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A term used as an abbreviation for the phrase "[[semi-group]] with [[identity element|identity]]" . Thus, a monoid is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647401.png" /> with an associative binary operation, usually called multiplication, in which there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647402.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647403.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647404.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647405.png" /> is called the identity (or unit) and is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647406.png" />. In any monoid there is exactly one identity. If the operation given on the monoid is commutative, it is often called addition and the identity is called the zero and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647407.png" />.
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Examples of monoids. 1) The set of all mappings of an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647408.png" /> into itself is a monoid relative to the operation of successive application (composition) of mappings. The identity mapping is the identity. 2) The set of endomorphisms of a [[universal algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m0647409.png" /> is a monoid relative to composition; the identity is the identity endomorphism. 3) Every [[group]] is a monoid.
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Every semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474010.png" /> without an identity can be imbedded in a monoid. For this it suffices to take a symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474011.png" /> not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474012.png" /> and give a multiplication on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474013.png" /> as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474015.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474016.png" />, and on elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474017.png" /> the operation is as before. Every monoid can be represented as the monoid of all endomorphisms of some universal algebra.
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A term used as an abbreviation for the phrase "[[semi-group]] with [[identity element|identity]]" . Thus, a monoid is a set  $  M $
 +
with an associative binary operation, usually called multiplication, in which there is an element  $  e $
 +
such that  $  ex = x = xe $
 +
for any $  x \in M $.  
 +
The element  $  e $
 +
is called the identity (or unit) and is usually denoted by  $  1 $.  
 +
In any monoid there is exactly one identity. If the operation given on the monoid is commutative, it is often called addition and the identity is called the zero and is denoted by  $  0 $.
  
An arbitrary monoid can also be considered as a [[category]] with one object. This allows one to associate with a monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474018.png" /> its dual (opposite, adjoint) monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474019.png" />. The elements of both monoids coincide, but the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474022.png" /> is put equal to the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474024.png" />.
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Examples of monoids. 1) The set of all mappings of an arbitrary set  $  S $
 +
into itself is a monoid relative to the operation of successive application (composition) of mappings. The identity mapping is the identity. 2) The set of endomorphisms of a [[universal algebra]]  $  A $
 +
is a monoid relative to composition; the identity is the identity endomorphism. 3) Every [[group]] is a monoid.
  
The development of the theory of monoids and adjoint functors has shown the utility of the definition of a monoid in so-called monoidal categories. Suppose given a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474025.png" /> equipped with a bifunctor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474026.png" />, an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474027.png" /> and natural isomorphisms
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Every semi-group  $  P $
 +
without an identity can be imbedded in a monoid. For this it suffices to take a symbol  $  1 $
 +
not in $  P $
 +
and give a multiplication on the set  $  P \cup \{ 1 \} $
 +
as follows: $  1 \cdot 1 = 1 $,
 +
$  1 \cdot x = x = x \cdot 1 $
 +
for any  $  x \in P $,  
 +
and on elements from  $  P $
 +
the operation is as before. Every monoid can be represented as the monoid of all endomorphisms of some universal algebra.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474028.png" /></td> </tr></table>
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An arbitrary monoid can also be considered as a [[category]] with one object. This allows one to associate with a monoid  $  M $
 +
its dual (opposite, adjoint) monoid  $  M ^ { \mathop{\rm op} } $.  
 +
The elements of both monoids coincide, but the product of  $  x $
 +
and  $  y $
 +
in  $  M ^ { \mathop{\rm op} } $
 +
is put equal to the product  $  yx $
 +
in  $  M $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474029.png" /></td> </tr></table>
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The development of the theory of monoids and adjoint functors has shown the utility of the definition of a monoid in so-called monoidal categories. Suppose given a category  $  \mathfrak M $
 +
equipped with a bifunctor  $  \otimes : \mathfrak M \times \mathfrak M \rightarrow \mathfrak M $,
 +
an object  $  Z $
 +
and natural isomorphisms
  
satisfying coherence conditions. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474030.png" /> is called a monoid in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474031.png" /> if there are morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474033.png" /> such that the following diagrams are commutative:
+
$$
 +
\alpha _ {ABC} : ( A \otimes B ) \otimes C  \rightarrow  A \otimes ( B \otimes C ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474034.png" /></td> </tr></table>
+
$$
 +
\lambda _ {A} : Z \otimes A  \rightarrow  A ,\  \rho _ {A} : A \otimes Z  \rightarrow  A ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474035.png" /></td> </tr></table>
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satisfying coherence conditions. An object  $  M $
 +
is called a monoid in the category  $  \mathfrak M $
 +
if there are morphisms  $  \mu : M \otimes M \rightarrow M $
 +
and  $  \epsilon : Z \rightarrow M $
 +
such that the following diagrams are commutative:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474036.png" /> is taken to be the [[category of sets]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474037.png" /> the [[Cartesian product]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474038.png" /> a one-point set, and the isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474041.png" /> are chosen in the natural way (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474043.png" />), then the second definition of a monoid turns out to be equivalent to the original definition.
+
$$
 +
 
 +
$$
 +
 
 +
If $  \mathfrak M $
 +
is taken to be the [[category of sets]], $  \otimes $
 +
the [[Cartesian product]], $  Z $
 +
a one-point set, and the isomorphisms $  \alpha $,  
 +
$  \lambda $
 +
and $  \rho $
 +
are chosen in the natural way ( $  \alpha (( a , b ) , c )= ( a , ( b , c )) $,
 +
$  \lambda ( z , a ) = a = \rho ( a , z ) $),  
 +
then the second definition of a monoid turns out to be equivalent to the original definition.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For monoidal categories, and particularly the coherence conditions that the isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064740/m06474045.png" /> must satisfy cf. [[#References|[1]]], Chapt. 7, Sects. 1-2.
+
For monoidal categories, and particularly the coherence conditions that the isomorphisms $  \alpha _ {ABC }  $,  
 +
$  \lambda _ {A} $
 +
must satisfy cf. [[#References|[1]]], Chapt. 7, Sects. 1-2.

Latest revision as of 13:58, 7 June 2020


A term used as an abbreviation for the phrase "semi-group with identity" . Thus, a monoid is a set $ M $ with an associative binary operation, usually called multiplication, in which there is an element $ e $ such that $ ex = x = xe $ for any $ x \in M $. The element $ e $ is called the identity (or unit) and is usually denoted by $ 1 $. In any monoid there is exactly one identity. If the operation given on the monoid is commutative, it is often called addition and the identity is called the zero and is denoted by $ 0 $.

Examples of monoids. 1) The set of all mappings of an arbitrary set $ S $ into itself is a monoid relative to the operation of successive application (composition) of mappings. The identity mapping is the identity. 2) The set of endomorphisms of a universal algebra $ A $ is a monoid relative to composition; the identity is the identity endomorphism. 3) Every group is a monoid.

Every semi-group $ P $ without an identity can be imbedded in a monoid. For this it suffices to take a symbol $ 1 $ not in $ P $ and give a multiplication on the set $ P \cup \{ 1 \} $ as follows: $ 1 \cdot 1 = 1 $, $ 1 \cdot x = x = x \cdot 1 $ for any $ x \in P $, and on elements from $ P $ the operation is as before. Every monoid can be represented as the monoid of all endomorphisms of some universal algebra.

An arbitrary monoid can also be considered as a category with one object. This allows one to associate with a monoid $ M $ its dual (opposite, adjoint) monoid $ M ^ { \mathop{\rm op} } $. The elements of both monoids coincide, but the product of $ x $ and $ y $ in $ M ^ { \mathop{\rm op} } $ is put equal to the product $ yx $ in $ M $.

The development of the theory of monoids and adjoint functors has shown the utility of the definition of a monoid in so-called monoidal categories. Suppose given a category $ \mathfrak M $ equipped with a bifunctor $ \otimes : \mathfrak M \times \mathfrak M \rightarrow \mathfrak M $, an object $ Z $ and natural isomorphisms

$$ \alpha _ {ABC} : ( A \otimes B ) \otimes C \rightarrow A \otimes ( B \otimes C ) , $$

$$ \lambda _ {A} : Z \otimes A \rightarrow A ,\ \rho _ {A} : A \otimes Z \rightarrow A , $$

satisfying coherence conditions. An object $ M $ is called a monoid in the category $ \mathfrak M $ if there are morphisms $ \mu : M \otimes M \rightarrow M $ and $ \epsilon : Z \rightarrow M $ such that the following diagrams are commutative:

$$ $$

If $ \mathfrak M $ is taken to be the category of sets, $ \otimes $ the Cartesian product, $ Z $ a one-point set, and the isomorphisms $ \alpha $, $ \lambda $ and $ \rho $ are chosen in the natural way ( $ \alpha (( a , b ) , c )= ( a , ( b , c )) $, $ \lambda ( z , a ) = a = \rho ( a , z ) $), then the second definition of a monoid turns out to be equivalent to the original definition.

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2] S. MacLane, "Categories for the working mathematician" , Springer (1971)

Comments

For monoidal categories, and particularly the coherence conditions that the isomorphisms $ \alpha _ {ABC } $, $ \lambda _ {A} $ must satisfy cf. [1], Chapt. 7, Sects. 1-2.

How to Cite This Entry:
Monoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monoid&oldid=49310
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article