Real coextensions as a tool for constructing triangular norms

Thomas Vetterlein

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arXiv · 1808.10324

Real coextensions as a tool for constructing triangular norms

Thomas Vetterlein

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Abstract

We present in this paper a universal method of constructing left-continuous triangular norms (l.-c. t-norms). The starting point is an arbitrary, possibly finite, totally ordered monoid fulfilling the conditions that are characteristic for l.-c. t-norms: commutativity, negativity, and quanticity. We show that, under suitable conditions, we can extend this structure by substituting each element for a real interval. The process can be iterated and if the final structure obtained in this way is order-isomorphic to a closed real interval, its monoidal operation can, up to isomorphism, be identified with a l.-c. t-norm. We specify the constituents needed for the construction in an explicit way. We furthermore illustrate the method on the basis of a number of examples.

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BibTeX RIS

Thomas Vetterlein. 2018-08-30. Real coextensions as a tool for constructing triangular norms. https://arxiv.org/abs/1808.10324

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