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t-norms induced by metrics on Boolean algebras

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Abstract

Let d ν be the metric associated with a strictly positive submeasure ν on some Boolean algebra \(\mathcal{P}\) . If d ν is bounded from above by 1, E ν=1−d ν is a (fuzzy) similarity relation on \(\mathcal{P}\) at least w.r.t. the Lstrok ukasiewicz t-norm, but possibly also w.r.t. numerous further t-norms.

In this paper, we show that under certain assumptions on \(\mathcal{P}\) and ν, we may associate with ν in a natural way a continuous t-norm w.r.t. which E ν 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

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Correspondence to Thomas Vetterlein.

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Vetterlein, T. t-norms induced by metrics on Boolean algebras. Soft Comput 10, 995–1000 (2006). https://doi.org/10.1007/s00500-005-0026-6

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  • DOI: https://doi.org/10.1007/s00500-005-0026-6

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