\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
In a previous paper we introduced a generalized model for
translation invariant (TI) operators. In this model we considered
the space, $\Phi$, of all maps from an abelian group $G$ to
$\Omega\cup\{-\infty\}$, called LG-fuzzy sets, where $\Omega$ is a
complete lattice-ordered group; and we defined TI operators on
this space. Also, in that paper, we proved strong reconstruction
theorem to show the consistency of this model. This theorem states
that for an order-preserving TI operator $Y$ one can explicitly
compute $Y(A)$, for any $A$, from a specific subset of $\Phi$
called the base of $Y$. \\ \ \ \ \ In this paper duality is
considered in the same general framework, and in this regard,
continuous TI operators are studied. This kind of operators are
characterized in terms of critical points of their kernels. A
critical point is defined to be a point at which there exists a
net which converges to $-\infty$, when $-\infty$ is the value of a
nontrivial limit-function of the kernel at that point. A critical
point is called to be of first type if for every such net there
exists some other point at which this net converges to $+\infty$,
and it is called to be of second type otherwise. As the main
result of this paper we prove that a well-defined isotone TI
operator is continuous if and only if it has no critical point of
second type. Moreover, in this case a TI operator has a continuous
extension by duality.
\end{document}