Additive preorders of semirings

Semirings come from two major sources: classical ring theory and ordered algebraic structures. Every semiring $R$ is associated with an additive preorder: the natural increasing action preorder of the commutative additive monoid of $R$. This has $a \sqsubseteq b \Leftrightarrow \exists c : a+c = b$. This breaks up semirings into two basic classes: those whose additive preorders are symmetric and those whose preorder are antisymmetric.

Those semirings with symmetric preorder are precisely rings. To see this, notice that all J-total monoids are groups. on the other hand, it can be shown that semirings with antisymmetric preorder are ordered algebraic structures. This divides semirings along the lines that they most apppear in applications. Beyond these there are all the in between cases. The additively ordered semiring $\mathbb{N}$ is a special case, because although it is additively J-trivial it is also cancellative so that it can be embedded in a ring $\mathbb{Z}$. Most additively ordered semirings, such as the idempotent semirings emerging from quantales, cannot be embedded in rings under any circumstances. Numerical semigroups are also cancellative, commutative, and J-trivial, so they are another source of addition of semirings of this type.

Semirings with antisymmetric addition also emerge from classical ring theory. The ideals of a commutative ring form an idempotent semiring by the addition and multiplication of ideals. Radical ideals form a bounded distributive lattice $Spec(R)$ which means that they can also be represented as semirings under their lattice operations.

Theorem. let $R$ be a semiring with additive preorder $\sqsubseteq$. Then addition and multiplication are both monotone over this preorder.

Proof. (1) suppose $a \sqsubseteq b$ then $a+c = b$. Let $d$ be another element, then $(d+a)+c = (d+b)$ so $d+a \subseteq d+b$ by $c$.

(2) suppose $a \subseteq b$ then $a+c = b$. Let $d$ be another element, then $d(a+c) = da + dc = db$ so $da\subseteq db$ by $dc$. $\square$

By the preceding theorem, every semiring is a preordered algebraic structure. This is even true in the case of rings, it is just that the additive preorder on rings is the complete relation, and every map to a complete relation is monotone. Complete preorders are maximal in the hom class comparison ordering induced by underlying set functor.

In the more interesting case, when the semiring is not a ring, this turns any semiring into a preordered algebraic structure. Further, every semiring with antisymmetric preorder is an ordered algebraic structure. $\mathbb{N}$ is a good first example: both addition and multiplication are monotone over the additive ordering of the natural numbers. This is the subject of a corollary.

Corollary. let $R$ be a semiring with additive partial order $\sqsubseteq$. Then $R$ is an ordered semiring with respect to its additive ordering.

This covers the second basic case of semirings besides rings: those emerging from ordered algebraic structures. This idea neatly divides semirings into two basic classes: rings and ordered semirings. The other semirings have a mix of symmetry and antisymmetry of some kind.

Idempotent semirings are a very promising case, because they allow us to define all kinds of algebraic operations on sets: such as the arithmetic of ideals of semigroups and rings and the composition of morphism systems of a category. The use of idempotent semirings in category theory gets around the use of partial operations. There are also links between idempotent semirings, quantales, and hyperoperations that could be of use in number theory.