Representation of a Boolean algebra by its triangular norms

Abstract

Given a complete and atomic Boolean algebra $B$, there exists a family $\tau_{\gamma}$ of triangular norms on $B$ such that, under the partial ordering of triangular norms, $\tau_{\gamma}$ is a Boolean algebra isomorphic to $B$, where $\gamma$ is the set of all atoms in $B$. In other words, as we have shown in this note, every complete and atomic Boolean algebra can be represented by its own triangular norms. What we have not shown in this paper is our belief that $\tau_{\gamma}$ is not unique for $B$ and that, for such a representation, $B$ needs neither to be complete, nor to be atomic.