On filters and upper sets in CI-algebras
CI-algebras are a generalization of BE-algebras and dual BCK/BCI/BCH-algebras. In this paper filters of CI-algebras are considered. Given a subset of a CI-algebra, the least filter containing it is constructed. An equivalent condition of the filters using the notion of upper sets is provided.
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Інститут прикладної математики і механіки НАН України
2011
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| Цитувати: | On filters and upper sets in CI-algebras / B. Piekart, A. Walendziak // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 1. — С. 109–115. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1547652019-06-16T01:31:51Z On filters and upper sets in CI-algebras Piekart, B. Walendziak, A. CI-algebras are a generalization of BE-algebras and dual BCK/BCI/BCH-algebras. In this paper filters of CI-algebras are considered. Given a subset of a CI-algebra, the least filter containing it is constructed. An equivalent condition of the filters using the notion of upper sets is provided. 2011 Article On filters and upper sets in CI-algebras / B. Piekart, A. Walendziak // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 1. — С. 109–115. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:06F35, 03G25 http://dspace.nbuv.gov.ua/handle/123456789/154765 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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CI-algebras are a generalization of BE-algebras and dual BCK/BCI/BCH-algebras. In this paper filters of CI-algebras are considered. Given a subset of a CI-algebra, the least filter containing it is constructed. An equivalent condition of the filters using the notion of upper sets is provided. |
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Piekart, B. Walendziak, A. |
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Piekart, B. Walendziak, A. On filters and upper sets in CI-algebras Algebra and Discrete Mathematics |
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Piekart, B. Walendziak, A. |
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Piekart, B. |
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On filters and upper sets in CI-algebras |
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On filters and upper sets in CI-algebras |
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On filters and upper sets in CI-algebras |
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On filters and upper sets in CI-algebras |
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On filters and upper sets in CI-algebras |
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on filters and upper sets in ci-algebras |
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Інститут прикладної математики і механіки НАН України |
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On filters and upper sets in CI-algebras / B. Piekart, A. Walendziak // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 1. — С. 109–115. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT piekartb onfiltersanduppersetsincialgebras AT walendziaka onfiltersanduppersetsincialgebras |
| first_indexed |
2025-07-14T06:52:26Z |
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2025-07-14T06:52:26Z |
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1837604218661240832 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 11 (2011). Number 1. pp. 109 – 115
c© Journal “Algebra and Discrete Mathematics”
On filters and upper sets in CI-algebras
Bożena Piekart and Andrzej Walendziak
Communicated by V. V. Kirichenko
Abstract. CI-algebras are a generalization of BE-algebras
and dual BCK/BCI/BCH-algebras. In this paper filters of CI-
algebras are considered. Given a subset of a CI-algebra, the least
filter containing it is constructed. An equivalent condition of the
filters using the notion of upper sets is provided.
1. Introduction
In 1966, Y. Imai and K. Iséki [3] introduced the notion of a BCK-algebra.
There exist several generalizations of BCK-algebras, such as BCI-algebras
[4], BCH-algebras [2], BCC-algebras [8], BH-algebras [5], d-algebras [12],
etc. In [6], H. S. Kim and Y. H. Kim introduced the notion of a BE-algebra
as a dualization of a generalization of a BCK-algebra. They defined and
studied the concept of a filter in BE-algebras. This concept was also
investigated in [10] and [7]. As a generalization of BE-algebras, B. L.
Meng [9] introduced the notion of CI-algebras and discussed its important
properties.
In this paper, we consider filters in CI-algebras. Given a subset of a
CI-algebra, we make the least filter containing it. We provide an equivalent
condition of the filters using the notion of upper sets.
2. Preliminaries
Definition 2.1. ([9]) A CI-algebra is an algebra (X; ∗, 1) of type (2, 0)
satisfying the following axioms:
2000 Mathematics Subject Classification: 06F35, 03G25.
Key words and phrases: CI-algebra, filter, upper set.
110 On filters and upper sets in CI-algebras
(CI-1) x ∗ x = 1,
(CI-2) 1 ∗ x = x,
(CI-3) x ∗ (y ∗ z) = y ∗ (x ∗ z) .
A CI-algebra X is said to be a BE-algebra if for all x ∈ X
(BE) x ∗ 1 = 1.
Throughout this paper X will denote a CI-algebra. We introduce a
relation ≤ on X by x ≤ y if and only if x ∗ y = 1.
Example 2.2. Let X = {1, a, b, c} and ∗ be defined by the following
table:
∗ 1 a b c
1 1 a b c
a 1 1 1 c
b 1 1 1 c
c c c c 1
Then (X, ∗, 1) is a CI-algebra, which is not a BE-algebra.
For any x1, . . . , xn, a ∈ X, we define
n
∏
i=1
xi ∗ a = xn ∗ (· · · ∗ (x1 ∗ a) · · · ).
Proposition 2.3. ([9]) For any x, y ∈ X we have
(a) y ∗ ((y ∗ x) ∗ x) = 1,
(b) 1 6 x ⇒ x = 1.
Definition 2.4. ([11]) A CI-algebra X is said to be transitive if for all
x, y, z ∈ X,
y ∗ z 6 (x ∗ y) ∗ (x ∗ z).
It is easily seen that the CI-algebra X of Example 2.2 is transitive.
Consider the following example.
Example 2.5. Let X = {1, a, b, c, d} and ∗ be defined by the following
table:
∗ 1 a b c d
1 1 a b c d
a 1 1 b c d
b 1 1 1 1 d
c 1 a c 1 d
d d d d d 1
Then (X, ∗, 1) is a CI-algebra. Since b∗a = 1 and (c∗b)∗(c∗a) = c∗a = a,
X is not transitive.
B. Piekart, A. Walendziak 111
Lemma 2.6. ([11]) If a CI-algebra X is transitive, then for all x, y, z ∈ X,
x 6 y implies z ∗ x 6 z ∗ y.
Lemma 2.7. Let X be a transitive CI-algebra and let x, y ∈ X such
that x ∗ y = 1. Then for all a1, . . . , an ∈ X,
∏n
i=1
ai ∗ x = 1 implies
∏n
i=1
ai ∗ y = 1.
Proof. We have x 6 y and from Lemma 2.6 we see that
1 =
n
∏
i=1
ai ∗ x 6
n
∏
i=1
ai ∗ y.
Applying Proposition 2.3 (b) we conclude that
∏n
i=1
ai ∗ y = 1.
3. Filters
Following [9], a filter of X is a subset F of X such that for all x, y ∈ X:
(F1) 1 ∈ F ,
(F2) if x ∗ y ∈ F and x ∈ F , then y ∈ F .
By Fil(X) we denote the set of all filters in X. It is obvious that {1}, X ∈
Fil(X).
Example 3.1. Consider the CI-algebra X of Example 2.2. It is easy to
check that Fil(X) = {{1}, {1, a, b}, X}.
Proposition 3.2. If Fi (i ∈ I) are filters of X, then
⋂
i∈I Fi is a filter
of X.
Proof. Straightforward.
Proposition 3.3. Let F be a subset of X containing 1. Then F ∈ Fil(X)
if and only if for any a, b ∈ F and x ∈ X, a ∗ (b ∗ x) = 1 implies x ∈ F .
Proof. (⇐) Since 1 ∈ F , the condition (F1) holds. Suppose that a ∗ x ∈ F
and a ∈ F . By Proposition 2.3 (a), a ∗ [(a ∗ x) ∗ x] = 1. Then x ∈ F and
hence (F2) is true. Therefore F is a filter of X.
(⇒) Let F ∈ Fil(X). Assume a, b ∈ F and x ∈ X such that a∗(b∗x) =
1. From (F1) we obtain a ∗ (b ∗ x) ∈ F . Applying (F2) twice we have
x ∈ F .
By induction we easily obtain
Corollary 3.4. Let F be a subset of X containing 1. Then F ∈ Fil(X) if
and only if for any a1, . . . , an ∈ F and x ∈ X,
∏n
i=1
ai ∗ x = 1 implies
x ∈ F .
112 On filters and upper sets in CI-algebras
Definition 3.5. For every subset A ⊆ X, the smallest filter of X which
contains A, that is, the intersection of all filters F ⊇ A, is said to be the
filter generated by A, and will be denoted [A). Obviously, [∅) = {1}.
Theorem 3.6. Let A be a nonvoid subset of a transitive CI-algebra X.
Then
[A) = {x ∈ X : x = 1 or
∏n
i=1
ai ∗ x = 1 for some a1, . . . , an ∈ A}.
Proof. Set F = {x ∈ X : x = 1 or
∏n
i=1
ai∗x = 1 for some a1, . . . , an ∈ A}.
Since a ∗ a = 1 for all a ∈ A, we obtain A ⊆ F . Obviously, 1 ∈ F . Let
x ∗ y ∈ F and x ∈ F . To prove that y ∈ F , we will consider three cases.
Case 1: x = 1.
Then y = 1 ∗ y ∈ F.
Case 2: x ∗ y = 1 and x 6= 1.
Since x ∈ F and x 6= 1, we conclude that
∏n
i=1
ai ∗ x = 1 for some
a1, . . . , an ∈ A. From Lemma 2.7 it follows that
∏n
i=1
ai ∗y = 1. Therefore
y ∈ F .
Case 3: x ∗ y 6= 1 and x 6= 1.
Then there are a1, . . . , an, b1, . . . , bm ∈ A such that
∏n
i=1
ai ∗ (x ∗ y) = 1
and
∏m
j=1
bj ∗ x = 1. Applying (CI-3) we deduce that x ≤
∏n
i=1
ai ∗ y.
From Lemma 2.6 we see that
1 =
m
∏
j=1
bj ∗ x 6
m
∏
j=1
bj ∗
(
n
∏
i=1
ai ∗ y
)
.
By Proposition 2.3 (b),
∏m
j=1
bj ∗ (
∏n
i=1
ai ∗ y) = 1. Hence y ∈ F , and so
F is a filter of X.
Suppose now that U is any filter of X containing A. Let x ∈ F . If x = 1,
then obviously x ∈ U . Assume that x 6= 1. Then there are a1, . . . , an ∈ A
such that
∏n
i=1
ai ∗ x = 1. Since A ⊆ U , it follows that a1, . . . , an ∈ U .
Therefore x ∈ U by Corollary 3.4. Thus F ⊆ U and hence F = [A).
Let F1, F2 ∈ Fil(X). We define the meet of F1 and F2 (denoted by
F1 ∧ F2) by F1 ∧ F2 = F1 ∩ F2 and the join of F1 and F2 (denoted by
F1 ∨ F2) by F1 ∨ F2 = [F1 ∪ F2). We note that (Fil(X);∧,∨) is a lattice.
Moreover, by Proposition 3.2 we have
Theorem 3.7. (Fil(X);∧,∨) is a complete lattice.
4. Upper sets
For any x, y ∈ X, we define
A(x, y) = {z ∈ X : z = 1 or x ∗ (y ∗ z) = 1}
B. Piekart, A. Walendziak 113
and
A(x) = {z ∈ X : z = 1 or x ∗ z = 1}.
Applying (CI-2) we conclude that A(x) = A(1, x).
The set A(x) (resp. A(x, y)) is called an upper set of x (resp. of x and
y). We say that a subset A of X is an upper set of X if A = A(x, y) for
some x, y ∈ X. By US(X) we denote the set of all upper sets in X.
Remark 4.1. By (CI-3), A(x, y) = A(y, x) for all x, y ∈ X.
Example 4.2. Let X = {1, a, b} and ∗ be defined by the following table:
∗ 1 a b
1 1 a b
a a 1 1
b a 1 1
.
Then (X, ∗, 1) is a CI-algebra. For x, y ∈ X, we have
A(x, y) =
{
X if x 6= y and (x = 1 or y = 1)
{1} otherwise.
Since Fil(X) = {{1}, X}, we see that Fil(X) = US(X).
In general, not every filter is an upper set and not every upper set is a
filter. Indeed, we consider the following example.
Example 4.3. Let X be the CI-algebra of Example 2.2. We have (see Ex-
ample 3.1) Fil(X) = {{1}, {1, a, b}, X}. It is easy to check that US(X) =
{{1}, {1, a, b}, {1, c}}. Therefore X is not an upper set of X and {1, c} is
not a filter in X.
Lemma 4.4. For every x, y ∈ X,
(a) x ∈ A(x),
(b) 1 ∈ A(x, y) and 1 ∈ A(x),
(c) if y ∗ 1 = 1, then A(x) ⊆ A(x, y),
(d) if y ∗ 1 6= 1, then A(x)− {1} ⊆ X −A(x, y),
(e) if A(x) is a filter of X and y ∈ A(x), then A(x, y) ⊆ A(x).
Proof. (a) Let x ∈ X. Since x ∗ x = 1, we have x ∈ A(x).
(b) By the definition of upper sets.
(c) Let y ∗ 1 = 1 and let z ∈ A(x). If z = 1, then obviously z ∈ A(x, y).
Suppose that x ∗ z = 1. Hence y ∗ (x ∗ z) = y ∗ 1 = 1 and therefore
z ∈ A(y, x) = A(x, y). Consequently, A(x) ⊆ A(x, y).
(d) Let y ∗ 1 6= 1 and z ∈ A(x) − {1}. Then x ∗ z = 1 and applying
(CI-3) we get x ∗ (y ∗ z) = y ∗ (x ∗ z) = y ∗ 1 6= 1. Thus z /∈ A(x, y) and
we conclude that A(x)− {1} ⊆ X −A(x, y).
114 On filters and upper sets in CI-algebras
(e) Let A(x) be a filter of X and y ∈ A(x). If z ∈ A(x, y), then
z = 1 or x ∗ (y ∗ z) = 1. In the first case z = 1 ∈ A(x), in the second
one x ∗ (y ∗ z) ∈ A(x). Since A(x) is a filter and x, y ∈ A(x), we obtain
z ∈ A(x).
Theorem 4.5. Let F be a nonvoid subset of a CI-algebra X. Then F is a
filter of X if and only if A(x, y) ⊆ F for all x, y ∈ F .
Proof. Suppose that F is a fillter of X. Let x, y ∈ F and z ∈ A(x, y).
Then z = 1 or x ∗ (y ∗ z) = 1. Obviously z = 1 ∈ F . If x ∗ (y ∗ z) = 1, then
applying twice (F2) we obtain z ∈ F. Hence A(x, y) ⊆ F.
Now let A(x, y) ⊆ F for all x, y ∈ F . Since F 6= ∅, there exists z ∈ F.
By definition, 1 ∈ A(z, z) ⊆ F and therefore (F1) holds. Let x ∗ y ∈ F
and x ∈ F . By (CI-1), (x ∗ y) ∗ (x ∗ y) = 1 and hence y ∈ A(x ∗ y, x) ⊆ F .
Thus (F2) also holds and consequently, F is a filter of X.
Proposition 4.6. If F is a filter of X, then F =
⋃
x,y∈F A(x, y).
Proof. Let F be a filter. From Theorem 4.5 it follows that A(x, y) ⊆ F for
all x, y ∈ F . Hence
⋃
x,y∈F A(x, y) ⊆ F.
Now let z ∈ F . By Lemma 4.4 (a),
z ∈ A(z) = A(1, z) ⊆
⋃
x,y∈F
A(x, y).
Then F ⊆
⋃
x,y∈F A(x, y).
Proposition 4.7. If F is a filter of X, then F =
⋃
x∈F A(x).
Proof. Let F be a filter and let z ∈ F . By Lemma 4.4 (a), z ∈ A(z) ⊆
⋃
x∈F A(x). Therefore F ⊆
⋃
x∈F A(x). From Theorem 4.5 we conclude
that A(x) = A(1, x) ⊆ F for all x ∈ F . Hence
⋃
x∈F A(x) ⊆ F and
consequently, F =
⋃
x∈F A(x). �
References
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B. Piekart, A. Walendziak 115
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Contact information
B. Piekart Institute of Mathematics and Physics, University
of Siedlce, 3 Maja 54, 08–110 Siedlce, Poland
E-Mail: bpiekart@interia.pl
A. Walendziak Warsaw School of Information Technology,
Newelska 6, 01–447 Warszawa, Poland
E-Mail: walent@interia.pl
Received by the editors: 24.07.2010
and in final form 18.03.2011.
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