TITLE: Are there algebraic structures with more than one neutral element and/or more than one inverse element?
QUESTION [12 upvotes]: I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than one inverse element?

REPLY [3 votes]: This just came to my mind:
Take a set $X$ with two elements $a$ and $b$. We want to equip this with an interior multiplication that is associative ($X$ is then called a semigroup), and such that $a$ is neutral on the right but not neutral on the left. Then we already know three of four values:
$$aa=a,\hspace{20pt}ba=b\hspace{20pt}ab=a\hspace{20pt}bb=?$$
Now, associativity requires that $(ba)b=bb=b(ab)=ba=b$. So
$$aa=a,\hspace{20pt}ba=b\hspace{20pt}ab=a\hspace{20pt}bb=b$$ Or in other words, $(x,y)$ gets mapped to the first entry, so the multiplication is projection on the first coordinate. It is then easy to show that this is indeed associative. As we see, both element $a$ and $b$ are neutral on the right.