- : ''M'' × ''M'' → ''M'', obeying the following axioms:
- Associativity : for all ''a'', ''b'', ''c'' in M, (''a''---''b'')---''c'' = ''a''---(''b''---''c'')
- Identity Element : there exists an element ''e'' in M, such that for all ''a'' in M, ''a''---''e'' = ''e''---''a'' = ''a''.
One often sees the additional axiom
- Closure : for all ''a'', ''b'' in M, ''a''---''b'' is in M
though, strictly speaking, this isn't necessary as it is implied by the notion of a ''binary operation''.
Alternatively, a monoid is a Semigroup with an Identity Element .
A monoid satisfies all the axioms of a Group with the exception of having Inverses . A monoid with inverses is the same thing as a group.
A monoid whose operation is Commutative is called a (or, less commonly, an '''abelian monoid''').
- Every Singleton Set {''x''} gives rise to a one-element (trivial) monoid. For fixed ''x'' this monoid is unique, since the monoid axioms require that ''x''---''x'' = ''x'' in this case.
- Every Group is a monoid and every Abelian Group a commutative monoid.
- Every Semilattice is an Idempotent commutative monoid.
- Any Semigroup ''S'' may be turned into a monoid simply by adjoining an element ''e'' not in ''S'' and defining ''ee'' = ''e'' and ''es'' = ''s'' = ''se'' for all ''s'' ∈ ''S''.
- The Natural Number s, , form a commutative monoid under addition (identity element Zero ), or multiplication (identity element One ).
- The elements of any Unital Ring , with addition or multiplication as the operation.
- --- The Integer s, Rational Number s, Real Number s or Complex Number s, with addition or multiplication as operation.
- --- The set of all ''n'' by ''n'' Matrices over a given ring, with Matrix Addition or Matrix Multiplication as the operation.
- The set of all finite 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--- and is called the over Σ.
- Fix a monoid ''M'', and consider its .
- Let ''S'' be a Set . The set of all functions ''S'' → ''S'' forms a monoid under Function Composition . The identity is just the Identity Function . If ''S'' is finite with ''n'' elements, the monoid of functions on ''S'' is finite with ''n''''n'' elements.
- Generalizing the previous example, let ''C'' be a Category and ''X'' an object in ''C''. The set of all Endomorphism s of ''X'', denoted End''C''(''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 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 ''c=na+mb'' where ''n'' is the integer ''≥ 0'' and ''m=0,1,'' or ''2''. We have ''3b=a+b''.
- Let be a cyclic monoid of order ''n'', IE . Then for some . In fact, each such ''k'' gives a distinct monoid of order ''n'', and every cyclic monoid is isomorphic to one of these.
Moreover, ''f'' can be considered as a function on the points given by
:
or, equivalently
:
Multiplication of elements in is then given by function composition.
Note also that when then the function ''f'' is a permutation of
and gives the unique Cyclic Group of order ''n''.
Directly from the definition, one can show that the identity element ''e'' is unique. Then it is possible to define ''. In that sense, every monoid contains a group.
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
- ''b'' = ''a'' holds even though ''b'' is not the identity element. Such a monoid cannot be embedded in a group,
- ) has the '' Cancellation Property '' (or is '' Cancellative '') if for all ''a'', ''b'' and ''c'' in ''M'', ''a''---''b'' = ''a''---''c'' always implies ''b'' = ''c'' and ''b''---''a'' = ''c''---''a'' always implies ''b'' = ''c''. A commutative monoid with the cancellation property can always be embedded in a group. That's how the integers (a group with operation +) are constructed from the natural numbers (a commutative monoid with operation + and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.
If a monoid has the cancellation property and is ''finite'', then it is in fact a group.
An inverse monoid, is a monoid where for every ''a'' in ''M'', there exists a unique a-1 in ''M'' such that a=aa-1a and a-1=a-1aa-1.
A submonoid of a monoid ''G'', is a subset ''H'' of ''G'' containing the unit element, and such that, if ''x'',''y''∈''H'' then ''xy''∈''H''. It is then clear that ''H'' is itself a monoid, under the binary operation induced by that of ''G''.
- ) and (''M''′, @) is a function ''f'' : ''M'' → ''M''′ such that
- ''f''(''x''---''y'') = ''f''(''x'')@''f''(''y'') for all ''x'', ''y'' in ''M''
- ''f''(''e'') = ''e''′
where ''e'' and ''e''′ are the identities on ''M'' and ''M''′ respectively.
Not every magma homomorphism is a monoid homomorphism since it may not preserve the identity. Contrast this with the case of ensure that every magma homomorphism between groups preserves the identity. For monoids this isn't always true and it is necessary to state it as a separate requirement.
A Bijective monoid homomorphism is called a monoid Isomorphism . Two monoids are said to be isomorphic if there is an isomorphism between them.
Monoids can be viewed as a special class of 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. That is,
A monoid is, essentially, the same thing as a category with a single object.
- ), 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 ---.
Likewise, monoid homomorphisms are just Functor s between single object categories. 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.
Monoids, just like other algebraic structures, also form their own category, , whose objects are monoids and whose morphisms are monoid homomorphisms.
There is also a notion of Monoid Object which is an abstract definition of what is a monoid in a category.
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