So one monoid there would be where plus is and, and the zero is true, and the associativity law still holds, the identity law still hold there.

因此，一个单群例子是， 其中“加” 操作是“与”（AND）， 零元素是“真” （TRUE）， 并且结合然成立，同一然成立。

We have semi-groups, monoids, and semi-rings.

我们有半群、半群和半环。

So a monoid is the same thing as a semigroup, but it also has a notion of a zero.

所以半群和半群是一回事，但它还具有零元。

So, and then you can take that semigroup interface and you can add some additional features to it to turn it into a monoid.

因此，你可以获取这个半群接口，并为其添加一些额外功能，从而将其转变为半群。

But anything that works on monoids, so an algorithm that programs against this mono interface, would only work with lists, but it would not work with non-empty lists because they don't support that abstract interface.

但是任何针对单群操作东西， 就是说，一个针对这种单群接口编写算法，只能用于列表， 但不能用于非空列表， 因为非空列表不支持那个抽象接口。

And then this semi-ring thing is kind of like a double monoid where we have two associative operations that we can perform and two different identity elements, each of which corresponds to the operation.

然后这个半环东西有点像双单群， 我们有两个可执行结合运算和两个不同单位元素，每个单位元素都对应一个运算。