Editing Semiring (section)

=== Commutative semirings ===
A semiring is called a '''[[commutative]] semiring''' if also the multiplication is commutative.{{sfnp|Lothaire|2005|p=212|ps=}} Its axioms can be stated concisely: It consists of two commutative monoids <math>\langle +, 0\rangle</math> and <math>\langle \cdot, 1\rangle</math> on one set such that <math>a\cdot 0 = 0</math> and <math>a\cdot (b+c)=a\cdot b + a\cdot c</math>.

The [[Center (ring theory)|center]] of a semiring is a sub-semiring and being commutative is equivalent to being its own center.

The commutative semiring of natural numbers is the [[initial object]] among its kind, meaning there is a unique structure preserving map of <math>{\mathbb N}</math> into any commutative semiring.

The bounded distributive lattices are [[partial order|partially ordered]], commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their [[Duality theory for distributive lattices|duals]].