On smooth nullnorms on bounded lattices and their extensions to smooth uninorms

Document Type : Original Manuscript

Authors

1 Dept. Mathematics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Bratislava, Slovakia

2 Dept. Mathematics, Slovak University of Technology, Radlisnkeho 11, 810 05 Bratislava, Slovakia

10.22111/ijfs.2026.10079

Abstract

The paper deals with smooth uninorms and smooth nullnorms on finite lattices. These operations were first introduced on the unit interval and later generalized to bounded lattices. One of the most important and most studied properties is continuity. However, when working on finite lattices (finite structures, in general), the topological continuity is not very meaningful. For this reason, Mayor and Torrens (2005) introduced several notions that replace continuity. Smoothness and divisibility are among them. We study smooth nullnorms on finite lattices looking for conditions under which, when the nullnorm is defined on a lattice L, it is possible to extend the smooth nullnorm to a smooth uninorm defined on the horizontal sum of L and a finite chain. We provide necessary and sufficient conditions for such an extension to yield a smooth uninorm. Finally, we give a necessary and sufficient condition under which a smooth uninorm is also a divisible uninorm.

Keywords

References

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Iranian Journal of Fuzzy Systems
Volume 23, Issue 5
September and October 2026
Pages 1-7

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