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，一的例子是， 中“加” 操作是“与”（AND）， 零元素是“真” （TRUE）， 并且结合律仍然成立，同一律也仍然成立。

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

我们有半、半和半环。

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

所以半和半是一回事，但它还具有零元的概念。

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，你可以获取这半接口，并为添加一些额外的功能，从而变为半。

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.

但是任何针对操作的东西， 也就是说，一针对这种接口编写的算法，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.

然后这半环的东西有点像双， 我们有两可执行的结合运算和两不同的位元素，每位元素都对应一运算。