T-norms and t-conorms continuous around diagonals
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
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 assume its triadic expansion , i.e., , where for all . Then 0 corresponds to the expansion where for all , and 1 corresponds to the expansion where for all . The set of pure Cantor points is the set C such that each has a triadic expansion containing only 0 and 2. Let S be the set of points 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 , 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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