How should one refer to the smallest remainders generated by a modulus within DH or DLP? It's my understanding that the integer base and exponents chosen to create the initial public keys in DH are from the remainders of a modulus. 

For example, if the value of the modulus is $N=11$, a set of remainders "produced" is  {0,1,2,3,4,5,6,7,8,9,10}, however, I could also refer to the remainders of $N$ as 11,12,13,14,15,16,17,18,19,20,21. 

Is there a way to differentiate by name the remainders {0,1,2,3,4,5,6,7,8,9,10}? 

For example, is it accurate to refer to {0,1,2,3,4,5,6,7,8,9,10} as the canonical remainders of $\pmod N$? If yes, then can I refer to a particular base used as the canonical base and the exponent used the canonical exponent? Or is there a different terminology for these elements?
The space $\mathbb Z/n\mathbb Z$ is usually taken to consist of equivalence classes modulo $n$ called residue classes—equivalence classes of integers under the equivalence relation $a \sim b$ if and only if $n$ divides $a - b$.  For example, $\mathbb Z/3\mathbb Z$ consists of the three equivalence classes

\begin{align}
  3\mathbb Z &amp;= \{\dots, -3, 0, 3, 6, \dots\}, \\
  1 + 3\mathbb Z &amp;= \{\dots, -2, 1, 4, 7, \dots\}, \\
  2 + 3\mathbb Z &amp;= \{\dots, -1, 2, 5, 8, \dots\}.
\end{align}

Residue classes are sometimes also called cosets in group theory.

Any set of distinct representatives of all the equivalence classes modulo $n$ is called a complete residue system modulo $n$.  For example, $\{0,1,2\}$ is a complete residue system modulo 3, as is $\{99,-26,-1\}$.

Usually if we want to choose particular representatives for computation, we choose the least nonnegative residues like $\{0,1,2\}$, where each number is taken to represent the equivalence class it is an element of.

There are also more exciting systems like Montgomery residues in radix $r$, where a coset $a + n\mathbb Z$ is represented by the integer $a \cdot r^{-1} \bmod n$, where $r^{-1}$ is an integer such that $r \cdot r^{-1} \equiv 1 \pmod n$.  For instance, in radix 8, modulo 5 we have the representatives

\begin{align}
  0 &amp;\mapsto 0 + 5\mathbb Z, \\
  2 &amp;\mapsto 1 + 5\mathbb Z, \\
  4 &amp;\mapsto 2 + 5\mathbb Z, \\
  1 &amp;\mapsto 3 + 5\mathbb Z, \\
  3 &amp;\mapsto 4 + 5\mathbb Z.
\end{align}

This peculiar-looking function has the property that we can compute a representative of its image by $$\rho(x) = \bigr(x + n\cdot[(x \bmod r) \cdot n' \bmod r]\bigr)/r,$$ where the only divisor involved in the computation is $r$.  Here $n' n \equiv 1 \pmod r$, and the result is either the least nonnegative residue modulo $n$, or the next one greater, so you can get the least nonnegative residue with a single conditional subtraction.  The form $a \cdot r^{-1}$ is preserved by addition and subtraction, $(a \pm b) \cdot r^{-1} = a \cdot r^{-1} \pm b \cdot r^{-1}$, and while it is not preserved by multiplication, it can be restored by evaluating $\rho$: $$(a \cdot b) \cdot r^{-1} = \rho\bigl((a \cdot r^{-1}) \cdot (b \cdot r^{-1})\bigr).$$  This technique is called Montgomery multiplication.  When $r$ is a natural machine word size like $2^{32}$, this representation can be considerably faster without timing side channels than reduction modulo a general odd $n$.