Soft Comput (2009) 13:591–596
DOI 10.1007/s00500-008-0336-6
ORIGINAL PAPER
Fuzzy logic models in a category of fuzzy relations
Jiˇ rí Moˇ ckoˇ r
Published online: 28 June 2008
© Springer-Verlag 2008
Abstract We investigate interpretations ‖ψ ‖
E
of formulas
ψ in a first order fuzzy logic in models E which are based
on objects of a category SetR() which consists of -sets,
i.e. sets with similarity relations with values in a complete
MV-algebra and with morphisms defined as special fuzzy
relations between -sets. The interpretations ‖ψ ‖
E
are then
morphisms in a category SetR() from some -set to the
object (, ↔). We define homomorphisms between models
in a category SetR() and we prove that if ϕ : E
1
→ E
2
is a
(special) homomorphism of models in a category SetR()
then there is a relation between interpretations ‖ψ ‖
E
i
of a
formula ψ in models E
i
.
Keywords Sets with similarities · MV-algebras · Category
of fuzzy relations · Fuzzy logic · Models of fuzzy logic
1 Introduction
In a fuzzy set theory the category SetF() of sets with simi-
larity relations defined over an MV -algebra = (, →,
⊗, ∨, ∧) is of principal importance (Moˇ ckoˇr2004, 2006,
2007a,b, 2008). This category consists of objects ( A,δ)
(called -sets), where A is a set and δ is a similarity relation,
i.e. a map δ : A × A → such that
(a) δ(x , x ) = 1 = 1
,
Supported by MSM6198898701, grant 201/07/0191 of GA
ˇ
CR and
grant 1M0572.
J. Moˇ ckoˇr(B )
Institute for Research and Applications of Fuzzy Modeling,
University of Ostrava, 30. dubna 22, 70103 Ostrava 1,
Czech Republic
e-mail: Jiri.Mockor@osu.cz
(b) δ(x , y ) = δ( y , x ),
(c) δ(x , y ) ⊗ δ( y , z ) ≤ δ(x , z ).
A morphism f : ( A,δ) → ( B,γ) in SetF() is a map f :
A → B such that γ( f (x ), f ( y )) ≥ δ(x , y ) for all x , y ∈ A.
We say that f is a strong morphism, if γ( f (x ), f ( y )) =
δ(x , y ) for all x , y .
From historical point of view there is another category
consisting of sets with similarity relations defined. This cate-
gory SetR() is an analogy of a category of sets with rela-
tions between sets as morphisms. Objects of this category
SetR() are the same as in the category SetF() and mor-
phism f : ( A,δ) → ( B,γ) are maps f : A × B → such
that
(a) (∀x , z ∈ A)(∀ y ∈ B ) δ(z , x ) ⊗ f (x , y ) ≤ f (z , y ),
(b) (∀x ∈ A)(∀ y , z ∈ B ) f (x , y ) ⊗ γ( y , z ) ≤ f (x , z )
(c) (∀x ∈ A)1 =
{ f (x , y ) : y ∈ B }.
If f : ( A,δ) → ( B,γ) and g : ( B,γ) → (C, ω) are two
morphisms then their composition is a function g ◦ f : A ×
C → such that
g ◦ f (x , z ) =
y∈B
( f (x , y ) ⊗ g( y , z )).
It is well known (see Moˇ ckoˇ r 2004, Lemma 1) that there
exists a functor F : SetF() → SetR() which is an identity
function on objects and for a morphism f : ( A,δ) → ( B γ),
F ( f )(a, b) = γ( f (a), b) holds for all a ∈ A, b ∈ B .
In papers (Moˇ ckoˇr2007b, 2008) we investigated models
of first order fuzzy logic which are based on a generalization
of fuzzy sets, namely on the category SetF() of sets with
similarity relations. An interpretation ‖ψ ‖ of a formula ψ in a
model based on the category SetF() was defined as a fuzzy
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