Variables in  $\mathcal{T}_{LIA}$ are of integer sort ($\mathbb{Z}$).
The functions of $\mathcal{T}_{LIA}$ are $+$ and $-$ and the predicates are 
$=, \neq, <, >, \leq,$ and  $\geq$. The axioms withing $\mathcal{T}_{LIA}$
define the meaning for these functions and predicates.

Therefore, for the theory of Linear Integer Arithmetic $\mathcal{T}_{LIA}$ we have:
\begin{itemize}
\item $\Sigma = \{ ... ..., -3,-2,-1,0,1,2,3...,=,+,-,\neq, <, >, \leq,\geq\} $
\item $\mathcal{A}$ defines the usual meaning to all symbols. (Constant number symbols are mapped to the corresponding value in $\mathbb{Z}$,  $+$ is interpreted as the function $0+0\rightarrow 0$, $0+1\rightarrow 1$, etc.).
\end{itemize}