什么是同构和同构

问题描述

我试图在编程的背景下理解同构和同构，并且需要一些帮助.

I tried to understand isomorphism and homomorphisms in context of programming and need some help.

在FPiS书中，它解释了:

In the book FPiS it explains:

让我们从同构开始:

"foo".length + "bar".length == ("foo" + "bar").length

在这里，长度是String to Int中的一个函数，它保留了类半球形结构.

Here, length is a function from String to Int that preserves the monoid structure.

• 为什么是同态的?

• Why is that a homomorphisms?

为什么保留monoid结构?

Why it preserve the monoid structure?

例如list上的map是否具有同态功能?

Is for example map on list function a homomorphisms?

关于同构，我有以下解释是我从书中摘录的:

About isomorphism, I have following explaination that I took it from a book:

为什么(false, ||), (true, &&)和String and List[Char] monoids with concatenation是同构的?

推荐答案

按定义.

由于上面的表达式中的==.

Because of the == in the expression above.

是的.将"foo"和"bar"替换为两个列表，将.length替换为.map(f).这样就很容易看到(证明)该方程式成立.

Yes. Replace "foo" and "bar" by two lists, and .length by .map(f). It's then easy to see (and prove) that the equation holds.

按定义.证明是微不足道的，留作练习. (提示:采用同构的定义，用具体对象替换所有抽象对象，证明所得到的数学表达式正确)

By definition. The proof is trivial, left as an exercise. (Hint: take the definition of an isomorphism, replace all abstract objects with concrete objects, prove that the resulting mathematical expression is correct)

编辑:这是您在评论中要求的几个定义:

Edit: Here are the few definitions you asked in comments:

• 同态:从一个集合到另一个集合的转换，在第二个集合中保留了第一个集合之间的关系.正式f: A → B，其中A和B都具有*操作，使得f(x * y) = f(x) * f(y).

• Homomorphism: a transformation of one set into another that preserves in the second set the relations between elements of the first. Formally f: A → B where bothA and B have a * operation such that f(x * y) = f(x) * f(y).

Monoid:具有单个关联二进制运算和一个恒等式元素的代数结构.正式(M, *, id)是Monoid iff (a * b) * c == a * (b * c) && a * id == a && id * a == a for all a, b, c in M.

Monoid: algebraic structure with a single associative binary operation and an identity element. Formally (M, *, id) is a Monoid iff (a * b) * c == a * (b * c) && a * id == a && id * a == a for all a, b, c in M.

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