[en] We investigate the class of binary associative and quasitrivial operations on a given finite set. Here the quasitriviality property (also known as conservativeness) means that the operation always outputs one of its input values. We also examine the special situations where the operations are commutative and nondecreasing, in which cases the operations reduce to discrete uninorms (which are discrete fuzzy connectives playing an important role in fuzzy logic).
Interestingly, associative and quasitrivial operations that are nondecreasing are characterized in terms of total and weak orderings through the so-called single-peakedness property introduced in social choice theory by Duncan Black.
We also address and solve a number of enumeration issues: we count the number of binary associative and quasitrivial operations on a given finite set as well as the number of those operations that are commutative and/or nondecreasing.
Disciplines :
Mathematics Computer science
Author, co-author :
Couceiro, Miguel; [LORIA, CNRS - Inria Nancy Grand Est - Université de Lorraine]
Devillet, Jimmy ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Marichal, Jean-Luc ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
yes
Language :
English
Title :
Associative and quasitrivial operations on finite sets: characterizations and enumeration
Publication date :
02 July 2018
Event name :
International Symposium on Aggregation and Structures (ISAS 2018)
Event organizer :
José Luis García-Lapresta Miguel Martínez-Panero David Pérez-Román
