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 = \Z \union \{+,-\} \union \{=, \neq, <, \leq,>,\geq\} $
\item $\mathcal{A}$ defines the usual meaning to all symbols:
  \begin{itemize}
    \item Constant symbols are mapped to the corresponding value in $\Z$.
    \item $+$ is interpreted as the function $0+0 \rightarrow 0, 0+1 \rightarrow 1, \ldots$. $-$ follows it analogous interpretation.
    \item The predicate symbols are interpreted as their respective comparison operator.
  \end{itemize}
\end{itemize}