Definition:Additive Semiring
Definition
An additive semiring is a semiring with a commutative distributand.
That is, an additive semiring is a ringoid $\struct {S, *, \circ}$ in which:
- $(1): \quad \struct {S, *}$ forms a commutative semigroup
- $(2): \quad \struct {S, \circ}$ forms a semigroup.
Additive Semiring Axioms
An additive semiring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:
\((\text A 0)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a * b \in S \) | Closure under $*$ | |||||
\((\text A 1)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a * b} * c = a * \paren {b * c} \) | Associativity of $*$ | |||||
\((\text A 2)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a * b = b * a \) | Commutativity of $*$ | |||||
\((\text M 0)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | Closure under $\circ$ | |||||
\((\text M 1)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | Associativity of $\circ$ | |||||
\((\text D)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | $\circ$ is distributive over $*$ | |||||
\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c} \) |
These criteria are called the additive semiring axioms.
Note on Terminology
The term additive semiring was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ to describe this structure.
As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.
Most of the literature simply calls this a semiring; however, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the term semiring is reserved for more general structures, not imposing that the distributand be commutative.
Also see
- Definition:Ringoid (Abstract Algebra)
- Definition:Semiring (Abstract Algebra)
- Definition:Rig
- Definition:Ring (Abstract Algebra)
Sources
- Weisstein, Eric W. "Semiring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Semiring.html