Elsevier

Fuzzy Sets and Systems

Volume 299, 15 September 2016, Pages 105-112
Fuzzy Sets and Systems

T-norms and t-conorms continuous around diagonals

https://doi.org/10.1016/j.fss.2015.11.001Get rights and content

Abstract

Triangular norms and conorms with continuous diagonals are discussed. In literature we can find examples of non-continuous t-norms with continuous diagonals, however, their deeper study is still missing. In this paper we introduce a sufficient condition under which a t-norm with continuous diagonal is continuous. Moreover, we show that an Archimedean t-norm continuous on the boundary is continuous. Several illustrative examples are also included.

Section snippets

Introduction and basic notions

Triangular norms and conorms [1], [5] are applied in many domains and therefore knowledge of the structure of the class of t-norms (t-conorms) is very important. An open question whether a t-norm with continuous diagonal is continuous was answered negatively in [7], [9], [11]. In this paper we introduce a sufficient condition under which a t-norm with continuous diagonal is necessarily continuous. First, we introduce some basic notions.

A triangular norm (see [1], [5], [8], [10]) is a function T:

Triangular norms with continuous diagonals

Several examples of non-continuous t-norms with continuous diagonals can be found in [7], [9], [11].

Example 1

For every x[0,1] assume its triadic expansion (xn)nN, i.e., x=nNxn3n, where xn{0,1,2} for all nN. Then 0 corresponds to the expansion where xn=0 for all nN, and 1 corresponds to the expansion where xn=2 for all nN. The set of pure Cantor points is the set C such that each xC has a triadic expansion containing only 0 and 2. Let S be the set of points x[0,1] such that for the triadic

Conclusions

We have studied t-norms with continuous diagonals and shown that each t-norm T with a continuous diagonal which is continuous around (x,x), where x is an idempotent point of T, is continuous. We have also shown that an Archimedean t-norm continuous on the boundary of the unit square is continuous. Similar results can be obtained also for t-conorms by duality. In the future work we would like to focus on uninorms with continuous diagonals.

Acknowledgement

This work was supported by grants VEGA 2/0049/14, APVV-0178-11 and Program Fellowship of SAS.

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