Using the notions of left and right derivations in
The notion of derivations arising in analytic theory is extremely helpful in exploring the structures and properties of algebraic systems. Several authors [1, 2] studied derivations in rings and near rings. Jun and Xin [3] applied the notion of derivation in ring and near ring theory to BCI-algebras. In [4], the concept of derivation for lattices was introduced and some of its properties are investigated. Alshehri [5] introduced the notion of ranked bigroupoids and discussed (X, *, ω)-self-(co)derivations. Recently, Jun et al. [6] obtained further results on derivations of ranked bigroupoids, and Jun et al. [7] introduced the notion of generalized coderivations in ranked bigroupoids.
The notions of BCK-algebras and BCI-algebras were introduced by Iseki and his colleague [8, 9]. The class of BCK-algebras is a proper subclass of the class of BCI-algebras. We refer useful textbooks for BCK-algebras and BCI-algebras to [10–12]. Neggers and Kim [13] introduced the notion of d-algebras which is another useful generalization of BCK-algebras, and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs.
In this paper we discuss left and right derivations in d/BCK-algebras, especially in excision-d-algebras, and cycles and polynomial elements in d/BCK-algebras related to (r, l)-derivations.
A d-algebra [13] is a non-empty set X with a constant 0 and a binary operation “*” satisfying the following axioms:
(I) x * x = 0,
(II) 0 * x = 0,
(III) x * y = 0 and y * x = 0 imply x = y, for all x, y ∈ X. For more information on d-algebras we refer to [14–17].
A BCK-algebra is a d-algebra X satisfying the following additional axioms:
(IV) ((x * y) * (x * z)) * (z * y) = 0,
(V) (x * (x * y)) * y = 0 for all x, y, z ∈ X.
If (X, *, 0) is a BCK-algebra, then
(VI) (x * y) * z = (x * z) * y,
(VII) x ≤ y implies x * z ≤ y * z,
(VIII) x ≤ y implies z * y ≤ z * x for all x, y, z ∈ X.
Let X:= {0, 1, 2, 3, 4} be a set with the following table:
^{*} | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 0 | 1 |
2 | 2 | 2 | 0 | 3 | 0 |
3 | 3 | 3 | 2 | 0 | 3 |
4 | 4 | 4 | 1 | 1 | 0 |
Then (X, *, 0) is a d-algebra which is not a BCK-algebra.
A BCI-algebra is an algebra (X, *, 0) having the conditions (I), (III), (IV) and (V). It is well known that every BCK-algebra is a BCI-algebra. A BCI-algebra is said to be p-semisimple if x * (x * y) = y for all x, y ∈ X. Let X be a d-algebra and x ∈ X. X is said to be edge if for any x ∈ X, x*X = {x, 0}. It is known that if X is an edge d-algebra, then x *0 = x for any x ∈ X [17].
Let (X, *, 0) be a d-algebra and ∅︀ ≠ I ⊆ X. I is called a d-subalgebra of X if x * y ∈ I whenever x ∈ I and y ∈ I. I is called a BCK-ideal of X if it satisfies:
(D_{0}) 0 ∈ I,
(D_{1}) x * y ∈ I and y ∈ I imply x ∈ I.
I is called a d-ideal of X if it satisfies (D_{1}) and (D_{2}) x ∈ I and y ∈ X imply x * y ∈ I, i.e., I * X ⊆ I.
It is known that I is a right ideal of X if it satisfies the condition (D_{2}).
Let (X, *, 0) be a d/BCI-algebra and let x∧y:= y *(y *x) for all x, y ∈ X. A map d: X → X is said to be an (r, l)-derivation if d(x*y) = (x*d(y))∧(d(x) *y) for all x, y ∈ X. Similarly, a map d: X → X is said to be an (l, r)-derivation if d(x * y) = (d(x) * y) ∧ (x * d(y)) for all x, y ∈ X.
Let (X, *, 0) be an edge d-algebra. If d: X → X is an (r, l)-derivation (or an (l, r)-derivation), then d(0) = 0.
If d: X → X is an (r, l)-derivation, then
If we let y:= x in (
If we take x:= 0 in (
Similarly, we prove that d(0) = 0 for any (l, r)-derivation d.
Let (X, *, 0) be an edge d-algebra. If d: X → X is an (r, l)-derivation, then (x * d(x)) ∧ (d(x) * x) = 0 for all x ∈ X.
It follows immediately from (
Note that (a * b) ∧ (b * a) is not unusual in d-algebras.
Let (X, *, 0) be a d-algebra. If d: X → X is an (r, l)-derivation, then d(x * d(x)) = 0 for all x ∈ X.
If we take y:= d(x) in (
Using the “idea” that d(x) represents the boundary of x, then x * d(x) is said to be x with its boundary removed and thus we may think of x * d(x) = y as an open element. Also, if d(x) = 0, then one usually thinks of x as being a cycle, i.e., we observe that, by Proposition 3.3, every open element in a d-algebra is a cycle.
Let (X, *, 0) be a d-algebra and let d: X → X be an (r, l)-derivation. If x is a cycle, then x * y is also a cycle for any y ∈ X, i.e., if J is the set of all cycles in X, then J forms a right ideal of X.
If x is a cycle, then d(x) = 0. Since d is an (r, l)-derivation, we have d(x * y) = (x * d(y)) ∧ (d(x) * y) = (x * d(y)) ∧ (0 * y) = (x * d(y)) ∧ 0 = 0, proving that x * y is a cycle, for any y ∈ X.
Note that the set J of all cycles in Proposition 3.4 is a dsubalgebra of X.
Let (X, *, 0) be a d-algebra and let d: X → X be an (r, l)-derivation (or an (l, r)-derivation). An element x ∈ X is said to be geometric if d^{2}(x) = 0. Note that every cycle is a geometric element.
Let (X, *, 0) be a d-algebra and let d: X → X be an (r, l)-derivation. Then any boundary of a geometric element is a cycle.
Straightforward.
If Z_{d}X,G_{d}X,B_{d}X denotes the set of cycles, geometric elements and boundaries of geometric elements, respectively, then we have inclusions:
If (X, *, 0) is a BCK-algebra, then
for any x, y, a ∈ X.
Given x, y, a ∈ X, we claim that (x*a)*y ≤ x*(y*a). By Theorem 2.1(VI), we obtain ((x * a) * y) * (x * (y * a)) = [(x*a) * (x*(y * a))] *y. By using (IV) and Theorem 2.1(VII), (VI), we have
which proves [(x * a) * y] * [x * (y * a)] = 0. Using the claim, we prove that
This proves the theorem.
Let (X, *, 0) be a BCK-algebra and let a ∈ X. If we define d_{a}: X → X by d_{a}(x):= x * a, then it is an (r, l)-derivation on X.
Given x, y ∈ X, by (
Let (X, *, 0) be a p-semisimple BCI-algebra and let a ∈ X. Then d_{a}(x * y) = x * d_{a}(y) for all x, y ∈ X.
Given x, y ∈ X, we have d_{a}(x * y) = (x * d_{a}(y)) ∧ (d_{a}(x)*y) = (d_{a}(x)*y)*[(d_{a}(x)*y)*(x*d_{a}(y))] = x*d_{a}(y), proving the corollary.
A d-algebra (X, *, 0) is said to be an excision-d-algebra if, for any a ∈ X, d_{a} is an (r, l)-derivation on X. We define a set d_{X} by
By Corollary 4.2, every BCK-algebra is an excision-d-algebra. It is not yet known that there are examples of excision-d-algebras which are not BCK-algebras.
Let (X, *, 0) be aBCK-algebra and let a, b ∈ X. Then d_{a} ∘ d_{b} = d_{b} ∘ d_{a} where “∘” is the usual composition of functions.
Given x ∈ X, we have (d_{a} ∘ d_{b})(x) = d_{a}(d_{b}(x)) = d_{a}(x * b) = (x * b) * a = (x * a) * b = (d_{b} ∘ d_{a})(x), proving the proposition.
Given a groupoid (X, *), we define a set RLDer(X) by
If (X, *, 0) is an excision-d-algebra, then there is an injection ξ: X → d_{X}.
If we define ξ: X → d_{X} by ξ(a):= d_{a}, then it is injective. In fact, if ξ(a) = ξ(b) for any a, b ∈ X, then d_{a} = d_{b} and hence x * a = x * b for all x ∈ X. If we take x:= b, then b * a = b * b = 0. If we take x:= a, then a * b = a * a = 0. Since (X, *, 0) is a d-algebra, we obtain a = b.
Give some conditions for ξ to be a groupoid homomorphism from (X, *) to (d_{X}, ∘).
If ξ is a groupoid homomorphism, then we obtain the following relation:
An excision-d-algebra (X, *) is said to be complete if d is an (r, l)-derivation on X, then there exists a ∈ X such that d = d_{X}.
What (if any) complete excision-d-algebras are there?
Are Boolean algebras complete excision-d-algebras?
Let (X, *, 0) be a d-algebra. An element x ∈ X is said to be polynomial if there exists n ∈
Let (X, *, 0) be an edge d-algebra and let d be an (r, l)-derivation on X. If we define
Then E_{d}(X) is a d-subalgebra of (X, *, 0).
By Proposition 3.1, we have d(0) = 0, i.e., 0 ∈ E_{d}(X). If x, y ∈ E_{d}(X), then d(x) = x and d(y) = y. It follows that
proving that x * y ∈ E_{d}(X).
Let (X, *, 0) be a d-algebra and let x ∈ X. If x is a cycle of an (r, l)-derivation d_{a} for some a ∈ X, then a ∧ x = 0.
If x is a cycle of an (r, l)-derivation d_{a} for some a ∈ X, then d_{a}(x) = 0. It follows that x * a = x and hence a ∧ x = x * (x * a) = x * x = 0.
Let (X, *, 0) be a d-algebra and let d be an (r, l)-derivation on X. If we define
Then P_{d}(X) ∩ E_{d}(X) = {0}.
If x ∈ P_{d}(X) ∩ E_{d}(X), then there exists n ∈
Proposition 5.3 shows that 0 is the only element which is both a cycle and a polynomial element in d-algebras.
Let (X, *, 0) be an edge d-algebra and let d be an (r, l)-derivation on X. If (x * y) * z = (x * z) * y for all x, y, z ∈ X and if d(y) ∧ x ≤ y for all x, y ∈ X, then (E_{d}(X), *) is a right ideal of (X, *, 0).
Given x ∈ E_{d}(X) and y ∈ X, we have
proving that x * y ∈ E_{d}(X).
Let (X, *, 0) be a BCK-algebra and let d be an (r, l)-derivation on X such that d(x) = x for some x ∈ X. Then (x * y) * (d(y) * y) ≤ d(x * y) for all y ∈ X.
Since (X, *, 0) is a BCK-algebra, we obtain (x * y) * (x * d(y)) ≤ d(y) * y. By applying Theorem 2.1(VIII), we obtain
proving the proposition.
In this paper, we proposed some properties on cycles and polynomial elements in d/BCK-algebra. We could get the results of 5 proposions in Section 5.
No potential conflict of interest relevant to this article was reported.
^{*} | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 0 | 1 |
2 | 2 | 2 | 0 | 3 | 0 |
3 | 3 | 3 | 2 | 0 | 3 |
4 | 4 | 4 | 1 | 1 | 0 |
E-mail: yhkim@chungbuk.ac.kr