t-norms induced by metrics on Boolean algebras

Let dν be the metric associated with a strictly positive submeasure ν on some boolean algebra P. If dν is bounded from above by 1, Eν = 1 − dν is a (fuzzy) similarity relation on 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 P and ν, we may associate with ν in a natural way a continuous t-norm w.r.t. which Eν 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.