Devillet, Jimmy[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]

Couceiro, Miguel[LORIA, CNRS - Inria Nancy Grand Est - Université de Lorraine, France]

Marichal, Jean-Luc[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]

3-Oct-2017

International

Séminaire de MAthématique et LOgique pour l'exTraction et le traitEment de Connaissances (MALOTEC)

03-10-2017

Miguel Couceiro

Amedeo Napoli

LORIA, CNRS - Inria Nancy Grand Est - Université de Lorraine, Nancy

France

[en] We investigate the class of binary associative and quasitrivial operations on a given finite set. Here quasitriviality (also known as conserva-tiveness) 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 the latter case, these operations reduce to discrete uninorms, which are discrete fuzzy connectives that play an important role in fuzzy logic. As we will see nondecreasing, associative and quasitrivial operations are chara-cterized in terms of total and weak orderings through the so-called single-peakedness property introduced in social choice theory by Duncan Black. This will enable visual interpretaions of the above mentioned algebraic properties. Motivated by these results, we will also address a number of counting issues: we enumerate all binary associative and quasitrivial operations on a given finite set as well as of those operations that are commutative, are nondecreasing, have neutral and/or annihilator elements. As we will see, these considerations lead to several, previously unknown, integer sequences.