Category:Semirings

From ProofWiki
Jump to navigation Jump to search

This category contains results about Semirings.


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

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


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.