Nullary operation ($c\in S$)

A nullary operation on a set $S$ has no inputs: formally it is a function $S^0\to S$. The Cartesian power $S^0$ is the singleton consisting of the empty tuple, so such a function selects exactly one distinguished element $c\in S$.

The zero and identity elements of a unital ring can be encoded as nullary operation symbols. A homomorphism preserving those operations must carry their selected constants to the corresponding constants in the target structure. This is why constants fit naturally into the same signature language as unary operations and binary operations.

There is no nullary operation on the empty set: its source $S^0$ still has one element, while its target has none. Allowing constants therefore affects whether empty algebraic structures are possible.