Commutative monoid - Knowlpedia

Definition

A commutative monoid is a monoid $(A,\cdot,1)$ such that

ab=ba

for all $a,b\in A$ . A commutative monoid with zero also has an absorbing element $0$ , so $0a=0$ for every $a\in A$ .

Conventions

“Abelian monoid” is a common synonym. Additively written commutative monoids use $(A,+,0)$ ; multiplicative notation is standard in blueprint theory. In that setting $0$ and $1$ are distinguished and usually required to be different unless the trivial object is explicitly allowed.

Examples

The nonnegative integers under addition form a commutative monoid.
The underlying multiplicative monoid of a commutative ring consists of all ring elements and has $0$ as an absorbing element. If the ring has no zero divisors, its nonzero elements are also closed under multiplication.
Every abelian group is a commutative monoid, but a commutative monoid need not have inverses.

References

Nicolas Bourbaki, Algebra I: Chapters 1–3, Springer, 1989. Publisher record. Relevant: Chapter I, algebraic structures with one composition law.