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arXiv · 1810.06406

Generators of aggregation functions and fuzzy connectives

Radomír Halaš·Radko Mesiar·Jozef Pócs
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Abstract

We show that the class of all aggregation functions on $[0,1]$ can be generated as a composition of infinitary sup-operation $\bigvee$ acting on sets with cardinality not exceeding $\mathfrak{c}$, $b$-medians $\mathsf{Med}_b$, $b\in[0,1[$, and unary aggregation functions $1_{]0,1]}$ and $1_{[a,1]}$, $a\in ]0,1]$. Moreover, we show that we cannot relax the cardinality of argument sets for suprema to be countable, thus showing a kind of minimality of the introduced generating set. As a by product, generating sets for fuzzy connectives, such as fuzzy unions, fuzzy intersections and fuzzy implications are obtained, too.

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Radomír Halaš, Radko Mesiar, Jozef Pócs. 2018-10-09. Generators of aggregation functions and fuzzy connectives. https://doi.org/10.1109/tfuzz.2016.2544353

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