Category:Semirings

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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.