Rough Approximations in Algebras of a Non-associative Generalization
of the Łukasiewicz Infinite Valued Logic
Jiˇr´ı Rach˚unek
1
and Dana
ˇ
Salounov´a
2
1
Department of Algebra and Geometry, Faculty of Sciences, Palack´y University, 17. listopadu 12, Olomouc, Czech Republic
2
Department of Mathematical Methods in Economy, Faculty of Economics, V
ˇ
SB–Technical University Ostrava,
Sokolsk´a 33, Ostrava, Czech Republic
Keywords:
Commutative Basic Algebra, MV-algebra, Non-associative Logic, Łukasiewicz Logic, Classical Rough Set,
Approximation Space, Ideal, Congruence.
Abstract:
Commutative basic algebras are non-associative generalizations of MV-algebras. They are an algebraic coun-
terpart of a non-associative propositional logic which generalizes the Łukasiewicz infinite valued logic and
which is related to reasoning under uncertainty. The paper investigates approximation spaces in commutative
basic algebras based on their ideals.
1 INTRODUCTION
Rough sets were introduced by Pawlak (Pawlak,
1982) in 1982 to give a new mathematical approach to
vagueness. The key idea is that our knowledge about
the properties of the objects of a given universeof dis-
course may be inadequate or incomplete in the sense
that the objects of this universe can be observed only
within the accuracy of indiscernibility relations. Re-
call that in the classical rough set theory, subsets are
approximated by means of pairs of ordinary sets, so-
called lower and upper approximations, which are e.g.
composed by some classes of given equivalences.
It is known that the basic (fuzzy) logic B L is the
logic of continuous t-norms and their residua (H´ajek,
1998). That means, if a continuous t-norm & is
considered as the truth function of conjunction and
its residuum → is the truth function of implication,
then each evaluation of propositional variables by
truth values from [0,1] extends to the evaluation of
each formula. (See (H´ajek, 1998), (Botur and Halaˇs,
2009).) In all these logics the conjunction & is asso-
ciative, i.e., for arbitrary formulas φ,ψ, χ, the formula
φ & (ψ & χ) ←→ (φ & ψ) & χ is provable.
But there are situations where the associativity of
& need not be satisfied. Let we have expert systems
where we need estimate for the degree of certainty of
conjunctionand disjunction of statements S
1
, ..., S
n
of
which they are not completely sure. This uncertainty
is described by the probabilities p
i
assigned to the
statements S
i
. The conclusion C of an expert system
usually depends on several statements S
i
. Then, e.g.,
the probability p(S
1
& S
2
) of S
1
& S
2
can take differ-
ent values depending on whether S
1
and S
2
are inde-
pendent or correlated. It is known that for given p
1
=
p(S
1
) and p
2
= p(S
2
), possible values of p(S
1
& S
2
)
form an interval p = [p
−
, p
+
] ⊆ [0, 1], where p
−
=
max(p
1
+ p
2
− 1, 0) and p
+
= min(p
1
, p
2
). (See
(Kreinovich, 2004) or (Botur and Halaˇs, 2009).)
Therefore we can use such interval estimates to
get an interval p(C) of possible values of p(C). But
the interval p(C) can be too large. Then in such sit-
uations it is reasonable to select a point within this
interval as an estimate for p(S
1
& S
2
), e.g., a mid-
point of this interval. That means, we can evaluate
S
1
& S
2
:=
1
2
· max(p
1
+ p
2
− 1, 0) +
1
2
· min(p
1
, p
2
).
(See (Botur and Halaˇs, 2009).) It is obvious that op-
eration & is not associative.
Hence we can see that in such situations we need
to have a propositional logic which generalizes fuzzy
logics, e.g. Łukasiewicz, G¨odel or product logic, such
that the conjunction is not necessarily associative.
In (Botur and Halaˇs, 2009), the authors proposed
a logic foundation for fuzzy reasoning with non-
associative conjunction in the form of a new for-
mal deductive system L
CBA
. This logic is very close
to the Łukasiewicz logic (differs just in this non-
associativity of the conjunction). The authors have
shown that L
CBA
is algebraizable logic in the sense of
(Blok and Pigozzi, 1989) and that its equivalent al-
gebraic semantics is the variety of commutative basic
algebras. Since MV-algebras are an algebraic coun-
158
Rach˚unek J. and Šalounová D..
Rough Approximations in Algebras of a Non-associative Generalization of the Łukasiewicz Infinite Valued Logic.
DOI: 10.5220/0005129401580162
In Proceedings of the International Conference on Fuzzy Computation Theory and Applications (FCTA-2014), pages 158-162
ISBN: 978-989-758-053-6
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

terpart of the Łukasiewicz logic, commutative basic
algebras are appropriate non-associative generaliza-
tions of MV-algebras.
MV-algebras are an algebraic semantics of a logic
with truth values from the real interval [0, 1] and thus
it is natural that rough sets in MV-algebras were in-
troduced and investigated. (See (Rasouli and Davvaz,
2010).) The corresponding approximate spaces are
based on congruences or, equivalently, on ideals of
MV-algebras.
In the paper we introduce and study approximate
spaces in commutative basic algebras. Analogously
as in MV-algebras, congruences correspond to ideals
and so we deal with approximate spaces based on ide-
als of these algebras.
2 PRELIMINARIES
An algebra A = (A; ⊕, ¬,0) of type h2,1, 0i is called
a basic algebra (Chajda et al., 2009) if for any x, y, z ∈
A:
(1) x⊕ 0 = x;
(2) ¬¬x = x;
(3) ¬(¬x⊕ y) ⊕ y = ¬(¬y⊕ x) ⊕ x;
(4) ¬(¬(¬(x⊕ y) ⊕ y) ⊕ z) ⊕ (x⊕ z) = ¬0.
If the groupoid (A; ⊕) is commutative then
(A; ⊕,¬,0) is a commutative basic algebra.
Put 1 := ¬0. Let ≤ be the binary relation on A
such that
x ≤ y :⇐⇒ ¬x⊕ y = 1.
Then ≤ is an order and the ordered set (A; ≤) is a
bounded lattice, where 0 is the least and 1 the greatest
element, and for the lattice operations we have
x∨ y = ¬(¬x⊕ y) ⊕ y, x∧ y = ¬(¬x∨ ¬y).
The class of basic algebras contains certain classes
of algebras of many-valued and quantum logics. For
example, MV-algebras, orthomodular lattices and lat-
tice effect algebras can be viewed as particular cases
of basic algebras (see (Chajda et al., 2009)).
In what follows, we will deal with commutative
basic algebras. Recall that in such a case the lat-
tice (A; ∨,∧) is distributive (Chajda et al., 2009).
Moreover, every finite commutative basic algebra is
an MV-algebra (Botur and Halaˇs, 2008), but there
are commutative basic algebras which are not MV-
algebras. (Recall that MV-algebras are just associa-
tive commutative basic algebras.)
Define, for any x, y ∈ A,
x⊖ y := ¬(¬x⊕ y).
For the fundamental properties of commutative
basic algebras see (Botur and Halaˇs, 2008), (Botur
and Halaˇs, 2009) or (Botur et al., 2012).
Let A be a commutative basic algebra and
/
0 6= I ⊆
A. Then I is called
(a) a preideal of A if
(i) x, y ∈ I =⇒ x⊕ y ∈ I;
(ii) x ∈ I, y ∈ A, y ≤ x =⇒ y ∈ I;
(b) an ideal of A if I is the 0-class of some congruence
on A.
(See (Krˇn´avek and K¨uhr, 2011) or (Botur et al.,
2012).)
Every ideal of A is a preideal of A but not con-
versely (Krˇn´avek and K¨uhr, 2011). Ideals of A are
exactly kernels of congruences and since the variety
of commutative basic algebras is congruence regular,
any ideal I is the 0-class of a unique congruence θ
I
on
A. Then (x,y) ∈ θ
I
iff x⊖ y, y⊖ x ∈ I. Hence we will
denote the quotient algebra A/θ
I
also in the form A/I.
Let P (A) and I (A) be the set of preideals and ide-
als of A, respectively. Then by (Krˇn´avek and K¨uhr,
2011), (P (A), ⊆) is a distributive complete lattice and
(I (A), ⊆) is its complete sublattice.
An additive term is a commutative basic algebra
term in which the symbol ¬ does not occure. If A
is a commutative basic algebra and
/
0 6= B ⊆ A, then
the preideal hBi generated by B contains exactly those
elements a ∈ A such that a ≤ τ(b
1
, . .. , b
n
) for some
n-ary additive term τ and b
1
, . .. , b
n
∈ B.
Now we recall some basic notions of the theory
of classical approximation spaces. An approxima-
tion space is a pair (S,θ) where S is a set and θ
an equivalence on S. For any approximation space
(S,θ), by the upper rough approximation in (S,θ)
we will mean the mapping
Apr : P (S) −→ P (S)
such that Apr(X) := {x ∈ S : x/θ ∩ X 6=
/
0} and by
the lower rough approximation in (S,θ) the mapping
Apr
: P (S) −→ P (S) such that Apr(X) := {x ∈ S :
x/θ ⊆ X}, for any X ⊆ S. (x/θ is the class of S/θ
containing x.)
If
Apr(X) = Apr(X) then X is called a definable
set, otherwise X is called a rough set.
3 APPROXIMATIONS INDUCED
BY IDEALS
In this section we introduce and investigate special
approximation spaces (A,θ) such that A is the uni-
verse of a commutative basic algebra and θ is a con-
gruence on this basic algebra.
RoughApproximationsinAlgebrasofaNon-associativeGeneralizationoftheŁukasiewiczInfiniteValuedLogic
159

If A = (A; ⊕,¬,0) is a commutative basic alge-
bra, θ a congruence on A and I = I
θ
the correspond-
ing ideal, then Apr
I
(X) and
Apr
I
(X) will denote the
lower and upper rough approximation of any X ⊆ A
in the approximation space (A,θ).
Proposition 3.1. Let A be a commutative basic alge-
bra, I an ideal of A and a, b ∈ A. Then a/I = b/I if
and only if there are x, y ∈ I such that a = (b⊕ y) ⊖ x.
Proof. Let a = (b ⊕ y) ⊖ x, where x, y ∈ I. Then
b ⊖ a = b ⊖ ((b⊕ y) ⊖ x) ≤ b ⊖ (b ⊖ x) = b ∧ x ≤ x,
thus b⊖ a ∈ I.
Further, a⊖ b = ((b⊕ y) ⊖ x) ⊖ b ≤ (b⊕ y) ⊖ b =
y∧ ¬b ≤ y ∈ I, hence a⊖ b ∈ I.
Therefore a/I = b/I.
Conversely, let a/I = b/I, i.e. x = b⊖ a, y = a ⊖
b ∈ I. We have y ⊕ b = (a ⊖ b) ⊕ b = a ∨ b = (b ⊖
a)⊕ a = x⊕a, thus (y⊕b) ⊖x = (x⊕a)⊖ x = a∧¬x.
At the same time x = b ⊖ a ≤ 1⊖ a = ¬a, that means
x ≤ ¬a, hence a ≤ ¬x, thus a∧ ¬x = a.
Therefore we get a = (b⊕ y) ⊖ x. 
If A is a commutative basic algebra and
/
0 6=
X, Y ⊆ A, denote by hX,Yi the preideal of A gener-
ated by X ∪Y.
If τ = τ(z
1
, ..., z
n
) is an additive term, then by
l(τ) we will mean the number of occurencies of the
variables z
1
, . .. , z
n
in τ.
Lemma 3.2. (Botur et al., 2012) Let A be a commu-
tative basic algebra and I an ideal of A. Then, for all
a,b ∈ A, a⊕ (b⊕ I) = (a⊕ b) ⊕ I.
Theorem 3.3. Let I be an ideal of a commutative ba-
sic algebra A and
/
0 6= X, Y ⊆ A. Then
Apr
I
(hX,Yi) ⊆ hApr
I
(X), Apr
I
(Y)i.
If A is linearly ordered then
Apr
I
(hX,Yi) = hApr
I
(X), Apr
I
(Y)i.
Proof. If a ∈
Apr
I
(hX,Yi) then a/I ∩ hX,Yi 6=
/
0. Let b ∈ a/I ∩ hX,Yi and b ≤ τ(z
1
, . .. , z
n
)
where τ is an n-ary additive term and z
i
∈ X ∪
Y, i = 1, ... ,n. Suppose that τ
1
and τ
2
are n-
ary additive terms such that l(τ
1
), l(τ
2
) < l(τ) and
τ(z
1
, . .. , z
n
) = τ
1
(z
1
, . .. , z
n
) ⊕ τ
2
(z
1
, . .. , z
n
). Since
a/I = b/I, there are x, y ∈ I such that a = (b⊕ y) ⊖ x.
Hence a = (b ⊕ y) ⊖ x ≤ b ⊕ y ≤ (τ
1
(z
1
, . .. , z
n
) ⊕
τ
2
(z
1
, . .. , z
n
))⊕y = τ
1
(z
1
, . .. , z
n
)⊕(τ
2
(z
1
, ..., z
n
)⊕
u) where u ∈ I. Since (τ
2
(z
1
, . .. , z
n
) ⊕ u)/I =
τ
2
(z
1
, . .. , z
n
)/I ⊕ u/I = τ
2
(z
1
, . .. , z
n
)/I and z
i
∈
Apr
I
(X) ∪ Apr
I
(Y), i = 1,. .., n, we obtain a ∈
h
Apr
I
(X), Apr
I
(Y)i.
Suppose A is linearly ordered. Let a ∈
h
Apr
I
(X), Apr
I
(Y)i, a ≤ τ(v
1
,.. .,v
n
), where τ is an
n-ary additive term, v
i
∈ Apr
I
(X) ∪ Apr
I
(Y), i =
1,..., n. Let w
i
∈ v
i
/I ∩ X, provided v
i
∈ X, and
w
i
∈ v
i
/I ∩Y, provided v
i
∈ Y, and let z ∈ a/I. Sup-
pose a/I 6= τ(w
1
,.. .w
n
)/I. Since A is linearly or-
dered, z < τ(w
1
,.. .w
n
), hence z ∈ hX,Yi. Therefore
a ∈
Apr
I
(hX,Yi). 
Theorem 3.4. Let I be an ideal of a commutative ba-
sic algebra A and
/
0 6= X, Y ⊆ A. Then
hApr
I
(X), Apr
I
(Y)i ⊆ Apr
I
(hX,Yi).
Proof. Let a ∈ hApr
I
(X), Apr
I
(Y)i. Suppose
a ≤ τ(z
1
, . .. , z
n
), where τ is an n-ary additive term
and z
i
∈ Apr
I
(X) ∪ Apr
I
(Y), i = 1,... ,n. Let b ∈ a/I.
Then there are x,y ∈ I with b = (a⊕x)⊖y. If τ
1
and τ
2
are n-ary additive terms such that l(τ
1
), l(τ
2
) < l(τ)
and τ(z
1
, ..., z
n
) = τ
1
(z
1
, . .. , z
n
) ⊕ τ
2
(z
1
, ..., z
n
),
then b = (a ⊕ x) ⊖ y ≤ a ⊕ x ≤ τ(z
1
, . .. , z
n
) ⊕ x =
(τ
1
(z
1
, . .. , z
n
)⊕τ
2
(z
1
, ..., z
n
))⊕x = τ
1
(z
1
, ..., z
n
)⊕
(τ
2
(z
1
, . .. , z
n
) ⊕ u), where u ∈ I.
We have (τ
2
(z
1
, . .. , z
n
) ⊕ u)/I =
τ
2
(z
1
, . .. , z
n
)/I = τ
2
(z
1
/I, ... , z
n
/I) ⊆ hX,Yi,
because z
i
/I ⊆ X ∪Y, i = 1,.. .,n.
Analogously τ
1
(z
1
/I, ..., z
n
/I) ⊆ hX,Yi, thus
also τ(z
1
/I, ..., z
n
/I) ⊆ hX,Yi, i.e. b ∈ hX,Yi.
Therefore a ∈ Apr
I
(hX,Yi). 
Theorem 3.5. Let A be a linearly ordered commuta-
tive basic algebra, I an ideal of A and X 6=
/
0 a convex
subset of A. Then also Apr
I
(X) and
Apr
I
(X) are con-
vex.
Proof. Let x,y ∈ Apr
I
(X), z ∈ A, x ≤ z ≤ y and
x/I 6= z/I 6= y/I. Suppose a ∈ z/I. The congruence
θ
I
has convex classes, hence for any elements x
1
∈
x/I, y
1
∈ y/I and z
1
∈ z/I we have x
1
< z
1
< y
1
, thus
z
1
∈ Apr
I
(X), and therefore z ∈ Apr
I
(X). That means
Apr
I
(X) is convex
Let now x,y ∈
Apr
I
(X) and z ∈ A such that x ≤ z ≤
y. Let x
1
∈ x/I ∩X, y
1
∈ y/I ∩X and x/I 6= z/I 6= y/I.
If z
1
∈ z/I, then x
1
< z
1
< y
1
. Since x
1
,y
1
∈ X, we
get z
1
∈ z/I ∩ X, therefore z ∈
Apr
I
(X). That means
Apr
I
(X) is convex. 
Let A be a commutative basic algebra. If B ⊆ A,
put ¬B := {¬b : b ∈ B}.
Theorem 3.6. Let A be a commutative basic algebra,
I an ideal of A and
/
0 6= X ⊆ A. Then
a) ¬
Apr
I
(X) = Apr
I
(¬X);
b) ¬Apr
I
(X) = Apr
I
(¬X).
Proof. a) Let x ∈ ¬
Apr
I
(X). Then ¬x ∈ Apr
I
(X),
thus ¬x/I ∩ X 6=
/
0. Let y ∈ ¬x/I ∩ X. Then ¬x ⊖
y, y⊖ ¬x ∈ I, hence also x⊖ ¬y, ¬y⊖ x ∈ I and ¬y ∈
¬X. Therefore x ∈
Apr
I
(¬X), and so ¬Apr
I
(X) ⊆
Apr
I
(¬X).
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
160

Let x ∈ Apr
I
(¬X) and y ∈ x/I ∩ ¬X. Then x ⊖
y, y ⊖ x ∈ I, hence also ¬x ⊖ ¬y, ¬y ⊖ ¬x ∈ I, and
¬y ∈ X. Thus ¬x/I ∩ X 6=
/
0, so ¬x ∈
Apr
I
(X), and
consequentlyx ∈ ¬Apr
I
(X). That means Apr
I
(¬X) ⊆
¬
Apr
I
(X).
b) Let x ∈ ¬Apr
I
(X). Then ¬x ∈ Apr
I
(X), that
means ¬x/I ⊆ X. Thus x/I ⊆ ¬X, hence x ∈
Apr
I
(¬X).
Let x ∈ Apr
I
(¬X), i.e. x/I ⊆ ¬X. Hence ¬x/I ⊆
X, therefore x ∈ ¬Apr
I
(X). 
Lemma 3.7. (Botur et al., 2012, Lemma 2.7) If A is
a commutative basic algebra and I is a preideal of A,
then the following are equivalent:
(i) I is an ideal of A;
(ii) (a⊕ (b⊕x))⊖ (a⊕ b) ∈ I for all a,b ∈ A, x ∈ I.
Theorem 3.8. Let A be a linearly ordered commu-
tative basic algebra and I and J ideals of A. Then
Apr
I
(J) is an ideal of A.
Proof. Obviously 0 ∈
Apr
I
(J).
Let x ∈
Apr
I
(J), y ∈ A, y ≤ x. Let x
1
∈ x/I ∩ J.
Suppose that y
1
∈ y/I and y/I 6= x/I. Then y
1
< x
1
,
hence y
1
∈ J, and thus y ∈
Apr
I
(J).
Now, let x,y ∈ Apr
I
(J), x
1
∈ x/I ∩ J and y
1
∈
y/I ∩J. Then x
1
⊕y
1
∈ J and x
1
⊕y
1
∈ (x/I)⊕(y/I) =
(x⊕ y)/I. Therefore x⊕ y ∈
Apr
I
(J).
Let x,y ∈ A, a ∈ Apr
I
(J) and a
1
∈ a/I ∩ J. Then
((x ⊕ (y ⊕ a)) ⊖ (x ⊕ y))/I = (x/I ⊕ (y/I ⊕ a/I)) ⊖
(x/I ⊕ y/I) = (x/I ⊕ (y/I ⊕ a
1
/I)) ⊖ (x/I ⊕ y/I) =
(x ⊕ (y ⊕ a
1
)) ⊖ (x ⊕ y))/I and (x ⊕ (y ⊕ a
1
)) ⊖ (x ⊕
y) ∈ J. Hence (x⊕ (y⊕ a)) ⊖ (x⊕ y) ∈
Apr
I
(J).
Therefore by Lemma 3.7,
Apr
I
(J) is an ideal of A.

4 CONNECTIONS AMONG
APPROXIMATION SPACES
In this section we investigate approximation spaces
which are induced by different or special ideals.
Proposition 4.1. If I and J are ideals of a commuta-
tive basic algebra and
/
0 6= X ⊆ A, then
Apr
hI,Ji
(X) ⊆ hApr
I
(X), Apr
J
(X)i.
Proof. If a ∈ Apr
hI,Ji
(X), then a/hI, Ji ⊆ X,
thus also a/I, a/J ⊆ X. Hence a ≤ a ⊕ a ∈
hApr
I
(X), Apr
J
(X)i. 
Lemma 4.2. Let A
1
and A
2
be commutative basic al-
gebras, I an ideal of A
2
and f a homomorphism of A
1
into A
2
. Then f
−1
(I) is an ideal of A
1
.
Proof. Obviously f
−1
(I) is a preideal of A
1
. Let
x,y ∈ A
1
and a ∈ f
−1
(I). Then f((x⊕ (y⊕ a)) ⊖ (x⊕
y)) = ( f(x)⊕( f(y)⊕ f(a))⊖( f(x)⊕ f(y)) ∈ I, there-
fore f
−1
(I) is an ideal of A
1
. 
Theorem 4.3. Let A
1
and A
2
be commutative basic
algebras, f a homomorphism of A
1
into A
2
, I an ideal
of A
2
and
/
0 6= X ⊆ A
2
. Then
f
−1
(
Apr
I
(X)) = Apr
f
−1
(I)
( f
−1
(X)).
Proof. Let x ∈ A
1
. Then x ∈
Apr
f
−1
(I)
( f
−1
(X))
if and only if there exists z ∈ x/ f
−1
(I) ∩ f
−1
(X) iff
z⊖x, x⊖z ∈ f
−1
(I) iff f(z⊖x), f(x⊖x) ∈ I iff f(z)⊖
f(x), f(x) ⊖ f(z) ∈ I iff f(z)/I = f(x)/I.
We have f(z) ∈ f (x)/I, z ∈ f
−1
(X), then f(z) ∈
f(x)/I ∩ X, hence f(x)/I ∩ X 6=
/
0, and so f(x) ∈
Apr
I
(X).
That means x ∈
Apr
f
−1
(I)
( f
−1
(X)) if and only if
x ∈ f
−1
(
Apr
I
(X)). 
Theorem 4.4. Let A
1
and A
2
be commutative basic
algebras, f : A
1
−→ A
2
a homomorphism and
/
0 6=
X ⊆ A
1
. Then
f(
Apr
Ker( f)
(X)) = f(X).
Proof. Let x ∈ f(
Apr
Ker( f)
(X)) and y ∈
Apr
Ker( f )
(X) be such that x = f (y). Let z ∈
y/Ker(f) ∩ X. Then z⊖ y, y⊖ z ∈ Ker( f) and z ∈ X.
Hence f(z ⊖ y) = 0, f (y ⊖ z) = 0, so f(z) ⊖ f(y) =
0, f(y) ⊖ f(z) = 0. Therefore f(z) = f(y) = x, i.e.
f(z) ∈ f(X), and consequently f(
Apr
Ker( f)
(X)) ⊆
f(X).
The converse inclusion is obvious. 
Proposition 4.5. Let A be a commutative basic alge-
bra, I and J ideals of A and
/
0 6= X ⊆ A.
a) If A is linearly ordered, then
Apr
I
(X) ∩ Apr
J
(X) = Apr
I∩J
(X).
b) If X is definable with respect to I or J, or if A is
linearly ordered, then
Apr
I∩J
(X) = Apr
I
(X) ∩ Apr
J
(X).
Proof. a) Obvious.
b) Let X be definable, e.g., with respect to I.
Then
Apr
I
(X) ∩ Apr
J
(X) = X ∩ Apr
J
(X) = X ⊆
Apr
I∩J
(X).
The converse inclusion follows from the fact that
I ∩ J ⊆ I, J implies
Apr
I∩J
(X) ⊆ Apr
I
(X), Apr
J
(X).
For linearly ordered A it is obvious. 
RoughApproximationsinAlgebrasofaNon-associativeGeneralizationoftheŁukasiewiczInfiniteValuedLogic
161

5 CONCLUSIONS
It is known that there are situations concerning rea-
soning where the associativity of the logical con-
nection conjunction need not be satisfied. Recently,
a logic foundation for fuzzy reasoning with non-
associative conjunction, as a generalization of the
Lukasiewicz infinite valued logic, was proposed.
Commutative basic algebras are an algebraic seman-
tics of such logic. This paper introduces and inves-
tigates the concept of approximate spaces based on
ideals of commutative basic algebras and shows that
it is reasonable to study approximate spaces in non-
associative structures.
ACKNOWLEDGEMENTS
The first author was supported by the ESF Project
CZ.1.07/2.3.00/20.0051 and IGA PˇrF 2014016, the
second author was supported by ESF Project
CZ.1.07/2.3.00/20.0296.
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FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
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