# Category:Semirings

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This category contains results about Semirings.

A **semiring** is a ringoid $\struct {S, *, \circ}$ in which:

That is, such that $\struct {S, *, \circ}$ has the following properties:

\((\text A 0)\) | $:$ | \(\displaystyle \forall a, b \in S:\) | \(\displaystyle a * b \in S \) | |||||

\((\text A 1)\) | $:$ | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle \paren {a * b} * c = a * \paren {b * c} \) | |||||

\((\text M 0)\) | $:$ | \(\displaystyle \forall a, b \in S:\) | \(\displaystyle a \circ b \in S \) | |||||

\((\text M 1)\) | $:$ | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | |||||

\((\text D)\) | $:$ | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | |||||

\(\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |

These are called the **semiring axioms**.

## Pages in category "Semirings"

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