monoid同态到底是什么? [英] What is monoid homomorphism exactly?

问题描述

我已经从 Monoid形态，产品和副产品，并且无法理解100％.

I've read about monoid homomorphism from Monoid Morphisms, Products, and Coproducts and could not understand 100%.

作者说(强调原文):

length函数从String映射到Int ，同时保留 单体结构.这样的功能，可以将一个半身像映射到 另一个以这种保存方式被称为 monoid homomorphism .在 一般而言，对于Monoid M和N，同态f: M => N和所有值 x:M，y:M，以下等式成立:

The length function maps from String to Int while preserving the monoid structure. Such a function, that maps from one monoid to another in such a preserving way, is called a monoid homomorphism. In general, for monoids M and N, a homomorphism f: M => N, and all values x:M, y:M, the following equations hold:

f(x |+| y) == (f(x) |+| f(y))

f(mzero[M]) == mzero[N]

他的意思是说，由于数据类型String和Int是单面体，并且函数length映射String => Int保留单面体结构(Int是一个单面体)，因此称为单面体同态，对吧?

Does he mean that, since the datatypes String and Int are monoids, and the function length maps String => Int preserving the monoid structure (Int is a monoid), it is called monoid homomorphism, right?

推荐答案

他的意思是说，数据类型String和Int是等宽的.

Does he mean, the datatype String and Int are monoid.

否，String和Int都不是类半体动物. id半群是一个三元组(S，&oplus ;, e)，其中⊕是二元运算符⊕ :S× S→ S ，这样对于所有元素 a，b，c∈ S 都认为(a⊕ b)⊕ c = a⊕(b⊕ c)，而 e∈ S 是一个身份元素"，因此 a⊕ e = e⊕ a = a . String和Int是类型，因此基本上是一组值，但不是三元组.

No, neither String nor Int are monoids. A monoid is a 3-tuple (S, ⊕, e) where ⊕ is a binary operator ⊕ : S×S → S, such that for all elements a, b, c∈S it holds that (a⊕b)⊕c=a⊕(b⊕c), and e∈S is an "identity element" such that a⊕e=e⊕a=a. String and Int are types, so basically sets of values, but not 3-tuples.

文章说:

让我们将 String串联和 Int添加 有关系的示例 monoids .

Let's take the String concatenation and Int addition as example monoids that have a relationship.

因此，作者显然也提到了二进制运算符(在String情况下为(++)，在Int情况下为(+)).身份(在String情况下为空字符串，在Int情况下为0)保留为隐式；在非正式英语语篇中，将身份作为一种练习留给读者是很常见的.

So the author clearly also mentions the binary operators ((++) in case of String, and (+) in case of Int). The identities (empty string in case of String and 0 in case of Int) are left implicit; leaving the identities as an exercise for the reader is common in informal English discourse.

现在假设我们有两个单半体结构(M，&oplus ;, e _m )和(N，&otimes ;, e _n )，函数 f:M→ N (例如length)然后称为 monoid同态 [wiki] ，因为它认为 f(m ₁ ⊕ m ₂ )= f(m ₁ )⊗ f(m ₂ )用于所有元素 m ₁ ，m ₂ ∈ M 并且该映射还保留了标识元素: f(e _m )= e _n .

Now given that we have two monoid structures (M, ⊕, e_m) and (N, ⊗, e_n), a function f : M → N (like length) is then called a monoid homomorphism [wiki] given it holds that f(m₁⊕m₂)=f(m₁)⊗f(m₂) for all elements m₁, m₂∈M and that mapping also preserves the identity element: f(e_m)=e_n.

例如length :: String -> Int是一个单面体同态，因为我们可以考虑这些单面体(String，(++)，"")和(Int，，0).它认为:

For example length :: String -> Int is a monoid homomorphism, since we can consider the monoids (String, (++), "") and (Int, (+), 0). It holds that:

1. length (s1 ++ s2) == length s1 + length s2(对于所有String s s1和s2)；和
2. length "" == 0.

1. length (s1 ++ s2) == length s1 + length s2 (for all Strings s1 and s2); and
2. length "" == 0.

这篇关于monoid同态到底是什么?的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！