Commutative monoids as semimodules

The generalization of rings and modules to semirings and semimodules is justified by the fact we can use this new setting to make any commutative monoid into an $\mathbb{N}$ semimodule. Semimodules then can play a basic role in the theory of commutativity.

The basic construction:
Definition. let $R$ be a semiring and $M$ a commutative monoid, then $M$ is a (left) semimodule provided that it satisfies the following conditions:

• Action $r(m+n) = rm + rn$.
• Additivity $(r+s)m = rm + sm$.
• Multiplicativity $(rs)m = r(sm)$.
• Annihilation: $0m = r0 = 0$.

A unitary semimodule also has $1m = m$.

Theorem. let $M$ be a commutative monoid, then it is a $\mathbb{N}$ semimodule

Proof. (1) $(ab)^n = ababab...$ which by commutativity equals $a^n b^n$.
(2) $a^n a^m = \underbrace{a...}_{\text{n}}\underbrace{a...}_{\text{m}} = \underbrace{a...}_{\text{n+m}} = a^{n+m}$
(3) $(a^n)^m = \underbrace{(a^n)...}_{\text{m}} = a^{\underbrace{n...}_{\text{m}}} = a^{nm}$
(4) $a^0 = 1_M = (1_M)^n$. $\square$

This is a good first stepping stone towards a theory of commutative operations. The general idea is that commutative operations, as distinguished from non-commutative ones, are always define over some semiring: which provides coefficients, multiplicities, etc.

Guide to commutative operations:
The fact that commutative groups are $\mathbb{Z}$ modules means that different types of commutative operations are interpreted in different kinds of ways. This is described below.

• Commutative semigroups: let $S$ be a commutative semigroup, then we can adjoin an identity to it to get $S^1$ which is a $\mathbb{N}$ semimodule.
• Commutative monoids are $\mathbb{N}$ semimodules.
• Commutative groups are $\mathbb{Z}$ modules.

Commutative cancellative monoids can be embedded in $\mathbb{Z}$ modules. In particular, the free commutative $\mathbb{N}$ semimodule $F(S)$ can be embedded in the free commutative $\mathbb{Z}$ module $F^{\circ}(S)$. The techniques of module theory and semimodule are very useful in the theory of commutative operations.

Data structures:
The implementation of commutative operations in a computer algebra system requires the use of a number of different data structures:

• Multisets: $\mathbb{N}$ semimodules
• Signed multisets: $\mathbb{Z}$ modules
• Real valued sets: $\mathbb{R}$ modules

The use of modules and semimodules allows us to better describe the sort of data structures that are used in commutative monoids and commutative groups. Commutative monoids ($\mathbb{N}$ semimodules) deal with multisets, and commutative groups ($\mathbb{Z}$ modules) deal with computations over signed multisets.