半環 (rig)
semiring。rig
More sophisticatedly, we can say that, just as a ring is a monoid object in abelian groups, so a rig is a monoid object in abelian monoids and a semiring is a monoid object in abelian semigroups, where abelian groups, abelian monoids and abelian semigroups have suitable monoidal structures (they are not the cartesian ones).
Equivalently, a semiring is the hom-set of a category with a single object that is enriched in the category of abelian semigroups.
半圈 (semicategory)
a semigroup is (the hom-set of) a semicategory with a single object;
a semiring is (the hom-set of) a semicategory enriched in Ab with a single object.
乘法$ (R,\cdot)が monoidである。單位を$ 1と呼ぶ 左$ a\cdot(b+c)=(a\cdot b)+(a\cdot c)
右$ (a+b)\cdot c=(a\cdot c)+(b\cdot c)
加法$ +の單位 0 が乘法$ \cdotの吸收元である。$ 0\cdot a=a\cdot 0=0
例
$ (\max,+)代數の$ \maxを加法 (單位は$ -\infty)、$ +を乘法 (單位は$ 0) として$ \R\cup-\inftyをtropical 半環と定義できる 例