# T-norms induced by metrics on boolean algebras

Let $\displaystyle d_\nu$ be the metric associated with a strictly positive submeasure $\displaystyle \nu$ on some boolean algebra $\displaystyle \mathcal P$ . If $\displaystyle d_\nu$ is bounded from above by 1, $\displaystyle E_\nu = 1-d_\nu$ is a (fuzzy) similarity relation on $\displaystyle \mathcal P$ at least w.r.t.the Lukasiewicz t-norm, but possibly also w.r.t. numerous further t-norms.
In this paper, we show that under certain assumptions on $\displaystyle \mathcal P$ and $\displaystyle \nu$ , we may associate with $\displaystyle \nu$ in a natural way a continuous t-norm w.r.t. which $\displaystyle E_\nu$ is a similarity relation and which, in a certain sense, is the weakest such t-norm. Up to isomorphism, every continuous t-norm arises in this way.