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Commutative monoids
Abbreviation: CMon
Definition
A \emph{commutative monoid} is a monoids M=⟨M,⋅,e⟩M=⟨M,⋅,e⟩ such that
⋅⋅ is commutative: x⋅y=y⋅xx⋅y=y⋅x
Definition
A \emph{commutative monoid} is a structure M=⟨M,⋅,e⟩M=⟨M,⋅,e⟩, where ⋅⋅ is an infix binary operation, called the \emph{monoid product}, and ee is a constant (nullary operation), called the \emph{identity element}, such that
⋅⋅ is commutative: x⋅y=y⋅xx⋅y=y⋅x
⋅⋅ is associative: (x⋅y)⋅z=x⋅(y⋅z)(x⋅y)⋅z=x⋅(y⋅z)
ee is an identity for ⋅⋅: e⋅x=xe⋅x=x
Morphisms
Let MM and NN be commutative monoids. A morphism from MM to NN is a function h:M→Nh:M→N that is a homomorphism:
h(x⋅y)=h(x)⋅h(y)h(x⋅y)=h(x)⋅h(y), h(e)=eh(e)=e
Examples
Example 1: ⟨N,+,0⟩, the natural numbers, with addition and zero. The finitely generated free commutative monoids are direct products of this one.
Basic results
Properties
Finite members
f(1)=1f(2)=2f(3)=5f(4)=19f(5)=78f(6)=421f(7)=2637