How would you name the gate which maps $|x\rangle\rightarrow |(x+1)\text{mod}\hspace{1mm}n\rangle$

Question

Suppose you have a qudit with $n$ decoherent states.

How would you name the gate which maps $|x\rangle\rightarrow |(x+1)\text{mod}\hspace{1mm}n\rangle$

Answer 1

I am assuming you meant to say qudit with $n$-basis states. The gate you are describing would be a shift matrix, i.e., a generalized Pauli $X$.

As I have answered in your previous question (and also here), this would be a $X(1)$ gate.

$$X(1)|j\rangle = |(j +1) \text{mod} (n) \rangle \,,$$

where $\{|j\rangle\}$ are the basis states. (using different variable than one in your question to avoid confusion)