\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
In a previous paper we introduced a
generalized model for translation invariant (TI) operators. 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. In this paper thresholding is considered in the same general framework, and in this regard, positive TI operators are studied. The
threshold of an LG-fuzzy set $A$ at level $t$ is defined to be the set $\{g\in G:A(g) \geq t \}$. Also, a TI operator is defined to be positive if its kernel consists of elements which are positive except possibly
at some points which are $-\infty$. As the main result of this note, it is proved that a positive and isotone TI operator commutes with thresholding if and only if it has a crisp kernel (or base). Also, the theorem is used to derive a similar result for an operator whose kernel is bounded from below.
\end{document}