# Category:Semirings

This category contains results about Semirings.

A semiring is a ringoid $\left({S, *, \circ}\right)$ in which:

$(1): \quad \left({S, *}\right)$ forms a semigroup
$(2): \quad \left({S, \circ}\right)$ forms a semigroup.

That is, such that $\left({S, *, \circ}\right)$ has the following properties:

 $(A0)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a * b \in S$ $(A1)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)$ $(M0)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b \in S$ $(M1)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$ $(D)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right)$ $\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)$

These are called the semiring axioms.

## Pages in category "Semirings"

The following 2 pages are in this category, out of 2 total.