3 research outputs found
On lattices with a smallest set of aggregation functions
Given a bounded lattice with bounds and , it is well known that
the set of all -preserving polynomials of
forms a natural subclass of the set of aggregation functions on
. The main aim of this paper is to characterize all finite lattices for
which these two classes coincide, i.e. when the set is as small
as possible. These lattices are shown to be completely determined by their
tolerances, also several sufficient purely lattice-theoretical conditions are
presented. In particular, all simple relatively complemented lattices or simple
lattices for which the join (meet) of atoms (coatoms) is () are of this
kind.Comment: 18 page
Generators of aggregation functions and fuzzy connectives
We show that the class of all aggregation functions on can be
generated as a composition of infinitary sup-operation acting on sets
with cardinality not exceeding , -medians ,
, and unary aggregation functions and , . 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.Comment: 5 page
Generalized comonotonicity and new axiomatizations of Sugeno integrals on bounded distributive lattices
Two new generalizations of the relation of comonotonicity of lattice-valued
vectors are introduced and discussed. These new relations coincide on
distributive lattices and they share several properties with the comonotonicity
for the real-valued vectors (which need not hold for -valued vectors
comonotonicity, in general). Based on these newly introduced generalized types
of comonotonicity of -valued vectors, several new axiomatizations of
-valued Sugeno integrals are introduced. One of them brings a substantial
decrease of computational complexity when checking an aggregation function to
be a Sugeno integral.Comment: 22 page