Lattice ordered semimodules of multisets

The free commutative monoid $F(S)$ on a set $S$ consists of all multisets on $S$. This has a natural semiring action by $\mathbb{N}$, so $F(S)$ is a semimodule. The algebraic preorder of $F(S)$ is not only a partial order, since $F(S)$ is J-trivial, but it also a lattice.

Commutativity and semimodule theory:
A semiring action on a commutative monoid is defined by a structure preserving map from $R$ to $End(M)$. This requires that three conditions must be satisfied:

• Action: $(ab)^n = a^nb^n$
• Additivity $a^n a^m = a^(n+m)$
• Multiplicativity $(a^n)^m = a^{nm}$

Every commutative monoid is a $\mathbb{N}$ semimodule. In a non-commutative monoid we can get the last two conditions, but we cannot get the first. Thusly, the techniques of semimodule theory cannot be applied to non-commutative monoids.

Commutativity and lattice theory:
The algebraic preordering of the free commutative semigroup is a lattice. This provides a natural link between commutative operations and lattices. Additionally, it is a distributive lattice of multisets.

On the other hand, the algebraic preordering on a non-commutative free semigroup is in general not a lattice. It is a partial order which orders lists by consecutive subsequences. Thusly, the techniques of lattice theory cannot be applied to non-commutative monoids. It seems that the lattice ordered semimodule structure is something that can only be applied in the commutative case.

Examination of the ordered algebraic structure:
An ordered monoid is an internal monoid in the category of preorders and monotone maps. It is not hard to see that $(\mathbb{N},\le,+)$ is an ordered monoid, but it has the stronger property that its action is biextensive. This naturally trasfers to the free commutative monoid $(F(S),+)$.

• Monotone: $\forall a,b,c : a \leq b \implies a+c \leq b+c$
• Biextensive: $\forall a,b : a \leq a + b$

The commutative monoid of all multiset addition actions is a submonoid of the monoid of all increasing monotone actions on the distributive lattice of multisets. We can therefore say that $F(S)$ is not only an ordered semimodule, its actions are also increasing.

Linear maps:
Recall from basic list processing, that mapcat replaces each element of a list with another list and then concatenates them all together. The corresponding operation over multisets is the unordered mapcat, which takes any element of a multiset and replaces it with another multiset, then adds them altogether.

The linear maps of free commutative semimodules $m : F(S) \to F(T)$ are precisely those defined by some unordered mapcat function $f : S \to F(T)$ that takes each element of the underlying set to a multiset in $F(T)$. This is also a monotone map of ordered semimodules. Therefore, the category of ordered semimodules can be used to understand the maps of free commutative monoids.