TITLE: why is a nullary operation a special element, usually 0 or 1?
QUESTION [10 upvotes]: Does a nullary operation mean an operation not taking any argument?
Then why is a nullary operation a special element, usually 0 or 1, in an algebraic structure?
Thanks!

REPLY [13 votes]: An operation $\lambda$ with $n$ arguments, that is a $n$-ary operation, on the set $S$ is a function $S^n\to S$. Now we can view an element $x$ of $S^n$ either as an $n$-tuple or as a function $x:\{1,2,\ldots,n\}\to S$. Specializing to the case $n=0$, an element of $S^0$ can be seen as a function $x:\emptyset\to S$. There is exactly one such function that trivialy maps all elements of the empty set to elements of $S$. So $S^0$ has only one element. In the usual set-theoretic way of identifying a function with its graph, we have $S^0=\{\emptyset\}=\{0\}$.
So a nullary operation $\lambda:S^0\to S$ is determined by the value $\lambda(\emptyset)$, a single element of $S$. For convenience, we sometimes treat the nullary operation as if it would be this element $\lambda(\emptyset)$, and in this sense a nullary operation is a constant.