which the operation is commutative for some, but not all elements is a [[trace monoid]]; trace monoids commonly occur in the theory of [[concurrent computation]].
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| == Examples ==
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| * Every [[singleton set]] {''x''} closed under a binary operation • forms the trivial (one-element) monoid, which is also the [[trivial group]].
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| * Every [[group (mathematics)|group]] is a monoid and every [[abelian group]] a commutative monoid.
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| * Every bounded [[semilattice]] is an [[idempotent]] commutative monoid.
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| ** In particular, any bounded [[lattice (order)|lattice]] can be endowed with both a [[Join and meet|meet]]- and a [[Join and meet|join]]- monoid structure. The identity elements are the lattice's top and its bottom, respectively. Being lattices, [[Heyting algebra]]s and [[Boolean algebra (structure)|Boolean algebra]]s are endowed with these monoid structures.
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| * Any [[semigroup]] ''S'' may be turned into a monoid simply by adjoining an element ''e'' not in ''S'' and defining {{nowrap|1=''e'' • ''s'' = ''s'' = ''s'' • ''e''}} for all {{nowrap|''s'' ∈ ''S''}}. This conversion of any semigroup to the monoid is done by the [[free functor]] between the [[category of semigroups]] and the [[category of monoids]] {{Citation needed|date=January 2011}}.
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| ** Thus, an idempotent monoid (sometimes known as ''find-first'') may be formed by adjoining an identity element ''e'' to the [[left zero semigroup]] over a set ''S''. The opposite monoid (sometimes called ''find-last'') is formed from the [[right zero semigroup]] over ''S''.
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| *** Adjoin an identity ''e'' to the left-zero semigroup with two elements {''lt''; ''gt''}. Then the resulting idempotent monoid {{nowrap|{''lt''; ''e''; ''gt''}<nowiki/>}} models the [[lexicographical order]] of a sequence given the orders of its elements, with ''e'' representing equality.
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| * The [[natural number]]s, '''N''', form a commutative monoid under addition (identity element [[0 (number)|zero]]), or multiplication (identity element [[1 (number)|one]]). A submonoid of '''N''' under addition is called a [[numerical monoid]].
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| * The [[positive integer]]s, {{nowrap|'''N''' ∖ {0}<nowiki/>}}, form a commutative monoid under multiplication (identity element one).
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| * The elements of any [[unital algebra|unital]] [[ring (algebra)|ring]], with addition or multiplication as the operation.
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| ** The [[integer]]s, [[rational number]]s, [[real number]]s or [[complex number]]s, with addition or multiplication as operation.<ref>Jacobson (2009), p. 29, examples 1, 2, 4 & 5.</ref>
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| ** The set of all ''n'' by ''n'' [[matrix (mathematics)|matrices]] over a given ring, with [[matrix addition]] or [[matrix multiplication]] as the operation.
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| * The set of all finite [[string (computer science)|strings]] over some fixed alphabet Σ forms a monoid with [[string concatenation]] as the operation. The [[empty string]] serves as the identity element. This monoid is denoted Σ<sup>∗</sup> and is called the '''[[free monoid]]''' over Σ.
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| * Given any monoid ''M'', the ''opposite monoid'' M<sup>op</sup> has the same carrier set and identity element as ''M'', and its operation is defined by ''x'' •<sup>op</sup> {{nowrap|1=''y'' = ''y'' • ''x''}}. Any [[commutative monoid]] is the opposite monoid of itself.
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| * Given two sets ''M'' and ''N'' endowed with monoid structure (or, in general, any finite number of monoids, ''M''<sub>1</sub>, ..., ''M<sub>k</sub>''), their [[cartesian product]] {{nowrap|''M'' × ''N''}} is also a monoid (respectively, {{nowrap|''M<sub>1</sub>'' × ... × ''M<sub>k</sub>''}}). The associative operation and the identity element are defined pairwise.<ref>Jacobson (2009), p. 35</ref>
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| * Fix a monoid ''M''. The set of all functions from a given set to ''M'' is also a monoid. The identity element is a [[constant function]] mapping any value to the identity of ''M''; the associative operation is defined [[pointwise]].
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| * Fix a monoid ''M'' with the operation • and identity element ''e'', and consider its [[power set]] ''P''(''M'') consisting of all [[subset]]s of ''M''. A binary operation for such subsets can be defined by {{nowrap|1=''S'' • ''T'' = { ''s'' • ''t'' : ''s'' ∈ ''S'', ''t'' ∈ ''T'' }<nowiki/>}}. This turns ''P''(''M'') into a monoid with identity element {''e''}. In the same way the power set of a group ''G'' is a monoid under the [[product of group subsets]].
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| * Let ''S'' be a set. The set of all functions {{nowrap|''S'' → ''S''}} forms a monoid under [[function composition]]. The identity is just the [[identity function]]. It is also called the '''[[full transformation monoid]]''' of ''S''. If ''S'' is finite with ''n'' elements, the monoid of functions on ''S'' is finite with ''n''<sup>''n''</sup> elements.
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| * Generalizing the previous example, let ''C'' be a [[category (mathematics)|category]] and ''X'' an object in ''C''. The set of all [[endomorphism]]s of ''X'', denoted End<sub>''C''</sub>(''X''), forms a monoid under composition of [[morphism]]s. For more on the relationship between category theory and monoids see below.
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| * The set of [[homeomorphism]] [[Class (set theory)|classes]] of [[compact surface]]s with the [[connected sum]]. Its unit element is the class of the ordinary 2-sphere. Furthermore, if ''a'' denotes the class of the [[torus]], and ''b'' denotes the class of the projective plane, then every element ''c'' of the monoid has a unique expression the form {{nowrap|1=''c'' = ''na'' + ''mb''}} where ''n'' is the integer ≥ 0 and ''m'' = 0, 1, or 2. We have {{nowrap|1=3''b'' = ''a'' + ''b''}}.
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| * Let <math>\langle f\rangle</math> be a cyclic monoid of order ''n'', that is, <math>\langle f\rangle = \{f^0,f^1,\dots,f^{n-1}\}</math>. Then <math>f^n = f^k</math> for some <math>0 \le k \le n</math>. In fact, each such ''k'' gives a distinct monoid of order ''n'', and every cyclic monoid is isomorphic to one of these.
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| Moreover, ''f'' can be considered as a function on the points <math>\{0,1,2,\dots,n-1\}</math> given by
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| : <math>\begin{bmatrix}
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| 0 & 1 & 2 & \dots & n-2 & n-1 \\
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| 1 & 2 & 3 & \dots & n-1 & k\end{bmatrix}</math>
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| or, equivalently
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| :<math>f(i) := \begin{cases} i+1, & \mbox{if } 0 \le i < n-1 \\ k, & \mbox{if } i = n-1. \end{cases} </math>
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| Multiplication of elements in <math>\langle f\rangle</math> is then given by function composition.
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| Note also that when <math>k = 0</math> then the function ''f'' is a permutation of <math>\{0,1,2,\dots,n-1\}</math>
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| and gives the unique [[cyclic group]] of order ''n''. | |
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| == Properties ==
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| In a monoid, one can define positive integer powers of an element ''x'' : ''x''<sup>1</sup> = ''x'', and ''x''<sup>n</sup> = ''x'' • ... • ''x'' (''n'' times) for ''n'' > 1 . The rule of powers ''x''<sup>''n'' + ''p''</sup> = ''x''<sup>''n''</sup> • ''x''<sup>''p''</sup> is obvious.
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| From the definition of a monoid, one can show that the identity element ''e'' is unique. Then, for any ''x'', one can set ''x''<sup>0</sup> = ''e'' and the rule of powers is still true with nonnegative exponents.
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| It is possible to define [[inverse element|invertible elements]]: an element ''x'' is called invertible if there exists an element ''y'' such that {{nowrap|1=''x'' • ''y'' = ''e''}} and {{nowrap|1=''y'' • ''x'' = ''e''}}. The element ''y'' is called the inverse of ''x''. If ''y'' and ''z'' are inverses of ''x'', then by associativity {{nowrap|1=''y'' = (''zx'')''y'' = ''z''(''xy'') = ''z''}}. Thus inverses, if they exist, are unique.<ref>Jacobson, I.5. p. 22</ref>
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| If ''y'' is the inverse of ''x'', one can define negative powers of ''x'' by setting {{nowrap|1=''x''<sup>−1</sup> = ''y''}} and {{nowrap|1=''x''<sup>−''n''</sup> = ''y'' • ... • ''y''}} (''n'' times) for {{nowrap|''n'' > 1}}. And the rule of exponents is still verified for all ''n'', ''p'' rational integers. This is why the inverse of ''x'' is usually written ''x''<sup>−1</sup>. The set of all invertible elements in a monoid ''M'', together with the operation •, forms a [[group (mathematics)|group]]. In that sense, every monoid contains a group (possibly only the [[trivial group]] consisting of only the identity).
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| However, not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements ''a'' and ''b'' exist such that {{nowrap|1=''a'' • ''b'' = ''a''}} holds even though ''b'' is not the identity element. Such a monoid cannot be embedded in a group, because in the group we could multiply both sides with the inverse of ''a'' and would get that {{nowrap|1=''b'' = ''e''}}, which isn't true. A monoid {{nowrap|(''M'', •)}} has the [[cancellation property]] (or is [[cancellative]]) if for all ''a'', ''b'' and ''c'' in ''M'', {{nowrap|1=''a'' • ''b'' = ''a'' • ''c''}} always implies {{nowrap|1=''b'' = ''c''}} and {{nowrap|1=''b'' • ''a'' = ''c'' • ''a''}} always implies {{nowrap|1=''b'' = ''c''}}. A commutative monoid with the cancellation property can always be embedded in a group via the [[Grothendieck group|Grothendieck construction]]. That's how the additive group of the integers (a group with operation +) is constructed from the additive monoid of natural numbers (a commutative monoid with operation + and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.
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| If a monoid has the cancellation property and is ''finite'', then it is in fact a group. Proof: Fix an element ''x'' in the monoid. Since the monoid is finite, {{nowrap|1=''x''<sup>''n''</sup> = ''x''<sup>''m''</sup>}} for some {{nowrap|''m'' > ''n'' > 0}}. But then, by cancellation we have that {{nowrap|1=''x''<sup>''m'' − ''n''</sup> = ''e''}} where ''e'' is the identity. Therefore {{nowrap|1=''x'' • ''x''<sup>''m'' − ''n'' − 1</sup> = ''e''}}, so ''x'' has an inverse.
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| The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. obviously include the identity and not so obviously are closed under the operation). This means that the cancellative elements of any commutative monoid can be extended to a group.
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| It turns out that requiring the cancellative property in a monoid is not required to perform the Grothendieck construction – commutativity is sufficient. However, if the original monoid has an [[absorbing element]] then its Grothendieck group is the trivial group. Hence the homomorphism is, in general, not injective.
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| An '''inverse monoid''' is a monoid where for every ''a'' in ''M'', there exists a unique ''a''<sup>−1</sup> in ''M'' such that {{nowrap|1=''a'' = ''a'' • ''a''<sup>−1</sup> • ''a''}} and {{nowrap|1=''a''<sup>−1</sup> = ''a''<sup>−1</sup> • ''a'' • ''a''<sup>−1</sup>}}. If an inverse monoid is cancellative, then it is a group.
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| ==Acts and operator monoids==
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| {{main|monoid act}}
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| Let ''M'' be a monoid, with the binary operation denoted by • and the identity element denoted by ''e''. Then a (left) '''''M''-act''' (or left act over ''M'') is a set ''X'' together with an operation {{nowrap|⋅ : ''M'' × ''X'' → ''X''}} which is compatible with the monoid structure as follows:
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| * for all ''x'' in ''X'': {{nowrap|1=''e'' ⋅ ''x'' = ''x''}};
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| * for all ''a'', ''b'' in ''M'' and ''x'' in ''X'': {{nowrap|1=''a'' ⋅ (''b'' ⋅ ''x'') = (''a'' • ''b'') ⋅ ''x''}}.
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| This is the analogue in monoid theory of a (left) [[group action]]. Right ''M''-acts are defined in a similar way. A monoid with an act is also known as an '''[[operator monoid]]'''. Important examples include [[transition system]]s of [[semiautomata]]. A [[transformation semigroup]] can be made into an operator monoid by adjoining the identity transformation.
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| == Monoid homomorphisms ==
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| [[File:Exponentiation as monoid homomorphism svg.svg|thumb|x150px|Example monoid homomorphism {{nowrap|''x'' ↦ 2<sup>''x''</sup>}} from {{nowrap|{{color|#004000|('''N''', +, 0)}}}} to {{nowrap|{{color|#400000|('''N''', ×, 1)}}}}. It is injective, but not surjective.]]
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| A [[homomorphism]] between two monoids {{nowrap|(''M'', ∗)}} and {{nowrap|(''N'', •)}} is a function {{nowrap|''f'' : ''M'' → ''N''}} such that
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| * ''f''(''x'' ∗ ''y'') = ''f''(''x'') • ''f''(''y'') for all ''x'', ''y'' in ''M''
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| * ''f''(''e''<sub>''M''</sub>) = ''e''<sub>''N''</sub>,
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| where ''e''<sub>''M''</sub> and ''e''<sub>''N''</sub> are the identities on ''M'' and ''N'' respectively. Monoid homomorphisms are sometimes simply called '''monoid morphisms'''.
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| Not every [[semigroup homomorphism]] is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the element it maps the identity to will be an identity of the image of the mapping. In contrast, a semigroup homomorphisms between groups is always a [[group homomorphism]], as it necessarily preserves the identity. Since for monoids this isn't always true, it is necessary to state this as a separate requirement.
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| A [[bijective]] monoid homomorphism is called a monoid [[isomorphism]]. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.
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| ==Equational presentation==
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| {{main|Presentation of a monoid}}
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| Monoids may be given a '''presentation''', much in the same way that groups can be specified by means of a [[group presentation]]. One does this by specifying a set of generators Σ, and a set of relations on the [[free monoid]] Σ<sup>∗</sup>. One does this by extending (finite) [[binary relation]]s on Σ<sup>∗</sup> to monoid congruences, and then constructing the quotient monoid, as above.
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| Given a binary relation {{nowrap|''R'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup>}}, one defines its symmetric closure as {{nowrap|''R'' ∪ ''R''<sup>−1</sup>}}. This can be extended to a symmetric relation {{nowrap|''E'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup>}} by defining {{nowrap|''x'' ~<sub>''E''</sub> ''y''}} if and only if {{nowrap|1=''x'' = ''sut''}} and {{nowrap|1=''y'' = ''svt''}} for some strings {{nowrap|''u'', ''v'', ''s'', ''t'' ∈ Σ<sup>∗</sup>}} with {{nowrap|(''u'',''v'') ∈ ''R'' ∪ ''R''<sup>−1</sup>}}. Finally, one takes the reflexive and transitive closure of ''E'', which is then a monoid congruence.
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| In the typical situation, the relation ''R'' is simply given as a set of equations, so that <math>R=\{u_1=v_1,\cdots,u_n=v_n\}</math>. Thus, for example,
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| :<math>\langle p,q\,\vert\; pq=1\rangle</math>
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| is the equational presentation for the [[bicyclic monoid]], and
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| :<math>\langle a,b \,\vert\; aba=baa, bba=bab\rangle</math>
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| is the [[plactic monoid]] of degree 2 (it has infinite order). Elements of this plactic monoid may be written as <math>a^ib^j(ba)^k</math> for integers ''i'', ''j'', ''k'', as the relations show that ''ba'' commutes with both ''a'' and ''b''. | |
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| == Relation to category theory ==
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| {{Group-like structures}}
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| Monoids can be viewed as a special class of [[category theory|categories]]. Indeed, the axioms required of a monoid operation are exactly those required of [[morphism]] composition when restricted to the set of all morphisms whose source and target is a given object.<ref name=Awo10>{{cite book | zbl=1100.18001 | title=Category Theory | volume=49 | series=Oxford Logic Guides | first=Steve | last=Awodey | authorlink=Steve Awodey | publisher=[[Oxford University Press]] | year=2006 | isbn=0-19-856861-4 | page=10 }}</ref> That is,
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| :''A monoid is, essentially, the same thing as a category with a single object.''
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| More precisely, given a monoid {{nowrap|(''M'', •)}}, one can construct a small category with only one object and whose morphisms are the elements of ''M''. The composition of morphisms is given by the monoid operation •.
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| Likewise, monoid homomorphisms are just [[functor]]s between single object categories.<ref name=Awo10/> So this construction gives an [[equivalence of categories|equivalence]] between the [[category of monoids|category of (small) monoids]] '''Mon''' and a full subcategory of the category of (small) categories '''Cat'''. Similarly, the [[category of groups]] is equivalent to another full subcategory of '''Cat'''.
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| In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid.
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| Monoids, just like other algebraic structures, also form their own category, '''Mon''', whose objects are monoids and whose morphisms are monoid homomorphisms.<ref name=Awo10/>
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| There is also a notion of [[monoid (category theory)|monoid object]] which is an abstract definition of what is a monoid in a category. A monoid object in '''[[category of sets|Set]]''' is just a monoid.
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| == Monoids in computer science ==
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| In computer science, many [[abstract data types]] can be endowed with a monoid structure. In a common pattern, a [[sequence]] of elements of a monoid is "[[fold (higher-order function)|folded]]" or "accumulated" to produce a final value. For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can be [[parallelization|parallelized]] by employing a [[prefix sum]] or similar algorithm, in order to utilize multiple cores or processors efficiently.
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| Given a sequence of values of type ''M'' with identity element <math>\varepsilon</math> and associative operation <math>*</math>, the ''fold'' operation is defined as follows:
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| :<math>\mathrm{fold}: M^{*} \rarr M = l \mapsto \begin{cases} \varepsilon & \mbox{if } l = \mathrm{nil} \\ m * \mathrm{fold} \, l' & \mbox{if } l = \mathrm{cons} \, m \, l' \end{cases}</math>
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| In addition, any [[data structure]] can be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" a [[binary tree]] might differ depending on pre-order vs. post-order [[tree traversal]].
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| == See also ==
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| * [[Kleene algebra]]
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| * [[Monad (functional programming)]]
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| * [[Star height problem]]
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| * [[Vedic square]]
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| {{Div col end}}
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| | [[Group (mathematics)|Group]]
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| | Monoid
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| | [[Semigroup]]
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| | [[Magma (algebra)|Magma]]
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| | [[Operation_(mathematics)|Operation]]
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| |-
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| | [[Closure (mathematics)|Closure]]
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| | [[Associativity]]
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| |}
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| |-
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| | [[Identity element|Identity]]
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| |-
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| | [[Inverse element|Inverses]]
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| |}
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| |}
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| |}
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{citation | first=John M. | last=Howie | authorlink=John Mackintosh Howie | title=Fundamentals of Semigroup Theory | series=London Mathematical Society Monographs. New Series | volume=12 | year=1995 | publisher=Clarendon Press | location=Oxford | isbn=0-19-851194-9 | zbl=0835.20077 }}
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| * {{citation | last=Jacobson | first=Nathan | author-link=Nathan Jacobson | title=Lectures in Abstract Algebra | volume=I | publisher=D. Van Nostrand Company | year=1951 | isbn=0-387-90122-1}}
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| * {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189-1}}
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| * {{citation | zbl=0945.20036 | last1=Kilp | first1=Mati | last2=Knauer | first2=Ulrich | last3=Mikhalev | first3=Alexander V. | title=Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers| series=de Gruyter Expositions in Mathematics | volume=29 | location=Berlin | publisher=Walter de Gruyter | year=2000 | isbn=3-11-015248-7 }}
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| *{{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | others=Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R.; Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon | title=Combinatorics on words | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=17 | publisher=[[Cambridge University Press]] | year=1997 | doi = 10.1017/CBO9780511566097 | isbn=0-521-59924-5 | mr = 1475463 | zbl=0874.20040 }}
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| == External links ==
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| * {{springer|title=Monoid|id=p/m064740}}
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| * {{MathWorld | urlname=Monoid | title=Monoid}}
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| * {{PlanetMath| urlname=Monoid | title=Monoid | id=389}}
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| [[Category:Algebraic structures]]
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| [[Category:Category theory]]
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| [[Category:Semigroup theory]]
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