Andrzej Walendziak, Magdalena Wojciechowska

DEMONSTRATIO MATHEMATICA
Vol. XLIII
No 3
2010
Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa
BIPARTITE PSEUDO-BL ALGEBRAS
Abstract. The class of bipartite pseudo-BL algebras (denoted by BP) and the
class of strongly bipartite pseudo-BL algebras (denoted by BP0 ) are investigated. We
prove that the class BP0 is a variety and show that BP is closed under subalgebras
and arbitrary direct products but it is not a variety. We also study connections between
bipartite pseudo-BL algebras and other classes of pseudo-BL algebras.
1. Introduction
BL algebras were introduced by Hájek [9] in 1998. MV algebras introduced by Chang [1] are contained in the class of BL algebras. Georgescu and
Iorgulescu [6] introduced pseudo-MV algebras as a noncommutative generalization of MV algebras. In 2000, in a natural way, there were introduced
pseudo-BL algebras as a generalization of BL algebras and MV algebras. A
pseudo-BL algebra is a pseudo-MV algebra if and only if the pseudo-Double
Negation condition (pDN, for short) is satisfied, that is, (x− )∼ = (x∼ )− = x
for all x. Main properties of pseudo-BL algebras were studied in [2] and
[3]. Pseudo-BL algebras correspond to a pseudo-basic fuzzy logic (see [10]
and [11]).
Bipartite MV algebras were defined and studied by Di Nola, Liguori
and Sessa in [4]. Dymek [5] investigated bipartite pseudo-MV algebras.
Georgescu and Leuştean [8] introduced the class BP of pseudo-BL algebras
bipartite by some ultafilter and the subclass BP0 of pseudo-BL algebras
bipartite by all ultrafilters. In this paper we give some characterizations of
bipartite and strongly bipartite pseudo-BL algebras. We prove that the class
BP0 is a variety and show that BP is closed under subalgebras and arbitrary
direct products but it is not a variety. We also study connections between
bipartite pseudo-BL algebras and other classes of pseudo-BL algebras.
2000 Mathematics Subject Classification: 03G25, 06F05.
Key words and phrases: pseudo-BL algebra, filter, ultrafilter, (strongly) bipartite
pseudo-BL algebra.
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A. Walendziak, M. Wojciechowska-Rysiawa
2. Preliminaries
Definition 2.1. ([2]) Let (A, ∨, ∧, ⊙, →, , 0, 1) be an algebra of type
(2, 2, 2, 2, 2, 0, 0). The algebra A is called a pseudo-BL algebra if it satisfies
the following axioms, for any x, y, z ∈ A :
(C1)
(C2)
(C3)
(C4)
(C5)
(A, ∨, ∧, 0, 1) is a bounded lattice,
(A, ⊙, 1) is a monoid,
x⊙y ≤z ⇔x≤y →z ⇔y ≤x
z,
x ∧ y = (x → y) ⊙ x = x ⊙ (x
y),
(x → y) ∨ (y → x) = (x
y) ∨ (y
x) = 1.
Throughout this paper A will denote a pseudo-BL algebra. For any x ∈ A
and n = 0, 1, . . . , we put x0 = 1 and xn+1 = xn ⊙ x.
Proposition 2.2. ([2])The following properties hold in A for all x, y ∈ A :
(a) x ≤ y ⇔ x → y = 1 ⇔ x
(b) x ⊙ y ≤ x and x ⊙ y ≤ y.
y = 1,
Let us define x− = x → 0 and x∼ = x
0 for all x ∈ A.
Proposition 2.3. ([2]) The following properties hold in A for all x, y ∈ A:
(a) x ≤ (x− )∼ and x ≤ (x∼ )− ,
(b) x− ⊙ x = x ⊙ x∼ = 0,
(c) x ≤ y implies y − ≤ x− and y ∼ ≤ x∼ .
Definition 2.4. A nonempty set F is called a filter of A if the following
conditions hold:
(F1) If x, y ∈ F, then x ⊙ y ∈ F,
(F2) if x ∈ F, y ∈ A, x ≤ y then y ∈ F.
The filter F is called proper if F 6= A. The set of all filters of A is denoted by
F il(A). For every subset X ⊆ A, the smallest filter of A which contains X,
that is the intersection of all filters F ⊇ X, is said to be the filter generated
by X and will be denoted by [X).
Proposition 2.5. ([2]) If X ⊆ A, then
[X) = {y ∈ A : x1 ⊙ · · · ⊙ xn ≤ y for some n ≥ 1 and x1 , . . . , xn ∈ X}.
Definition 2.6. Let F be a proper filter of A.
(a) F is called prime iff for all x, y ∈ A, x ∨ y ∈ F implies x ∈ F or y ∈ F.
(b) F is called maximal ( or ultrafilter) iff whenever H is a filter such that
F ⊆ H ⊆ A, then either H = F or H = A.
Bipartite pseudo-BL algebras
489
We denote by Max (A) the set of ultrafilters of A.
Definition 2.7. A filter H of A is called normal if for every x, y ∈ A
x→y∈H⇔x
y ∈ H.
Proposition 2.8. ([2]) Any ultrafilter of A is a prime filter of A.
Proposition 2.9. ([2]) Any proper filter of A can be extended to an ultrafilter.
Following [8], for any F ⊆ A, we define two sets F∼∗ and F−∗ as follows:
F∼∗ = {x ∈ A : x ≤ f ∼ for some f ∈ F }
and
F−∗ = {x ∈ A : x ≤ f − for some f ∈ F }.
By Remark 1.13 of [8] we have
F∼∗ = {x ∈ A : x− ∈ F } and F−∗ = {x ∈ A : x∼ ∈ F }.
Lemma 2.10. If F is a proper filter of A, then:
(a)
(b)
(c)
(d)
F ∩ F∼∗ = ∅,
F ∩ F−∗ = ∅,
F∼∗ ⊆ A − F,
F−∗ ⊆ A − F.
Proof. (a) Suppose that x ∈ F ∩ F∼∗ . Then x ∈ F and x ≤ f ∼ for some
f ∈ F . Since F is a filter, from definition it follows that f ∈ F and f ∼ ∈ F.
Using Proposition 2.3 (b) we have 0 = f ⊙ f ∼ ∈ F. This contradicts the fact
that F is proper.
(b) Similar to (a).
(c) Let x ∈ F∼∗ . Then x− ∈ F. Suppose that x ∈ F. Applying Proposition
2.3 (b) we obtain x− ⊙ x = 0 ∈ F . This is a contradiction, because F is
proper.
(d) Similar to (c).
Proposition 2.11. Let F be a proper filter of A. Then the following
conditions are equivalent:
(a)
(b)
(c)
(d)
A = F ∪ F∼∗ = F ∪ F−∗ ,
F−∗ = F∼∗ = A − F,
∀x ∈ A (x ∈ F or (x− ∈ F and x∼ ∈ F )),
∀x ∈ A (x ∨ x− , x ∨ x∼ ∈ F ) and F is prime.
Proof. (a) ⇒ (b). Follows easily from Lemma 2.10 (a) and (b).
(b) ⇒ (c). Let x ∈ A − F . Therefore x ∈ F∼∗ = F−∗ . Hence x− ∈ F and
∼
x ∈ F.
(c) ⇔ (d). See Proposition 5.1 of [8].
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(c) ⇒ (a). Obvious.
Proposition 2.12. If F is a proper filter of A and one of the equivalent
conditions of Proposition 2.11 holds, then F is an ultrafilter.
Proof. Suppose that x ∈
/ F and let U = [F ∪ {x}). We show that A = U.
It suffices to prove that 0 ∈ U . Let x ∈ F∼∗ and hence x− ∈ F . Therefore
x− ∈ U . Consequently, 0 = x− ⊙ x ∈ U.
Let h : A → B be a homomorphism of pseudo-BL algebras. The set Ker(h) =
{x ∈ A : h(x) = 1} is called the kernel of h.
Proposition 2.13. ([8]) Let h : A → B be a homomorphism of pseudo-BL
algebras. Then:
(a) Ker(h) is a normal filter of A,
(b) A/Ker(h) ∼
= B.
Proposition 2.14. ([8]) If H is a normal filter of A, then there is a
bijection between the ultrafilters of A containing H and the ultrafilters of
A/H.
3. Bipartite pseudo-BL algebras
Definition 3.1. ([8]) A is called bipartite if A = F ∪ F∼∗ = F ∪ F−∗ for
some ultrafilter F .
Define the class BP as follows: A ∈ BP ⇔ A is bipartite.
Let us denote by sup(A) the set {x ∨ x− : x ∈ A} ∪ {x ∨ x∼ : x ∈ A}.
Proposition 3.2. ([8]) sup(A) = {x ∈ A : x ≥ x− or x ≥ x∼ }.
Proposition 3.3. Let sup(A) be a proper filter. Then A ∈ BP.
Proof. Suppose that sup(A) is a proper filter. By Proposition 2.9 there
exists an ultrafilter F of A such that sup(A) ⊆ F. From Proposition 2.8 we
conclude that F is prime. Applying Propositions 2.11 and 2.12 we deduce
that A ∈ BP.
Proposition 3.4. A ∈ BP ⇔ [sup(A)) 6= A.
Proof. ⇒: Assume that A ∈ BP and [sup(A)) = A. By Proposition 2.11,
there exists an ultrafilter F of A such that x ∨ x− , x ∨ x∼ ∈ F for all x ∈ A.
Then sup(A) ⊆ F . Consequently, A = [sup(A)) ⊆ F and hence A = F , a
contradiction.
⇐: Suppose that [sup(A)) 6= A. By Proposition 2.9, [sup(A)) can be
extended to an ultrafilter F. From Proposition 2.11 we have A = F ∪ F∼∗ =
F ∪ F−∗ . Thus A ∈ BP.
Proposition 3.5. If F = A−{0} is an ultrafilter of A, then A is bipartite.
Bipartite pseudo-BL algebras
491
Proof. Let x ∈ A. Then x ∈ F or x = 0. If x = 0, then x− = x∼ = 1 ∈ F.
By Proposition 2.11, A = F ∪ F−∗ = F ∪ F∼∗ , and hence A is bipartite.
Proposition 3.6. Any subalgebra of a bipartite pseudo-BL algebra is bipartite.
Proof. Let A ∈ BP and suppose that B is a subalgebra of A. Let F be
a proper filter of A satisfying the condition (d) of Proposition 2.11. Then
U = F ∩ B is a prime filter of B and sup B ⊆ U. By Propositions 2.11 and
2.12, U is an ultrafilter of B and B = U ∪ U∼∗ = U ∪ U−∗ . Hence B is a
bipartite pseudo-BL algebra.
Proposition
3.7. Let A and At (t ∈ T ) be pseudo-BL algebras and
Y
At . If At0 is bipartite for some t0 ∈ T , then A is bipartite.
A=
t∈T
Proof.Y
Let Ut0 be a prime filter of At0 such that sup(At0 ) ⊆ Ut0 . Let U =
As . It is obvious that U is a prime filter of A. For every x =
Ut0 ×
s6=t0
∼
∼
(at )t∈T ∈ A, x ∨ x− = (at ∨ a−
t )t∈T ∈ U and x ∨ x = (at ∨ at )t∈T ∈ U.
Therefore, A is bipartite.
Corollary
3.8. Let At (t ∈ T ) be bipartite pseudo-BL algebras. Then
Y
A=
At is a bipartite pseudo-BL algebra.
t∈T
Proposition 3.9. A homomorphic image of a bipartite pseudo-BL algebra
is not bipartite in general.
Proof. Let A = A1 × A2 , where A1 ∈ BP and A2 ∈
/ BP. We consider the
projection map π2 : A → A2 . Obviously π2 is a homomorphism from A onto
A2 . From Proposition 3.7 we see that A is bipartite but, by assumption, A2
is not bipartite.
Corollary 3.10. The class BP is not a variety.
4. Strongly bipartite pseudo-BL algebras
We define the class BP0 of pseudo-BL algebras as follows: A ∈ BP0 iff
A = F ∪ F∼∗ = F ∪ F−∗ for any ultrafilter F of A. Algebras from the class
BP0 are called strongly bipartite. Of course, BP0 ⊆ BP.
Proposition 4.1. The following conditions are equivalent:
(a) A is strongly bipartite,
(b) ∀F ∈ M ax(A)∀x ∈ A [x ∈
/ F ⇒ ∀n ∈ N((xn )− ∈ F and (xn )∼ ∈ F )].
Proof. (a) ⇒ (b). Let A ∈ BP0 and let F be an ultrafilter. Suppose that
x ∈ A − F. By Proposition 2.11, x− ∈ F and x∼ ∈ F. Applying Propositions
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A. Walendziak, M. Wojciechowska-Rysiawa
2.2 (b) and 2.3 (c) we have x− ≤ (xn )− and x∼ ≤ (xn )∼ for all n ∈ N. Then
(xn )− ∈ F and (xn )∼ ∈ F.
(b) ⇒ (a). Let the condition (b) be satisfied and F be an ultrafilter
of A. Suppose that x ∈
/ F. Then (xn )− ∈ F and (xn )∼ ∈ F for n ∈ N.
In particular, x− ∈ F and x∼ ∈ F. Thus the condition (c) of Proposition
2.11 holds. Consequently, A = F ∪ F−∗ = F ∪ F∼∗ . Therefore, A is strongly
bipartite.
Proposition 4.2. ([8]) The following conditions are equivalent:
(a) A is strongly bipartite,
\
(b) sup(A) ⊆ M(A), where M(A) = {F : F is an ultrafilter of A}.
In [3], there were defined two sets:
U (A) := {x ∈ A : (xn )∼ ≤ x for all n ∈ N}
and
V (A) := {x ∈ A : (xn )− ≤ x for all n ∈ N}.
Proposition 4.3. ([3]) M(A) ⊆ U (A) ∩ V (A).
Proposition 4.4. U (A) ∪ V (A) ⊆ sup(A).
Proof. Let x ∈ U (A). Then (xn )∼ ≤ x for all n ∈ N. In particular, x∼ ≤ x.
By Proposition 3.2, x ∈ sup(A). Thus U (A) ⊆ sup(A). Similarly, V (A) ⊆
sup(A).
From Propositions 4.3 and 4.4 we obtain
Corrolary 4.5. M(A) ⊆ sup(A).
Theorem 4.6. The following are equivalent:
(a) A ∈ BP0 ,
(b) sup(A) = U (A) = V (A) = M(A),
(c) ∀F ∈ M ax(A) sup(A) ⊆ F.
Proof. (a) ⇒ (b). We have
U (A) ⊆ sup(A)
(by Proposition 4.4)
⊆ M(A)
(by Proposition 4.2)
⊆ U (A)
(by Proposition 4.3).
Therefore sup(A) = U (A) = M(A). Similarly, sup(A) = V (A) = M(A).
(b) ⇒ (c). Obvious.
(c) ⇒ (a). Let (c) hold. Then sup(A) ⊆ M(A) and hence, by Proposition
4.2, A is strongly bipartite.
Proposition 4.7. Any subalgebra of strongly bipartite pseudo-BL algebra
is strongly bipartite.
Bipartite pseudo-BL algebras
493
Proof. Let A be strongly bipartite and B be a subalgebra of A. Let F be
an ultrafilter of B and F ′ be the filter generated by F in A. Then
F ′ = {y ∈ A : y ≥ x for some x ∈ F }
by Proposition 2.5. Suppose that 0 ∈ F ′ . Hence 0 ∈ F . This contradicts the
fact that F is proper. Then F ′ is proper too. By Proposition 2.9, there is
an ultrafilter U of A such that U ⊇ F ′ . It is easy to see that U ∩ B ∈ F il(B)
and U ∩ B ⊇ F . Since F ∈ M ax(B), it follows that U ∩ B = F. We obtain
sup(B) ⊆ sup(A) ⊆ U, because A is strongly bipartite. As B is a subalgebra
we have sup(B) ⊆ B. Consequently, sup(B) ⊆ U ∩ B = F . By Theorem 4.6,
B is strongly bipartite.
Proposition 4.8. The class BP0 is closed under direct products.
Y
Proof. Let A =
At , and At be bipartite for t ∈ T . Let F ∈ M ax(A).
t∈T
Then there is t0 ∈ T such that F = Ft0 ×
Y
As , where Ft0 ∈ M ax(At0 ).
s6=t0
Let x = (at )t∈T ∈ A. It is easily seen that x ∨ x− = (at ∨ a−
t )t∈T ∈ F and
x ∨ x∼ = (at ∨ a∼
)
∈
F.
Thus
sup(A)
⊆
F
for
each
F
∈
M ax(A), and
t∈T
t
therefore A ∈ BP0 by Theorem 4.6.
Proposition 4.9. Let A ∈ BP0 and h : A → B be a surjective homomorphism. Then B ∈ BP0 .
Proof. Write H = Ker(h). By Proposition 2.13, H is a normal filter and
B ∼
= A/H. From Proposition 2.14 it follows that every ultrafilter of A/H
has a form F/H, where F is an ultrafilter of A containing H. We have
sup(A/H) = {a/H ∨ (a/H)− : a ∈ A} ∪ {a/H ∨ (a/H)∼ : a ∈ A}
= {a ∨ a− /H : a ∈ A} ∪ {a ∨ a∼ /H : a ∈ A}
⊆ F/H,
because sup(A) ⊆ F. Consequently, B is strongly bipartite.
Propositions 4.7 – 4.9 yield
Theorem 4.10. The class BP0 is a variety.
Let B(A) denote the set of all complemented elements in the distributive
lattice L(A) = (A, ∨, ∧, 0, 1) of a pseudo-BL algebra A.
Proposition 4.11. ([3]) The following are equivalent:
(a) x ∈ B(A),
(b) x ∨ x− = 1,
(c) x ∨ x∼ = 1.
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A. Walendziak, M. Wojciechowska-Rysiawa
\
Write Mn (A) = {F : F is a normal ultrafilter of A}. Recall that A is
called semisimple iff Mn (A) = {1}.
Proposition 4.12. Let A be a semisimple pseudo-BL algebra. Then A is
strongly bipartite if and only if A = B(A).
Proof. Let A ∈ BP0 . Then sup(A) ⊆ M(A). It is easily seen that M(A) ⊆
Mn (A). Since A is semisimple, Mn (A) = {1}. Consequently, sup(A) = {1}.
Hence x ∨ x− = 1 for all x ∈ A and by Proposition 4.11, B(A) = A.
Assume now that x ∨ x− = 1 for all x ∈ A. Therefore sup(A) = {1}. Hence
sup(A) ⊆ F for all F ∈ M ax(A). From Theorem 4.6 it follows that A is
strongly bipartite.
A pseudo-BL algebra A is called good if it satisfies the following condition:
(a− )∼ = (a∼ )−
for all a ∈ A. We say that A is local if it has a unique ultrafilter. The order
of a ∈ A, in symbols ord(a), is the smallest natural number n such that
an = 0. If no such n exists, then ord(a) = ∞. A good pseudo-BL algebra A
is called perfect if it is local and for any a ∈ A,
ord(a) < ∞ ⇔ ord(a∼ ) = ∞.
Following [8], we define two sets:
D(A) = {a ∈ A : ord(a) = ∞} and D(A)∗ = {a ∈ A : ord(a) < ∞}.
It is obvious that D(A) ∩ D(A)∗ = ∅ and D(A) ∪ D(A)∗ = A.
Proposition 4.13. ([8]) The following conditions are equivalent:
(a) A is local;
(b) D(A) is the unique ultrafilter of A.
Proposition 4.14. ([8]) Let A be a local good pseudo-BL algebra. The
following are equivalent:
(a) A is perfect,
(b) D(A)∗ = D(A)∗− = D(A)∗∼ .
Proposition 4.15. Every perfect pseudo-BL algebra is strongly bipartite.
Proof. Let A be perfect. Then it is local, and so, by Proposition 4.13,
D(A) is the unique ultrafilter of A. We have A = D(A) ∪ D(A)∗ and from
Proposition 4.14 it follows that D(A)∗ = D(A)∗− = D(A)∗∼ . Consequently,
A is strongly bipartite.
Example 4.16. ([13]) Let a, b, c, d ∈ R, where R is the set of all real
numbers. We put by definition
(a, b) ≤ (c, d) ⇔ a < c or (a = c and b ≤ d).
Bipartite pseudo-BL algebras
495
For any x, y ∈ R, we define operations ∧ and ∨ as follows: x ∧ y =
min{x, y} and x ∨ y = max{x, y}. The meet and the join are defined on
R × R component-wise. Let
1
1
A=
, b : b ≥ 0 ∪ {(a, b) : < a < 1, b ∈ R} ∪ {(1, b) : b ≤ 0} .
2
2
For any (a, b), (c, d) ∈ A, we put:
1
, 0 ∨ (ac, bc + d) ,
(a, b) ⊙ (c, d) =
2
c d−b
1
,0 ∨
,
∧ (1, 0) ,
(a, b) → (c, d) =
2
a a
c ad − bc
1
,0 ∨
,
∧ (1, 0) .
(a, b)
(c, d) =
2
a
a
Then (A, ∨, ∧, ⊙, →, , 21 , 0 , (1, 0)) is a pseudo-BL algebra. Let (a, b) ∈ A.
We have
1
1
b
1
−
,0 =
,0 ∨
,−
∧ (1, 0)
(a, b) = (a, b) →
2
2
2a a
and
∼
(a, b) = (a, b)
1
,0
2
=
1
1
b
,0 ∨
,−
∧ (1, 0) .
2
2a 2a
It is easy to see that
((a, b)− )∼ = (a, b) = ((a, b)∼ )− .
Then A satisfies condition (pDN) and hence A is a good pseudo-BL algebra.
(Moreover, A is a pseudo-MV algebra.)
Let F = {(1, b) : b ≤ 0}. In [13], we proved that F is the unique ultrafilter of A. Consequently, A is local. Since F is normal (see [13]), we have
Mn (A) = {F } =
6 {(1, 0)}, and therefore A is not semisimple. Now we show
that condition (c) of Proposition 2.11 is not satisfied. Indeed, let x = 43 , 1 .
Then x ∈
/ F and
1
2 4
2 4
−
x =
∧ (1, 0) =
∈
/ F.
,0 ∨
,−
,−
2
3 3
3 3
Therefore, A is not bipartite and obviously it is not strongly bipartite.
Define
1
2
, b ∈ R : b ≥ 0 ∪ {(1, b) ∈ R2 : b ≤ 0}.
B=
2
It is easy to see that (B, ∨, ∧, ⊙, →, , 12 , 0 , (1, 0)) is a subalgebra of A.
The subset F is also the unique ultrafilter of B and hence B is local. Now we
496
A. Walendziak, M. Wojciechowska-Rysiawa
prove that B is perfect. By Proposition 4.13, D(B) = F . Let x = (a, b) ∈ B.
We have
ord(x) < ∞ ⇔ x ∈ B − F ⇔ x∼ ∈ F ⇔ ord(x∼ ) = ∞.
Thus B is perfect. From Proposition 4.15 it follows that B is strongly bipartite. Since A is not strongly bipartite, we see that A is not perfect by
Proposition 4.15.
Acknowledgments. The authors are highly grateful to referee for
her/his remarks and suggestions for improving the paper.
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Demonstratio Math. 42 (2009), 453-466.
Andrzej Walendziak
Magdalena Wojciechowska-Rysiawa
WARSAW SCHOOL OF INFORMATION TECHNOLOGY
UNIVERSITY OF PODLASIE
3 Maja 54
Newelska 6
PL-08110 SIEDLCE, POLAND
PL-01447 WARSZAWA, POLAND
E-mail: [email protected]
E-mail: [email protected]
Received March 23, 2009; revised version August 5, 2009.