Editing Monoid (section)

== Examples ==
* Out of the 16 possible [[truth table#Truth table for all binary logical operators|binary Boolean operator]]s, four have a two-sided identity that is also commutative and associative. These four each make the set {{math|{{mset|False, True}}}} a commutative monoid. Under the standard definitions, [[Logical conjunction|AND]] and [[Logical biconditional|XNOR]] have the identity {{math|True}} while [[Exclusive disjunction|XOR]] and [[Logical disjunction|OR]] have the identity {{math|False}}. The monoids from AND and OR are also [[idempotent]] while those from XOR and XNOR are not.
* The set of [[natural number]]s {{math|1='''N''' = {{mset|0, 1, 2, ...}}}} is a commutative monoid under addition (identity element [[0 (number)|{{math|0}}]]) or multiplication (identity element [[1 (number)|{{math|1}}]]). A submonoid of {{math|'''N'''}} under addition is called a [[numerical monoid]].
* The set of [[positive integer]]s {{math|'''N''' &setminus; {{mset|0}}}} is a commutative monoid under multiplication (identity element {{math|1}}).
* Given a set {{math|''A''}}, the set of subsets of {{math|''A''}} is a commutative monoid under intersection (identity element is {{math|''A''}} itself).
* Given a set {{math|''A''}}, the set of subsets of {{math|''A''}} is a commutative monoid under union (identity element is the [[empty set]]).
* Generalizing the previous example, every bounded [[semilattice]] is an [[idempotent]] commutative monoid.
** 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.
* Every [[singleton set]] {{math|{{mset|''x''}}}} closed under a binary operation {{math|•}} forms the trivial (one-element) monoid, which is also the [[trivial group]].
* Every [[group (mathematics)|group]] is a monoid and every [[abelian group]] a commutative monoid.
* Any [[semigroup]] {{math|''S''}} may be turned into a monoid simply by adjoining an element {{math|''e''}} not in {{math|''S''}} and defining {{math|1=''e'' • ''s'' = ''s'' = ''s'' • ''e''}} for all {{math|''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.{{sfn|ps=|Rhodes|Steinberg|2009|p=[https://books.google.com/books?id=8L0QIEj0PI4C&pg=PA22 22]}}
** Thus, an idempotent monoid (sometimes known as ''find-first'') may be formed by adjoining an identity element {{math|''e''}} to the [[left zero semigroup]] over a set {{math|''S''}}. The opposite monoid (sometimes called ''find-last'') is formed from the [[right zero semigroup]] over {{math|''S''}}.
*** Adjoin an identity {{math|''e''}} to the left-zero semigroup with two elements {{math|{{mset|lt, gt}}}}. Then the resulting idempotent monoid {{math|{{mset|lt, ''e'', gt}}}} models the [[lexicographical order]] of a sequence given the orders of its elements, with ''e'' representing equality.
* The underlying set of any [[ring (algebra)|ring]], with addition or multiplication as the operation.  (By definition, a ring has a multiplicative identity {{math|1}}.)
** The [[integer]]s, [[rational number]]s, [[real number]]s or [[complex number]]s, with addition or multiplication as operation.{{sfn|ps=|Jacobson|2009|p=29|loc=examples 1, 2, 4 & 5}}
** The set of all {{math|''n''}} by {{math|''n''}} [[matrix (mathematics)|matrices]] over a given ring, with [[matrix addition]] or [[matrix multiplication]] as the operation.
* The set of all finite [[string (computer science)|strings]] over some fixed alphabet {{math|Σ}} forms a monoid with [[string concatenation]] as the operation. The [[empty string]] serves as the identity element. This monoid is denoted {{math|Σ<sup>∗</sup>}} and is called the ''[[free monoid]]'' over {{math|Σ}}. It is not commutative if {{math|Σ}} has at least two elements.
* Given any monoid {{math|''M''}}, the ''opposite monoid'' {{math|''M''<sup>op</sup>}} has the same carrier set and identity element as {{math|''M''}}, and its operation is defined by {{math|1=''x'' •<sup>op</sup> ''y'' = ''y'' • ''x''}}. Any commutative monoid is the opposite monoid of itself.
* Given two sets {{math|''M''}} and {{math|''N''}} endowed with monoid structure (or, in general, any finite number of monoids, {{math|''M''<sub>1</sub>, ..., ''M<sub>k</sub>''}}), their [[Cartesian product]] {{math|''M'' × ''N''}}, with the binary operation and identity element defined on corresponding coordinates, called the [[direct product]], is also a monoid (respectively, {{math|''M''<sub>1</sub> × ⋅⋅⋅ × ''M''<sub>''k''</sub>}}).{{sfn|ps=|Jacobson|2009|p=35}}
* Fix a monoid {{math|''M''}}. The set of all functions from a given set to {{math|''M''}} is also a monoid. The identity element is a [[constant function]] mapping any value to the identity of {{math|''M''}}; the associative operation is defined [[pointwise]].
* Fix a monoid {{math|''M''}} with the operation {{math|•}} and identity element {{math|''e''}}, and consider its [[power set]] {{math|''P''(''M'')}} consisting of all [[subset]]s of {{math|''M''}}. A binary operation for such subsets can be defined by {{math|1=''S'' • ''T'' = {{mset| ''s'' • ''t'' : ''s'' ∈ ''S'', ''t'' ∈ ''T'' }}}}. This turns {{math|''P''(''M'')}} into a monoid with identity element {{math|{{mset|''e''}}}}. In the same way the power set of a group {{math|''G''}} is a monoid under the [[product of group subsets]].
* Let {{math|''S''}} be a set. The set of all functions {{math|''S'' → ''S''}} forms a monoid under [[function composition]]. The identity is just the [[identity function]]. It is also called the ''[[full transformation monoid]]'' of {{math|''S''}}. If {{math|''S''}} is finite with {{math|''n''}} elements, the monoid of functions on {{math|''S''}} is finite with {{math|''n''<sup>''n''</sup>}} elements.
* Generalizing the previous example, let {{math|''C''}} be a [[category (mathematics)|category]] and {{math|''X''}} an object of {{math|''C''}}. The set of all [[endomorphism]]s of {{math|''X''}}, denoted {{math|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.
* 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 {{math|''a''}} denotes the class of the [[torus]], and {{math|''b''}} denotes the class of the projective plane, then every element {{math|''c''}} of the monoid has a unique expression in the form {{math|1=''c'' = ''na'' + ''mb''}} where {{math|''n''}} is a positive integer and {{math|1=''m'' = 0, 1}}, or {{math|2}}. We have {{math|1=3''b'' = ''a'' + ''b''}}.
* Let {{math|{{angle bracket|{{itco|''f''}}}}}} be a cyclic monoid of order {{math|''n''}}, that is, {{math|1={{angle bracket|{{itco|''f''}}}} = {{mset|{{itco|''f''}}<sup>0</sup>, {{itco|''f''}}<sup>1</sup>, ..., {{itco|''f''}}<sup>''n''−1</sup>}}}}. Then {{math|1={{itco|''f''}}<sup>''n''</sup> = {{itco|''f''}}<sup>''k''</sup>}} for some {{math|0 ≤ ''k'' < ''n''}}. Each such {{math|''k''}} gives a distinct monoid of order {{math|''n''}}, and every cyclic monoid is isomorphic to one of these.<br/>Moreover, {{math|''f''}} can be considered as a function on the points {{math|{{mset|0, 1, 2, ..., ''n''−1}}}} given by
<math display="block">\begin{bmatrix}
 0 & 1 & 2 & \cdots & n-2 & n-1 \\
 1 & 2 & 3 & \cdots & n-1 & k\end{bmatrix}</math> or, equivalently <math display="block">f(i) := \begin{cases}
 i+1, & \text{if }  0 \le i < n-1  \\
 k,  & \text{if } i = n-1.
 \end{cases} </math>
Multiplication of elements in {{math|{{angle bracket|{{itco|''f''}}}}}} is then given by function composition. {{pb}} When {{math|1=''k'' = 0}} then the function {{math|''f''}} is a permutation of {{math|{{mset|0, 1, 2, ..., ''n''−1}}}}, and gives the unique [[cyclic group]] of order {{math|''n''}}.