Quasi-valuation maps based on positive implicative ideals in BCK-algebras
The notion of PI-quasi-valuation maps of a BCK-algebra is introduced, and related properties are investigated. The relationship between an I-quasi-valuation map and a PI-quasivaluation map is examined. Conditions for an I-quasi-valuation map to be a PI-quasi-valuation map are provided, and condition...
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irk-123456789-1883742023-02-27T01:27:39Z Quasi-valuation maps based on positive implicative ideals in BCK-algebras Young Bae Jun Kyoung Ja Lee Seok Zun Song The notion of PI-quasi-valuation maps of a BCK-algebra is introduced, and related properties are investigated. The relationship between an I-quasi-valuation map and a PI-quasivaluation map is examined. Conditions for an I-quasi-valuation map to be a PI-quasi-valuation map are provided, and conditions for a real-valued function on a BCK-algebra to be a quasi-valuation map based on a positive implicative ideal are founded. The extension property for a PI-quasi-valuation map is established. 2018 Article Quasi-valuation maps based on positive implicative ideals in BCK-algebras/ Young Bae Jun, Kyoung Ja Lee, Seok Zun Song // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 65–75. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC: 06F35, 03G25, 03C05. http://dspace.nbuv.gov.ua/handle/123456789/188374 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The notion of PI-quasi-valuation maps of a BCK-algebra is introduced, and related properties are investigated. The relationship between an I-quasi-valuation map and a PI-quasivaluation map is examined. Conditions for an I-quasi-valuation map to be a PI-quasi-valuation map are provided, and conditions for a real-valued function on a BCK-algebra to be a quasi-valuation map based on a positive implicative ideal are founded. The extension property for a PI-quasi-valuation map is established. |
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Young Bae Jun Kyoung Ja Lee Seok Zun Song Quasi-valuation maps based on positive implicative ideals in BCK-algebras Algebra and Discrete Mathematics |
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Young Bae Jun Kyoung Ja Lee Seok Zun Song |
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Young Bae Jun |
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Quasi-valuation maps based on positive implicative ideals in BCK-algebras |
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Quasi-valuation maps based on positive implicative ideals in BCK-algebras |
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Quasi-valuation maps based on positive implicative ideals in BCK-algebras |
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Quasi-valuation maps based on positive implicative ideals in BCK-algebras |
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Quasi-valuation maps based on positive implicative ideals in BCK-algebras |
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quasi-valuation maps based on positive implicative ideals in bck-algebras |
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Quasi-valuation maps based on positive implicative ideals in BCK-algebras/ Young Bae Jun, Kyoung Ja Lee, Seok Zun Song // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 65–75. — Бібліогр.: 11 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT youngbaejun quasivaluationmapsbasedonpositiveimplicativeidealsinbckalgebras AT kyoungjalee quasivaluationmapsbasedonpositiveimplicativeidealsinbckalgebras AT seokzunsong quasivaluationmapsbasedonpositiveimplicativeidealsinbckalgebras |
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“adm-n3” — 2018/10/20 — 9:02 — page 65 — #71
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 26 (2018). Number 1, pp. 65–75
c© Journal “Algebra and Discrete Mathematics”
Quasi-valuation maps based on positive
implicative ideals in BCK-algebras
Young Bae Jun, Kyoung Ja Lee and Seok Zun Song
Communicated by V. A. Artamonov
Abstract. The notion of PI-quasi-valuation maps of a
BCK-algebra is introduced, and related properties are investigated.
The relationship between an I-quasi-valuation map and a PI-quasi-
valuation map is examined. Conditions for an I-quasi-valuation map
to be a PI-quasi-valuation map are provided, and conditions for a
real-valued function on a BCK-algebra to be a quasi-valuation map
based on a positive implicative ideal are founded. The extension
property for a PI-quasi-valuation map is established.
1. Introduction
Logic appears in a ‘sacred’ form (resp., a ‘profane’) which is dominant
in proof theory (resp., model theory). The role of logic in mathematics
and computer science is twofold; as a tool for applications in both areas,
and a technique for laying the foundations. Non-classical logic including
many-valued logic, fuzzy logic, etc., takes the advantage of the classical
logic to handle information with various facets of uncertainty (see [11] for
generalized theory of uncertainty), such as fuzziness, randomness, and so
on. Non-classical logic has become a formal and useful tool for computer
science to deal with fuzzy information and uncertain information. Among
all kinds of uncertainties, incomparability is an important one which can
2010 MSC: 06F35, 03G25, 03C05.
Key words and phrases: (positive implicative) ideal, S-quasi-valuation map,
I-quasi-valuation map, PI-quasi-valuation map.
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66 Quasi-valuation maps based on PI-ideals
be encountered in our life. BCK and BCI-algebras are two classes of logical
algebras. They were introduced by Imai and Iséki (see [2–5]) and have
been extensively investigated by many researchers. It is known that the
class of BCK-algebras is a proper subclass of the class of BCI-algebras.
Neggers and Kim [10] introduced the notion of d-algebras which is another
useful generalization of BCK-algebras, and then they investigated several
relations between d-algebras and BCK-algebras as well as some other
interesting relations between d-algebras and oriented diagraphs. In [9],
Neggers et al. discussed the ideal theory in d-algebras. Neggers et al. [8]
introduced the concept of d-fuzzy function which generalizes the concept
of fuzzy subalgebra to a much larger class of functions in a natural way.
In addition they discussed a method of fuzzification of a wide class of
algebraic systems onto [0, 1] along with some consequences. In [6], Jun et
al. introduced the notion of quasi-valuation maps based on a subalgebra
and an ideal in BCK/BCI-algebras, and then they investigated several
properties. They provided relations between a quasi-valuation map based
on a subalgebra and a quasi-valuation map based on an ideal. In a BCI-
algebra, they gave a condition for a quasi-valuation map based on an ideal
to be a quasi-valuation map based on a subalgebra, and found conditions
for a real-valued function on a BCK/BCI-algebra to be a quasi-valuation
map based on an ideal. Using the notion of a quasi-valuation map based
on an ideal, they constructed (pseudo) metric spaces, and showed that
the binary operation ∗ in BCK-algebras is uniformly continuous. In this
paper, we introduce the notion of PI-quasi-valuation maps of a BCK-
algebra, and investigate related properties. We discuss the relationship
between an I-quasi-valuation map and a PI-quasi-valuation map. We
provide conditions for an I-quasi-valuation map to be a PI-quasi-valuation
map, and find conditions for a real-valued function on a BCK-algebra to
be a quasi-valuation map based on a positive implicative ideal. We finally
establish an extension property for a PI-quasi-valuation map.
2. Preliminaries
An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies
the following axioms:
(I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0),
(II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0),
(III) (∀x ∈ X) (x ∗ x = 0),
(IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y).
If a BCI-algebra X satisfies the following identity:
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Y. B. Jun, K. J. Lee, S. Z. Song 67
(V) (∀x ∈ X) (0 ∗ x = 0),
then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the
following conditions:
(a1) (∀x ∈ X) (x ∗ 0 = x),
(a2) (∀x, y, z ∈ X) (x∗y = 0 ⇒ (x∗z)∗ (y ∗z) = 0, (z ∗y)∗ (z ∗x) = 0),
(a3) (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y),
(a4) (∀x, y, z ∈ X) (((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y) = 0).
We can define a partial ordering 6 by x 6 y if and only if x ∗ y = 0.
A subset A of a BCK/BCI-algebra X is called an ideal of X if it satisfies
the following conditions:
(b1) 0 ∈ A,
(b2) (∀x, y ∈ X) (x ∗ y ∈ A, y ∈ A ⇒ x ∈ A).
A subset A of a BCK-algebra X is called a positive implicative ideal
of X if it satisfies (b1) and
(b3) (∀x, y, z ∈ X) ((x ∗ y) ∗ z ∈ A, y ∗ z ∈ A ⇒ x ∗ z ∈ A).
Proposition 2.1. [7] For a subset A of a BCK-algebra X, the following
are equivalent:
(1) A is a positive implicative ideal of X.
(2) A is an ideal, and for any x, y ∈ X, (x∗y)∗y ∈ A implies x∗y ∈ A.
We refer the reader to the books [1,7] for further information regarding
BCK/BCI-algebras.
3. Quasi-valuation maps based on a positive implicative
ideal
Definition 3.1 ([6]). Let X be a BCK/BCI-algebra. By a quasi-valuation
map of X based on a subalgebra (briefly S-quasi-valuation map of X), we
mean a mapping f : X → R which satisfies the following condition:
(∀x, y ∈ X) (f(x ∗ y) > f(x) + f(y)). (3.1)
Proposition 3.2 ([6]). For any S-quasi-valuation map f of a BCK-
algebra X, we have
(c1) (∀x ∈ X) (f(x) 6 0).
For any real-valued function f on a BCK/BCI-algebra X, we consider
the following conditions:
(c2) f(0) = 0.
(c3) f(x) > f(x ∗ y) + f(y) for all x, y ∈ X.
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68 Quasi-valuation maps based on PI-ideals
(c4) f(x ∗ y) > f(((x ∗ y) ∗ y) ∗ z) + f(z) for all x, y, z ∈ X.
(c5) f(x ∗ z) > f((x ∗ y) ∗ z) + f(y ∗ z) for all x, y, z ∈ X.
(c6) f(x ∗ y) > f((x ∗ y) ∗ y) for all x, y ∈ X.
(c7) f((x ∗ z) ∗ (y ∗ z) > f((x ∗ y) ∗ z) for all x, y, z ∈ X.
Definition 3.3 ([6]). Let X be a BCK/BCI-algebra. By a quasi-valuation
map of X based on an ideal (briefly I-quasi-valuation map of X), we mean
a mapping f : X → R which satisfies the conditions (c2) and (c3).
Definition 3.4. Let X be a BCK-algebra. By a quasi-valuation map
on X based on a positive implicative ideal (briefly PI-quasi-valuation map
of X), we mean a mapping f : X → R which satisfies the conditions (c2)
and (c5).
Example 3.5. Let X = {0, a, b} be a BCK-algebra with the ∗-operation
given by Table 1.
Table 1. ∗-operation.
∗ 0 a b
0 0 0 0
a a 0 0
b b b 0
Let f be a real-valued function on X defined by
f =
(
0 a b
0 0−2
)
.
Then f is a PI-quasi-valuation map of X.
Example 3.6. Let X = {0, a, b, c} be a BCK-algebra with the ∗-operation
given by Table 2.
Table 2. ∗-operation.
∗ 0 a b c
0 0 0 0 0
a a 0 0 a
b b a 0 b
c c c c 0
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Y. B. Jun, K. J. Lee, S. Z. Song 69
Let f be a real-valued function on X defined by
f =
(
0 a b c
0 0 0−7
)
.
Then f is a PI-quasi-valuation map of X.
Theorem 3.7. Let X be a BCK-algebra. Every PI-quasi-valuation map
of X is an I-quasi-valuation map of X.
Proof. Let f : X → R be a PI-quasi-valuation map on a BCK-algebra X.
If we take z = 0 in (c5) and use (a1), then we have the condition (c3).
Hence f is an I-quasi-valuation map of X.
The converse of Theorem 3.7 may not be true as shown by the following
example.
Example 3.8. Let X = {0, a, b, c} be a BCK-algebra with the ∗-operation
given by Table 2 and let g be a real-valued function on X defined by
g =
(
0 a b c
0−2−3 0
)
.
Then g is an I-quasi-valuation map of X, but not a PI-quasi-valuation
map of X since g(b ∗ a) = −2 < 0 = g((b ∗ a) ∗ a) + g(a ∗ a).
Example 3.9. Let X = {0, a, b, c} be a BCK-algebra with the ∗-operation
given by Table 3.
Table 3. ∗-operation
∗ 0 a b c
0 0 0 0 0
a a 0 0 0
b b b 0 0
c c c b 0
Let f be a real-valued function on X defined by
f =
(
0 a b c
0 0−3−4
)
.
Then f is an I-quasi-valuation map of X, but not a PI-quasi-valuation
map of X since f(c ∗ b) = −3 < 0 = f((c ∗ b) ∗ b) + f(b ∗ b).
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70 Quasi-valuation maps based on PI-ideals
We give conditions for an I-quasi-valuation map to be a PI-quasi-
valuation map. We first consider the following lemma.
Lemma 3.10. [6] For any I-quasi-valuation map f of X, we have the
following assertions:
(1) f is order reversing.
(2) f(x ∗ y) + f(y ∗ x) 6 0 for all x, y ∈ X.
(3) f(x ∗ y) > f(x ∗ z) + f(z ∗ y) for all x, y, z ∈ X.
Theorem 3.11. Let f be an I-quasi-valuation map of a BCK-algebra X.
If f satisfies the condition (c6), then f is a PI-quasi-valuation map of X.
Proof. Let f be an I-quasi-valuation map of X which satisfies the condition
(c6). Notice that ((x ∗ z) ∗ z) ∗ (y ∗ z) 6 (x ∗ z) ∗ y = (x ∗ y) ∗ z for all
x, y, z ∈ X. Since f is order reversing, it follows that
f(((x ∗ z) ∗ z) ∗ (y ∗ z)) > f((x ∗ y) ∗ z)
so from (c6) and (c3) that
f(x ∗ z) > f((x ∗ z) ∗ z) > f(((x ∗ z) ∗ z) ∗ (y ∗ z)) + f(y ∗ z)
> f((x ∗ y) ∗ z) + f(y ∗ z).
Therefore f is a PI-quasi-valuation map of X.
For any function f : X → R, consider the following set:
If := {x ∈ X | f(x) = 0}.
Lemma 3.12. [6] Let X be a BCK-algebra. If f is an I-quasi-valuation
map of X, then the set If is an ideal of X.
Lemma 3.13. [6] In a BCK-algebra, every I-quasi-valuation map is an
S-quasi-valuation map.
Lemma 3.14. Every PI-quasi-valuation map f of a BCK-algebra X
satisfies the condition (c6).
Proof. Let f be a PI-quasi-valuation map of X. Then f is an I-quasi-
valuation map of X by Theorem 3.7. If we take z = y in (c5), then
f(x ∗ y) > f((x ∗ y) ∗ y) + f(y ∗ y) = f((x ∗ y) ∗ y) + f(0) = f((x ∗ y) ∗ y)
for all x, y ∈ X. Thus the condition (c6) is valid.
Theorem 3.15. Let X be a BCK-algebra. If f is a PI-quasi-valuation
map of X, then the set If is a positive implicative ideal of X.
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Y. B. Jun, K. J. Lee, S. Z. Song 71
Proof. Suppose f is a PI-quasi-valuation map of X. Then f is an
I-quasi-valuation map of X by Theorem 3.7, and so If is an ideal of
X by Lemma 3.12. Let x, y ∈ X be such that (x ∗ y) ∗ y ∈ If . Then
f((x ∗ y) ∗ y) = 0 and so f(x ∗ y) > f((x ∗ y) ∗ y) = 0 by Lemma 3.14.
Using Lemma 3.13 and Proposition 3.2, we get f(x) 6 0 for all x ∈ X.
Thus f(x ∗ y) = 0 which means that x ∗ y ∈ If . Thus, by Proposition 2.1,
we conclude that If is a positive implicative ideal of X.
The following examples show that the converse of Theorem 3.15 may
not be true, that is, there exist a BCK-algebra X and a function f : X → R
such that
(1) f is not a PI-quasi-valuation map of X,
(2) If is a positive implicative ideal of X.
Example 3.16. Let X = {0, a, b, c, d} be a BCK-algebra with the ∗-
operation given by Table 4.
Table 4. ∗-operation.
∗ 0 a b c d
0 0 0 0 0 0
a a 0 a 0 a
b b b 0 b 0
c c a c 0 c
d d d d d 0
Let g be a real-valued function on X defined by
g =
(
0 a b c d
0 0−8 0−6
)
.
Then Ig = {0, a, c} is a positive implicative ideal of X. But g is not
a PI-quasi-valuation map of X since g(b ∗ c) = g(b) = −8 � −6 =
g((b ∗ d) ∗ c) + g(d ∗ c).
Proposition 3.17. Let X be a BCK-algebra. Then every PI-quasi-valua-
tion map f of X satisfies the condition (c7).
Proof. Let f be a PI-quasi-valuation map of X. Then f satisfies the
condition (c6) (see Lemma 3.14) and f is an I-quasi-valuation map f of X
(see Theorem 3.7). It follows from [6, Proposition 3.13] that f satisfies the
condition (c7).
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72 Quasi-valuation maps based on PI-ideals
Notice that an I-quasi-valuation map f of a BCK-algebra X does not
satisfy the condition (c7). In fact, consider a BCK-algebra X = {0, a, b, c}
in which the ∗-operation is given by the Table 5.
Table 5. ∗-operation.
∗ 0 a b c
0 0 0 0 0
a a 0 0 a
b b a 0 b
c c c c 0
Let f be a real-valued function on X defined by
f =
(
0 a b c
0−3−3−8
)
.
Then f is an I-quasi-valuation map of X. Since
f((b ∗ a) ∗ (a ∗ a)) = f(a ∗ 0) = f(a) = −3 < 0 = f((b ∗ a) ∗ a),
f does not satisfy the condition (c7).
Theorem 3.18. Let X be a BCK-algebra. If an I-quasi-valuation map f
of X satisfies the condition (c7), then it is a PI-quasi-valuation map of X.
Proof. Let f be an I-quasi-valuation map of X which satisfies the condi-
tion (c7). For any x, y, z ∈ X, we have
f(x ∗ z) > f((x ∗ z) ∗ (y ∗ z)) + f(y ∗ z) > f((x ∗ y) ∗ z) + f(y ∗ z)
by (c3) and (c7). Therefore f is a PI-quasi-valuation map of X.
Theorem 3.19. Let f be a real-valued function on a BCK-algebra X.
If f satisfies conditions (c2) and (c4), then f is a PI-quasi-valuation map
of X.
Proof. Assume that f satisfies conditions (c2) and (c4). Then
f(x) = f(x ∗ 0) > f(((x ∗ 0) ∗ 0) ∗ z) + f(z) = f(x ∗ z) + f(z)
for all x, z ∈ X. Hence f is an I-quasi-valuation map of X. Taking z = 0
in (c4) and using (a1) and (c2), we have
f(x ∗ y) > f(((x ∗ y) ∗ y) ∗ 0) + f(0) = f((x ∗ y) ∗ y)
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Y. B. Jun, K. J. Lee, S. Z. Song 73
for all x, y ∈ X. It follows from Theorem 3.11 that f is a PI-quasi-valuation
map of X.
Proposition 3.20. Every PI-quasi-valuation map f of a BCK-algebra X
satisfies the following implication for all x, y, a, b ∈ X:
(((x ∗ y) ∗ y) ∗ a) ∗ b = 0 ⇒ f(x ∗ y) > f(a) + f(b). (3.2)
Proof. Note that f is an I-quasi-valuation map of X by Theorem 3.7.
Assume that (((x ∗ y) ∗ y) ∗ a) ∗ b = 0 for all x, y, a, b ∈ X. Using [6,
Proposition 3.14], we have f((x ∗ y) ∗ y) > f(a) + f(b). It follows from
(III), (a1) and (c7) that
f(x∗y) = f((x∗y)∗0) = f((x∗y)∗ (y ∗y)) > f((x∗y)∗y) > f(a)+f(b).
This completes the proof.
Lemma 3.21. [6, Theorem 3.16] If a real-valued function f on X satisfies
the conditions (c2) and
(∀x, y, z ∈ X) ((x ∗ y) ∗ z = 0 ⇒ f(x) > f(y) + f(z)), (3.3)
then f is an I-quasi-valuation map of X.
Theorem 3.22. Let f be a real-valued function on a BCK-algebra X.
If f satisfies conditions (c2) and (3.2), then f is a PI-quasi-valuation map
of X.
Proof. Let x, y, z ∈ X be such that (x ∗ y) ∗ z = 0. Then
(((x ∗ 0) ∗ 0) ∗ y) ∗ z = 0.
It follows from (a1) and (3.2) that f(x) = f(x ∗ 0) > f(y) + f(z). Thus f
is an I-quasi-valuation map of X by Lemma 3.21. Since
(((x ∗ y) ∗ y) ∗ ((x ∗ y) ∗ y)) ∗ 0 = 0
for all x, y ∈ X, we have f(x ∗ y) > f((x ∗ y) ∗ y) + f(0) = f((x ∗ y) ∗ y)
by (3.2) and (c2). Therefore, by Theorem 3.11, f is a PI-quasi-valuation
map of X.
Proposition 3.23. Every PI-quasi-valuation map of a BCK-algebra X
satisfies the following implication for all x, y, z, a, b ∈ X:
(((x ∗ y) ∗ z) ∗ a) ∗ b = 0 ⇒ f((x ∗ z) ∗ (y ∗ z)) > f(a) + f(b). (3.4)
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74 Quasi-valuation maps based on PI-ideals
Proof. Let x, y, z, a, b ∈ X be such that (((x ∗ y) ∗ z) ∗ a) ∗ b = 0. Using
Propositions 3.17, Theorem 3.7 and [6, Proposition 3.14], we have
f((x ∗ z) ∗ (y ∗ z)) > f((x ∗ y) ∗ z) > f(a) + f(b)
which is the desired result.
Theorem 3.24. Let X be a BCK-algebra. If a real-valued function f on X
satisfies two conditions (c2) and (3.4), then f is a PI-quasi-valuation map
of X.
Proof. Let x, y, a, b ∈ X be such that (((x ∗ y) ∗ y) ∗ a) ∗ b = 0. Using (a1),
(III) and (3.4), we have
f(x ∗ y) = f((x ∗ y) ∗ 0) = f((x ∗ y) ∗ (y ∗ y)) > f(a) + f(b).
It follows from Theorem 3.22 that f is a PI-quasi-valuation map of X.
Theorem 3.25. (Extension Property) Let f and g be I-quasi-valuation
maps of a BCK-algebra X such that f(x) > g(x) for all x ∈ X. If g is a
PI-quasi-valuation map of X, then so is f .
Proof. Let x, y, z ∈ X. Using (a3), Proposition 3.17, (III) and (c2), we
have
f(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z))
= f(((x ∗ z) ∗ ((x ∗ y) ∗ z)) ∗ (y ∗ z))
= f(((x ∗ ((x ∗ y) ∗ z)) ∗ z) ∗ (y ∗ z))
> g(((x ∗ ((x ∗ y) ∗ z)) ∗ z) ∗ (y ∗ z))
> g(((x ∗ ((x ∗ y) ∗ z)) ∗ y) ∗ z)
= g(((x ∗ y) ∗ ((x ∗ y) ∗ z)) ∗ z)
= g(((x ∗ y) ∗ z) ∗ ((x ∗ y) ∗ z))
= g(0) = 0.
It follows from (c3) that
f((x ∗ z) ∗ (y ∗ z)) > f(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)) + f((x ∗ y) ∗ z)
= f((x ∗ y) ∗ z).
So from Theorem 3.18 we have that f is a PI-quasi-valuation map of X.
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Y. B. Jun, K. J. Lee, S. Z. Song 75
References
[1] Y. S. Huang, BCI-algebra, Science Press, China (2006).
[2] Y. Imai and K. Iséki, On axiom systems of propositional calculi. XIV, Proc. Japan
Acad. 42 (1966), 19–22.
[3] K. Iséki, An algebra related with a propositional calculus, Proc. Japan Acad. 42
(1966), 26–29.
[4] K. Iséki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125–130.
[5] K. Iséki and S. Tanaka, An introduction to theory of BCK-algebras, Math. Japonica
23 (1978), 1–26.
[6] Y. B. Jun, S. Z. Song and E. H. Roh, Quasi-valuation maps on BCK/BCI-algebras,
Filomat (submitted).
[7] J. Meng, Y. B. Jun, BCK-algebras, Kyungmoon Publisher, Seoul (1994).
[8] J. Neggers, A. Dvurečenskij and H. S. Kim, On d-fuzzy functions in d-algebras,
Found. Phys. 30 (2000), 1807–1816.
[9] J. Neggers, Y. B. Jun, H. S. Kim, On d-ideals in d-algebras, Math. Slovaca 49
(1999), 243–251.
[10] J. Neggers, H. S. Kim, On d-algebras, Math. Slovaca 49 (1999), 19–26.
[11] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outline, Inform.
Sci. 172, (2005), 1–40.
Contact information
Young Bae Jun Department of Mathematics Education,
Gyeongsang National University, Jinju 52828,
Korea
E-Mail(s): skywine@gmail.com
Kyoung Ja Lee Department of Mathematics Education,
Hannam University, Daejeon 34430, Korea
E-Mail(s): lsj1109@hotmail.com
Seok Zun Song Department of Mathematics,
Jeju National University, Jeju 63243, Korea
E-Mail(s): szsong@cheju.ac.kr
Received by the editors: 22.09.2016.
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