Monoid

in abstract algebra a Monoid is a quantity with quotable a without brackets linkage and a neutral element. An example are the natural numbers with the addition andthe number of 0 as neutral element.

definition

a Monoid is a Tripel <math> (M, *, e)< /math> consisting of a quantity <math> M< /math>, one internal two digit linkage <math> *: M \ times M \ tons of M, \; (A, b) \ mapsto a*b< /math> and an excellent element <math> e \ in M< /math> with the following characteristics:

1. Assoziativität of the linkage:

<math> \ forall A, b, C \ in M: (a*b) *c=a* (b*c)< /math>

2. e is neutral element:

<math> \ forall A \ in M: e*a=a*e=a< /math>

A Monoid is thusa half-group with neutral element.

If is shown by the context, which is the neutral element, a Monoid often also shortened as pair <math> (M, *)< /math> written.

The Assoziativität (part of 1. the definition) omitting clips justifies: Forthe binary operator * the term “ A is first ambiguous * b * C”. Because however the result is invariant concerning the analysis sequence specified by clasping, one can do here without the clips.

examples and counter examples

Each group is a Monoid.

Untermonoid

a subset <math> U \ subseteq M< /math> a Monoids <math> (M, *, e)< /math>, those the neutral element <math> e< /math> contains and relativethe linkage <math> *< /math> by <math> M< /math> is final (i.e. for all <math> u, v \ in U< /math> math <u*v> \ in U /math<)> is also called Untermonoid of math <M> /math< is>.

Monoid Homomorphismus

a Monoid Homomorphismus is defined as an illustration <math> f: A \ tons of B< /math> between two Monoiden <math> \ left (A,+_A, 0_A \ right)< /math>, <math> \ left (B, +_B, 0_B \ right)< /math>, to which applies:

• <math> \ forall x, y \ in A: f (x +_A y) = f (x) +_B f (y)< /math>,
• < math> f \ left (0_A \ right) = 0_B< /math>.

It concerns here thus an illustration, those with the linkages in A and B compatiblyis and the neutral element of A on the neutral element of B illustrates. A Monoid Homomorphismus is in the sense of abstract algebra a Homomorphismus between Monoiden.

The picture <math> f \ left (A \ right)< /math> a Monoid Homomorphismus <math> f: A \ tons of B< /math> is a Untermonoid a goalmonoids B.

The Monoid Homomorphismus is <math> f: A \ tons of B< /math> bijektiv, then one calls him a mono ID isomorphicism and the Monoide A and B isomorphic.

free Monoid

a Monoid <math> \ left (M, *, \ varepsilon \ right)< /math> means free, if a nonemptySubset <math> B \ subset M \ backslash \ {\ varepsilon \}< /math> existed and it to everyone <math> p \ in M \ backslash \ {\ varepsilon \}< /math> exactly a natural number <math> n> 1< /math> and tuple <math> (b_1, b_2, \ ldots, b_n)< a /math> from elements <math> B< /math> gives, so that

< math> p = b_1 *b_2 * \ ldots * b_n< /math>

applies. B is called then basis (producer) of the Monoids.

If A is any quantity, then the quantity of all finite consequences in the quantity forms A with the Hintereinanderschreiben of the consequences as linkage and the empty consequence <math> \ varepsilon< /math>as neutral element a Monoid, <math> (A^*, \ circ, \ varepsilon)< /math>. One calls this Monoid “the free Monoid " produced by A. If the quantity of A is finite, then one speaks usually of the alphabet A and of character strings over this alphabet.

The free Monoid A* over a quantity A a role plays within many ranges of theoretical computer science (e.g. Automata theory, formal language, regular expression). See also the article over the clover ash covering for to relatives a term.

The free Monoid A* over A the following universal characteristic fulfills: M is a Monoid and <math> f \ colon A \ tons of M< /math> any function, then gives it exactly a mono ID Homomorphismus <math> to T \ colon A^* \ tons of M< /math> with <math> T (A) = f \ (A \ right) /math< left> for all <math> A \ in A< /math>. M is thus isomorphic to the free Monoid A *. Such Homomorphismen is used in theoretical computer science for the definition of formal languages (as subsets of A *).

A Monoid <left> math \ (M, *, 1 \ right)< /math> a subset of A, so that itself everyoneElement of M clearly up to the order of the factors as product of elements from A to represent leaves, then one calls M freely commutatively with the producer A. Such Monoid is necessarily commutative. A free Monoid with oneat least zweielementigen producer is not commutative.

The free Monoid is like the free group an example of a free object in the category theory.

examples

• of a quantity of A is the quantity <math> \ operator name {Abb_ {fin}} (A, \ Bbb {N} _0)< /math> all illustrations of A inthe nonnegative whole numbers, which take a value only in finally many places not equal 0, with the component-wise addition a commutative Monoid. It is freely commutatively with the basic functions <math> \ chi_a (x) = \ delta_ {A, x}< /math> as producers (math <\> delta_ {A, x is}< /math> a crowning hitting a corner earth airworthiness directive).
• The Nullmonoid<math> \ ({0}, +, 0 \ right) /math< left> is free with the empty quantity as a producer. The Monoid <math> (\ mathbb {N} _0, +, 0)< /math> 1 is free} with the only producer {. Both Monoide are also freely commutative with the producer mentioned.
• The Monoid <math> (\ mathbb N, {\ cdot}, 1)< /math> is freely commutativeover the quantity of the prime numbers, is however no free Monoid.