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semirings_with_zero [2010/07/29 18:30]
127.0.0.1 external edit
semirings_with_zero [2019/03/28 16:22] (current)
jipsen
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-$\langle S,+,0\rangle $ is a [[commutative monoids]]+$\langle S,+,0\rangle $ is a [[commutative monoid]]
-$\langle S,\cdot\rangle$ is a [[semigroups]]+$\langle S,\cdot\rangle$ is a [[semigroup]]
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==Morphisms== ==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with zero. A morphism from $\mathbf{S}$ Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with zero. A morphism from $\mathbf{S}$
-to $\mathbf{T}$ is a function $h:Sarrow T$ that is a homomorphism: +to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:
$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$ $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
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$\begin{array}{lr} $\begin{array}{lr}
f(1)= &1\\ f(1)= &1\\
-f(2)= &1\\ +f(2)= &4\\ 
-f(3)= &\\ +f(3)= &22\\ 
-f(4)= &\\+f(4)= &283\\
f(5)= &\\ f(5)= &\\
f(6)= &\\ f(6)= &\\