Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have read here and in some other places that if $$(a \circ b) \star (c \circ d) = (a \star c) \circ (b \star d)$$ for all $a,b,c,d$ in a set with two binary operations $\circ$ and $\star$, then each one is a monoid homomorphism.

My problem is that I don't quite understand in what sense these are monoid homomorphisms, since neither is a function $M \to M'$, but instead $M^2 \to M$, which confuses me somewhat. So, what exactly are these homomorphisms? Is $M^2$ supposed to be a monoid? If so, how?

Thanks.

share|cite|improve this question
up vote 9 down vote accepted

If $M$ is a monoid, $M^2=M\times M$ is a monoid with coordinate-wise multiplication: that is, if $(a,b)$, $(a',b')\in M^2$, then the product is such that $$(a,b)\cdot(a',b')=(aa',bb').$$

Now suppose $M$ is a monoid, and that $M^2$ is endowed with this monoid structure. Suppose additionally that we have a function $\phi:M^2\to M$. It then makes sense to say that $\phi$ is a morphism of monoids.

Finally, it may well happen that the function $\phi$ defines itself a monoid structure on $M$. That is the situation of the lemma you have in mind, which is called the Eckmann-Hilton lemma.

share|cite|improve this answer

I realize that this is not exactly the answer you asked for, but...

The character of the morphisms doesn't depend on all the properties of a structure. For example, properties like associativity doesn't influence at all. What do influence is the outer structure, in the case of monoids the function $$M\times M\to M$$ To that outer structure morphisms always are characterized by $$f(xy)=f(x)f(y)$$ How come? Well, I don't know if there is a definite answer, but there are certain patterns:

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, here to be called a spotted set. Given two spotted sets, then a morphism $\alpha :(X,S)\longrightarrow(X^\prime,S^\prime)$ is a function $\alpha :X\longrightarrow X^\prime$ such that $x\in S\Rightarrow \alpha(x)\in S^\prime$. Call the category sSet.

Group-like structures as magmas and categories are characterized by relations $R\subseteq (X\times X)\times X$ and can obviously be expressed as spotted sets. Morphisms are functions $\alpha:(X\times X)\times X\longrightarrow(X^\prime\times X^\prime)\times X^\prime$ such that $((x,y),z)\in R \Rightarrow \alpha((x,y),z)\in R^\prime$. Functions $\alpha_1,\alpha_2,\alpha_3:X\longrightarrow X^\prime$ exists such that $\alpha((x,y),z)=((\alpha_1(x),\alpha_2(y)),\alpha_3(z))$ and if $\alpha$ is such that $\alpha_1=\alpha_2=\alpha_3$, then $\alpha_1$ correspond to group homomorphisms etc.

Action-like structures $R\subseteq (A\times X)\times X$. Here morphisms are functions $(A\times X)\times X\overset{\alpha}{\longrightarrow}(A\times X^\prime)\times X^\prime$ such that $((a,x),y)\in R \Rightarrow \alpha((a,x),y)\in R^\prime$. It exists functions $\alpha_0,\alpha_1,\alpha_2$ such that $\alpha((a,x),y)=((\alpha_0(a),\alpha_1(x)),\alpha_2(y))$. If $\alpha_0=1_A$ and $\alpha_1=\alpha_2$ this correspond to morphisms of actions.

Metric-like structures $R\subseteq (X\times X)\times B$. $((x,y),b)\in R \Rightarrow \alpha((x,y),b)\in R^\prime$.

Topological spaces. The spotted set is simply defined as $\tau\subseteq \mathcal{P}(X)$. Morphisms are functions $\mathcal{P}(X)\overset{\alpha}{\longrightarrow}\mathcal{P}(X^\prime)$ such that $\mathcal{O}\in\tau \Rightarrow \alpha(\mathcal{O})\in \tau^\prime$. If there is a function $f:X^\prime\longrightarrow X$ such that $\alpha = \mathcal{Q}(f)$, where $\mathcal{Q}$ is the contra-variant power set functor, this correspond to Top and $f$ is continuous with respect to the topologies.

Uniform spaces with a set of entourages $\phi\subseteq\mathcal{P}(X\times X)$. Morphisms are functions $\mathcal{P}(X\times X)\overset{\alpha}{\longrightarrow}\mathcal{P}(X^\prime\times X^\prime)$ such that $\mathcal{U}\in\phi \Rightarrow \alpha(\mathcal{U})\in \phi^\prime$. The condition on the $sSet$-morphism to correspond to a uniformly continuous function is similar as above.

Undirected graphs. $E\subseteq\mathcal{P}(X)$, $e\in E\Rightarrow \alpha(e)\in E^\prime$, where $\alpha$ is a function $\mathcal{P}(X)\rightarrow\mathcal{P}(X^\prime)$.

Multigraphs. Function $\varepsilon \subset E\times V^2$.

Matroids. $\mathcal{I}\subseteq \mathcal{P}(X)$. If there is a function $f:X^\prime\rightarrow X$ such that $\alpha=\mathcal{Q}(f)$ and $X=X^\prime$, then $\alpha$ correspond to maps of matroids with weak relation.

There are also other patterns.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.