Further Results on Derivations of Ranked Bigroupoids

Proposition 3.1. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X.

• (1)

  Assume that X satisfies axioms (b1), (b2), (b3), (a3), and (a4). If a map d : X → X is an (X, *, &)-self-derivation, then d(x) = d(x)&x for all x ∈ X.

• (2)

  If X satisfies two axioms (b1) and (a3) and a map d : X → X is an (X, *, &)-self-coderivation, then the following are equivalent:

  •  

    (2.1) d(0) = 0;

  •  

    (2.2) (∀x ∈ X)(d(x) = x&d(x)).

Proof. (1) Let x ∈ X. Using (b1) and (b2), we have

(2) Let d be an (X, *, &)-self-coderivation. If d(0) = 0, then

Corollary 3.2. Let (X, *, &) be a ranked bigroupoid in which (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X.

• (1)

  If a map d : X → X is an (X, *, &)-self-derivation, then d(x) = d(x)&x for all x ∈ X.

• (2)

  If a map d : X → X is an (X, *, &)-self-coderivation, then the following are equivalent:

  •  (2.1)

     d(0) = 0;

  •  (2.2)

    (∀x ∈ X) (d(x) = x&d(x)).

Lemma 3.3. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 in which three axioms (b2),(a3), and (a4) are valid and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X.

• (1)

  For every x ∈ X with x&0 = x, one has

  $$ ()

• (2)

  If an element a of X satisfies a&0 = a, then a&x = a for all x ∈ X.

Proof. (1) Let y ∈ X be such that y*x = 0. Then

(2) Since (a&x)*a = 0, it follows from (3.6) that a&x = a for all x ∈ X.

Corollary 3.4. Let (X, *, &) be a ranked bigroupoid in which (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X.

• (1)

  For every x ∈ X with x&0 = x, one has

  $$ ()

• (2)

  If an element a of X satisfies a&0 = a, then a&x = a for all x ∈ X.

Proposition 3.5. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 in which four axioms (b2),(b4), (a3), and (a4) are valid and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. If a map d : X → X is an (X, *, &)-self-coderivation, then 0*d(x) = d(x) for all x ∈ X with 0*x = x.

Proof. Let x ∈ X be such that 0*x = x. Since (0*d(x))&0 = 0*d(x), it follows from Lemma 3.3(2) that d(x) = d(0*x) = (0*d(x))&(d(0)*x) = 0*d(x).

Corollary 3.6. Let (X, *, &) be a ranked bigroupoid in which (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. If a map d : X → X is an (X, *, &)-self-coderivation, then 0*d(x) = d(x) for all x ∈ X with 0*x = x.

Example 3.7. Let (ℤ, −, &) be a ranked bigroupoid where ℤ is the set of all integers with the minus operation “−” and the minor operation “&” defined by x&y = y − (y − x) for all x, y ∈ ℤ. Let d be a self map of ℤ given by d(x) = x − 1 for all x ∈ ℤ. Then d is a (ℤ, −, &)-self-derivation since

Proposition 3.8. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. For an (X, *, &)-self-derivation d : X → X, if (X, *, 0) satisfies axioms (b2),(b5), and (b6), then d(x) = d(0)*(0*x) for all x ∈ X. Moreover, if d(0) = 0, then d is an identity map.

Proof. Assume that (X, *, 0) satisfies axioms (b2), (b5), and (b6). Then

Corollary 3.9. Let (X, *, &) be a ranked bigroupoid in which (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. If a map d : X → X is an (X, *, &)-self-derivation, then

• (1)

  d(0) = d(0)&0;

• (2)

  if (X, *, 0) is p-semisimple, then d(x) = d(0)*(0*x) for all x ∈ X;

• (3)

  if (X, *, 0) is p-semisimple and d(0) = 0, then d is an identity map.

Definition 3.10. Let (X, *, &) be a ranked bigroupoid with distinguished element 0. A self map d of (X, *, &) is said to be regular if d(0) = 0.

Example 3.11. Consider a ranked bigroupoid (X, *, &) in which X = {0, a, b, c, d, e} and binary operations “*” and “&” are defined by

Proposition 3.12. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X and 0*x = 0 for all x ∈ X. Then every (X, *, &)-self-derivation is regular. Moreover, if X satisfies the axioms (b1) and (a3) then every (X, *, &)-self-coderivation is regular.

Proof. Let d be an (X, *, &)-self-derivation. Then

Proposition 3.13. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X and two axioms (a3) and (b1) are satisfied. Let d be a self map of X and a ∈ X such that d(x)*a = 0 (resp., a*d(x) = 0) for all x ∈ X. If d is an (X, *, &)-self-derivation (resp., (X, *, &)-self-coderivation), then it is regular.

Proof. Assume that d is an (X, *, &)-self-derivation. For any x ∈ X, we have

Definition 3.14. Let (X, *, &) be a ranked bigroupoid with distinguished element 0. Let d be a self map of (X, *, &). A subset A of X is called a ranked *-subsystem of X if it satisfies the following:

• (r1)

  0 ∈ A,

• (r2)

  (∀x, y ∈ X)(x ∈ A, y*x ∈ A⇒y ∈ A).

Moreover, if a ranked *-subsystem A of X satisfies d(A)⊆A, then we say that A is ranked  d-invariant.

Example 3.15. Consider a ranked bigroupoid (X, *, &) in which X = {0, a, b, c, d, e} and binary operations “*”and “&” are defined by

Example 3.16. In Example 3.11, A = {0, a} is a ranked d-invariant *-subsystem of X.

Theorem 3.17. Let (X, *, &) be a ranked bigroupoid with distinguished element 0 in which three axioms (b1),(b2), and (a3) are valid and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. For an (X, *, &)-self-coderivation d, if d is regular then every ranked *-subsystem of X is ranked d-invariant.

Proof. Assume that d is regular and let A be a ranked *-subsystem of X. Then d(x) = x&d(x) for all x ∈ X by Proposition 3.1(2). Let y ∈ d(A). Then y = d(a) for some a ∈ A. Thus y*a = d(a)*a = (a&d(a))*a = 0 ∈ A, and so y ∈ A by (r2). Hence d(A)⊆A and A is ranked d-invariant.

Corollary 3.18. Let d be an (X, *, &)-self-coderivation where (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. If d is regular, then every ideal of X is ranked d-invariant.

Theorem 3.19. Let d be an (X, *, &)-self-coderivation where (X, *, &) is a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X and there does not exist a nonzero element x of X such that x*0 = 0. If every ranked *-subsystem of X is ranked d-invariant, then d is regular.

Proof. Assume that every ranked *-subsystem of X is ranked d-invariant. Note that A = {0} is a ranked *-subsystem of X. Thus d(A) = d({0})⊆{0}, and therefore d(0) = 0, that  is, d is regular.

Corollary 3.20. Let d be an (X, *, &)-self-coderivation where (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. Then d is regular if and only if every ranked *-subsystem of X is ranked d-invariant.

Proposition 3.21. Let (X, *, &) be a ranked bigroupoid where (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. For any α ∈ X, let d_α be a self map of X defined by d_α(x) = x*α for all x ∈ X. If X satisfies the following conditions:

• (1)

  ((x*y)*z)*(x*(y*z)) = 0 for all x, y, z ∈ X,

• (2)

  (∀x, y ∈ X) (x*y = 0⇒x = y),

Proof. If X satisfies two given conditions, then the following identity is valid (see [9]):

Corollary 3.22. Let (X, *, &) be a ranked bigroupoid where (X, *, 0) is a BCI-algebra and the minor operation & is defined by x&y = y*(y*x) for all x, y ∈ X. For any α ∈ X, let d_α be a self map of X defined by d_α(x) = x*α for all x ∈ X. If X satisfies (b1) and the following conditions:

• (1)

  ((x*y)*z)*(x*(y*z)) = 0 for all x, y, z ∈ X,

• (2)

  (x*y)*(x*z) = z*y for all x, y, z ∈ X,

Proof. If X satisfies both (b1) and the second condition, then X is a p-semisimple BCI-algebra (see [9]). Hence the second condition of Proposition 3.21 is valid. Therefore d_α is an abelian (X, *, &)-self-derivation.