Soft Computing (2018) 22:5043–5049
On invariant IF-state
· Beloslav Rieˇcan
Published online: 4 June 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
The Haar measure on invariant state for fuzzy sets is constructed in a locally compact space. Moreover, the invariant state is
studied on MV-algebra generated by a family of intuitionistic fuzzy sets, important as well as from the theoretical point of
view as from the applications.
Keywords Locally compact groups · Intuitionistic fuzzy sets · Invariant states · MV-algebras
In the classical measure theory, it is known theory about Haar
measure (Halmos 1950) stating that in every locally compact
Abelian group there exists a probability measure invariant
Let (G, +) be a locally compact Abelian topological
group, C be the family of all compact subsets of G, and σ(C)
be the σ-algebra generated by C. Then, there exists exactly
one probability measure P : σ(C) →[0, 1] such that
P(A + a) = P( A)
for any A ∈ σ(C) and any a ∈ G (see e.g. Halmos 1950).
The measure P isusually called the Haar measure or invariant
probability measure. Recall that in (Schwarz 1957)aversion
of the existence of invariant measure has been proved for
semigroups, and in (Stehlíková et al. 2011)forIP-loops.
In the paper, we will study the theory of invariant measures
on families of intuitionistic fuzzy sets (Atanassov 1999). The
theory of intuitionistic fuzzy sets is important from the theo-
retical point of view as well as from the point of applications.
Communicated by C. Kahraman.
Faculty of Natural Science, Matej Bel University, Tajovského
40, 97401 Banská Bystrica, Slovakia
Mathematical Institut, Slovak Academy of Sciences,
Štefánikova 49, Bratislava, Slovakia
For example, in Atanassov (1999), De et al. (2001), Szmidt
and Kacprzyk (2001) has been presented some applications
in biology and medicine, in
Cunderlíková (2008), Jureˇcková
and Rieˇcan (2005), Lašová (2010), Rieˇcan (2013), Rieˇcan
(2009) in theoretical physics, in
Durica (2010), Markechová
and Rieˇcan (to appear), Schwarz (1957) in informatics. From
the point of view of physical applications, very important
role plays the invariance of used transformations. Therefore,
in the present paper we construct an invariant probability
for the given IF-dynamical systems. Of course, the prob-
lem of an existence of an invariant measure is interesting and
actual also for the general point of view (see e. g. Markechová
et al. 2016; Stehlíková et al. 2011; Zadeh 1965). The main
problem is in the constructions of invariant states in general
IF-spaces. We will solve it using some representation pro-
cess in fuzzy spaces together with general maps deﬁning on
some topological groups. It seems that this original approach
leads to some useful results from theoretical points of view
as well as applications.
Recall that in the paper (Michalíková 2017) there was
considered a special case, the group (R, +), and the shift
:[0, 1) →[0, 1) given by the prescription T
x + a(mod1). Then, the theorem about the existence of an
invariant IF-states on real numbers was proved. In Sect. 2,
we will make an extension of this theory to locally com-
pact Abelian topological group and we will use more general
The basic theory of invariant measures was built on sets.
Since it is natural to consider fuzzy sets instead of sets (Zadeh
1965) in Sect. 3, we will discuss the properties of invariant
measure deﬁned on fuzzy sets, but we will look at this struc-
ture as a special case of IF-sets.