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Regular left-continuous t-norms

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Abstract

A left-continuous (l.-c.) t-norm ⊙ is called regular if there is an n<ω such that the map x ↦ x⊙a has, for any a∈[0,1], at most n discontinuity points, and if the function mapping every a∈[0,1] to the set \(\{x\in[0,1]\hspace {-0.15em}:\hspace{0.15em}\lim_{y\searrow x}y\odot a=x\}\) behaves in a specifically simple way. The t-norm algebras based on regular l.-c. t-norms generate the variety of MTL-algebras.

With each regular l.-c. t-norm, we associate certain characteristic data, which in particular specifies a finite number of constituents, each of which belongs to one out of six different types. The characteristic data determines the t-norm to a high extent; we focus on those t-norms which are actually completely determined by it. Most of the commonly known l.-c. t-norms are included in the discussion.

Our main tool of analysis is the translation semigroup of the totally ordered monoid ([0,1];≤,⊙,0,1), which consists of commuting functions from the real unit interval to itself.

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Authors and Affiliations

  1. Institute for Medical Expert and Knowledge-Based Systems, Medical University of Vienna, Spitalgasse 23, 1090, Wien, Austria

    Thomas Vetterlein

  2. European Centre for Soft Computing, C/ Gonzalo Gutiérrez Quirós, 33600, Mieres, Spain

    Thomas Vetterlein

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  1. Thomas Vetterlein
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Correspondence to Thomas Vetterlein.

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Communicated by Jean-Eric Pin.

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Vetterlein, T. Regular left-continuous t-norms. Semigroup Forum 77, 339–379 (2008). https://doi.org/10.1007/s00233-008-9103-3

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  • DOI: https://doi.org/10.1007/s00233-008-9103-3

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