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On the continuity points of left-continuous t-norms. (English) Zbl 1047.03016
The authors show that each left-continuous t-norm can be obtained as the completion of a continuous function on some dense subset of $$[0,1]^2$$. As a by-product, they prove that the set of discontinuity points of a left-continuous t-norm is a first-category set of measure zero. Further, they show that not all left-continuous t-norms are isomorphic to the completion of some continuous t-norm on $$Q\cap[0,1]$$ (the set of all rational numbers from the unit interval). However, they prove that at least all weakly cancellative ones possess this property. Finally, they characterize those continuous t-norms on $$Q\cap[0,1]$$ whose completion is continuous.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 06F05 Ordered semigroups and monoids
##### Keywords:
t-norm; continuity; left-continuity; residuated lattice
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