Monoid

In abstract algebra, a monoid is an algebraic structure with a binary operation, which is associative and has a neutral element, that is, it is a semigroup with a neutral element.

Formal definition

A monkey (A, ){displaystyle (A,circledcirc)} is an algebraic structure in which A{displaystyle A,} It's a set and {displaystyle circledcirc} is an internal binary operation A{displaystyle A,}:

:A× × AΔ Δ A(a,b) c=a b{displaystyle {begin{array}{rccl}circledcirc: strangerAtimes A hypolongrightarrow &Alongrightarrow (a,b) strangerlongmapsto &c=acircledcirc bend{array}}}}}}

Which satisfies the following three properties (the first is redundant with the definition):

It is easy to show that the neutral element is necessarily unique, so it is redundant to demand its uniqueness in this axiom or property. In essence, a monoid is a semigroup with a neutral element.

Commutativity

If in addition the commutative property is satisfied:

It is said to be a commutative or abelian monoid.

Examples

Concatenation of alphanumeric strings

Given a set A of alphanumeric characters, which we will call an alphabet, an alphanumeric string of the alphabet A is a sequence of elements of A in any order and of any length, if you take the set as:

A={d,e,f,g,5,8,9!{displaystyle A={d,e,f,g,5,8,9}}

Strings of the alphabet A, which we represent C(A) can be:

fdggdd {displaystyle langle fdggddrangle }

df5d8 {displaystyle langle df5d8rangle }

888 {displaystyle langle 888rangle }

eeefeffe {displaystyle langle eeefefferangle }

The empty string, the one with no characters, would be:

{displaystyle langle rangle }

We defined the operation {displaystyle ALES} concatenation of alphabet chains A like:

:C(A)× × C(A)→ → C(A)(a,b)→ → c=a b{displaystyle {begin{array}{rccl} ages: fakeC(A)times C(A) aliento &C(A) dream(a,b) strangerto &c=aexpends{array}}}}}

that we can represent, in the following ways:

• egdd dfdf → → egdddfdf {displaystyle langle egddranglelangle dfdfrangle ;to ;langle egdddfdfrangle }

• 589 gg → → 589gg {displaystyle langle 589ranglelangle ggrangle ;to ;langle 589ggrangle }

We can see that (C(A), ){displaystyle (C(A),78)} has monoid algebraic structure:

1.- It is an internal operation: for any two strings of the alphabet A their concatenation is a string of A:

Русский Русский a,b한 한 C(A):a b한 한 C(A){displaystyle forall a,bin C(A):quad aassociatedbin C(A)}.

2.- It is associative:

Русский Русский a,b,c한 한 C(A):a (b c)=(a b) c{displaystyle forall a,b,cin C(A):quad aassociated(bassociatedc)=(aassociatedb)associatedc;}

3.- It has neutral element: for every element a string characters A, there is the empty chain {displaystyle langle rangle } of ASo:

Русский Русский a한 한 C(A):consuming consuming : a=a =a{displaystyle forall ain C(A):quad exists ,langle rangle:quad langle rangle intangle rangle intangle rangle =a}

String concatenation is not commutative:

a,b한 한 C(A):a bI was. I was. b a{displaystyle a,bin C(A):quad aorganbneq bassociateda}

Being a, b of C(A) the concatenation of a with b is not equal to the concatenation of b with a.

Then the concatenation of alphanumeric strings is a noncommutative monoid.

Multiplying natural numbers

Starting from the set of natural numbers:

N={1,2,3,4,...... !{displaystyle mathbb {N} ={1,2,3,4,dots },}

and operation multiplication, we can see that: (N,× × ){displaystyle (mathbb {N}times)} He's a monkey.

1.- It is an internal operation: for any two natural numbers its multiplication is a natural number:

Русский Русский a,b한 한 N:a× × b한 한 N{displaystyle forall a,bin mathbb {N}:quad atimes bin mathbb {N} }.

2.- It is associative:

Русский Русский a,b,c한 한 N:a× × (b× × c)=(a× × b)× × c{displaystyle forall a,b,cin mathbb {N}:quad atimes (btimes c)=(atimes b)times c;}

3.- It has a neutral element: 1 in N is neutral for all natural numbers since it fulfills:

consuming consuming 1한 한 N:Русский Русский a한 한 N:1× × a=a× × 1=a{displaystyle exists ,1in mathbb {N}:quad forall ain mathbb {N}:quad 1times a=atimes 1=a}

4.- The multiplication of natural numbers is commutative:

Русский Русский a,b한 한 A:a× × b=b× × a{displaystyle forall a,bin A:quad atimes b=btimes a;}

The set of natural numbers, under the operation multiplication: (N,× × ){displaystyle (mathbb {N}times)}, has algebraic structure of monoid switching or abelian.

On category theory

Definition as category

A monoid can also be viewed as a particular type of category. Specifically, a monoid can be defined as a category with a single object.

Given a category C{displaystyle {mathsf {C}}} and its object A{displaystyle A}All the morbids of A{displaystyle A} in A{displaystyle A} form a set Hom (A,A){displaystyle operatorname {Hom} (A,A)}. On this set, the composition of morphisms defines an internal binary operation. Due to the axioms of the theory of categories, the composition of morphisms is associative and there must be a morphism identity 1A:A→ → A{displaystyle 1_{A}:Ato A}, so the whole Hom (A,A){displaystyle operatorname {Hom} (A,A)} equipped with the composition of morphisms constitutes a monoid.

In this way, any category with a single object A{displaystyle A} gives rise to a monoid by taking the set of morphisms Hom (A,A){displaystyle operatorname {Hom} (A,A)}. It is also possible to go in the opposite direction and define, from a monoid M{displaystyle M}a category with one object A{displaystyle A} such as Hom (A,A)=M{displaystyle operatorname {Hom} (A,A)=M}thus justifying the alternative definition of monoid in terms of categories.

Monoidal Category

A monoidal category is a category C{displaystyle {mathsf {C}}}, equipped with a bifuntor :C× × C→ → C{displaystyle otimes:{mathsf {C}}times {mathsf {C}}to {mathsf {C}}}}which satisfies properties similar to those of the binary operation in a monoid. Two examples are:

1. The category of sets with the union disjoined of sets and the empty set as neutral element.
2. Category VectK{displaystyle mathbf {Vect} _{mathbb {K} }} of vectorial spaces on a body K{displaystyle mathbb {K} } together with the tensorial product of vector spaces and K{displaystyle mathbb {K} } like the neutral element.