  Let $A$ be an algebra and $\sigma$ an automorphism of $A$. A linear map $d$
of $A$ is called a $\sigma$-derivation of $A$ if $d(xy) = d(x)y +
\sigma(x)d(y)$, for all $x, y \in A$. A linear map $D$ is said to be a
generalized $\sigma$-derivation of $A$ if there exists a $\sigma$-derivation
$d$ of $A$ such that $D(xy) = D(x)y + \sigma(x)d(y)$, for all $x, y \in A$. An
additive map $\Theta$ of $A$ is $\sigma$-centralizing if $\Theta(x)x -
\sigma(x)\Theta(x) \in Z(A)$, for all $x \in A$. In this paper, precise
descriptions of generalized $\sigma$-derivations and $\sigma$-centralizing maps
of triangular algebras are given. Analogues of the so-called commutative
theorems, due to Posner and Mayne, are also proved for the triangular algebra
setting.
