Generalized Derivations of BCC-Algebras

Abstract

The notion of generalized derivations of BCC-algebras is introduced, and some related properties are investigated. Also, we consider regular generalized derivations and the D-invariant on ideals of BCC-algebras. We also characterized KerD by generalized derivations.

1. Introduction

Imai and Iséki [1] defined a class of algebras of type (2,0) called 𝐵𝐶𝐾-algebras which generalizes on one hand the notion of algebra of sets with the set subtraction as the only fundamental nonnullary operation on the other hand the notion of implication algebra. The class of all 𝐵𝐶𝐾-algebras is a quasivariety. Iséki posed an interesting problem (solved by Wroński [2]) whether the class of 𝐵𝐶𝐾-algebras is a variety. In connection with this problem, Komori [3] introduced a notion of 𝐵𝐶𝐶-algebras, and Dudek [4] redefined the notion of 𝐵𝐶𝐶-algebras by using a dual form of the ordinary definition in the sense of Komori. Dudek and Zhang [5] introduced a new notion of ideals in 𝐵𝐶𝐶-algebras and described connections between such ideals and congruences. On the other hand, Jun and Xin [6] applied the notion of derivations in ring and near-ring theories to 𝐵𝐶𝐼-algebras, and they also introduced a new concept called a regular derivation in 𝐵𝐶𝐼-algebras and investigated related properties. They defined a 𝑑-derivation ideal and gave conditions for an ideal to be 𝑑-derivation. Prabayak and Leerawat [7] introduced the notion of derivation in 𝐵𝐶𝐶-algebras. In this paper, the notion of generalized derivations of 𝐵𝐶𝐶-algebras is introduced, and some related properties are investigated. Also, we consider regular generalized derivations and the 𝐷-invariant on ideals of 𝐵𝐶𝐶-algebras. We also characterized Ker𝐷 by generalized derivations.

2. Preliminaries

We review some definitions and properties that will be useful in our results.

Definition 1. Let 𝑋 be a set with a binary operation “∗” and a constant 0. Then (𝑋,∗,0) is called a 𝐵𝐶𝐶-algebra if the following axioms are satisfied for all 𝑥,𝑦,𝑧∈𝑋:(i)((𝑥∗𝑦)∗(𝑧∗𝑦))∗(𝑥∗𝑧)=0, (ii)0∗𝑥=0, (iii)𝑥∗0=𝑥, (iv)𝑥∗𝑥=0, (v)𝑥∗𝑦=0 and 𝑦∗𝑥=0⇒𝑥=𝑦.

Define a binary relation ≤ on 𝑋 by letting 𝑥∗𝑦=0 if and only if 𝑥≤𝑦. Then(𝑋,≤) is a partially ordered set.

In any 𝐵𝐶𝐶-algebra 𝑋 for all 𝑥,𝑦∈𝑋, the following properties hold:(1)(𝑥∗𝑦)∗𝑥=0, (2)𝑥≤𝑦 implies 𝑥∗𝑧≤𝑦∗𝑧,(3)𝑥≤𝑦 implies 𝑧∗𝑦≤𝑧∗𝑥.

Any 𝐵𝐶𝐾-algebra is a 𝐵𝐶𝐶-algebra, but there are 𝐵𝐶𝐶-algebras which are not 𝐵𝐶𝐾-algebras. Note that a 𝐵𝐶𝐶-algebra is a 𝐵𝐶𝐾-algebra if and only if satisfies(4)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦, or(5)(𝑥∗(𝑥∗𝑦))∗𝑦=0.

Definition 2. A nonempty subset 𝑆 of a 𝐵𝐶𝐶-algebra 𝑋 is called subalgebra of 𝑋 if 𝑥∗𝑦∈𝑆 whenever 𝑥,𝑦∈𝑆. For a 𝐵𝐶𝐶-algebra 𝑋, we denote 𝑥∧𝑦=𝑦∗(𝑦∗𝑥) for all 𝑥,𝑦∈𝑋.

Remark 3. Let 𝑋 be a 𝐵𝐶𝐶-algebra and 𝑥,𝑦∈𝑋; then, consider(1)0∧𝑥=0, (2)𝑥∧𝑦≤𝑦.

Definition 4. A 𝐵𝐶𝐶-algebra is said to be commutative if and only if satisfies for all 𝑥,𝑦∈𝑋, 𝑥∗(𝑥∗𝑦)=𝑦∗(𝑦∗𝑥); that is, 𝑥∧𝑦=𝑦∧𝑥.

Definition 5. Let 𝑋 be a 𝐵𝐶𝐶-algebra and 𝜙≠𝐼⊆𝑋. 𝐼 is called an ideal of 𝑋 if it satisfies the following conditions:(i)0∈𝐼, (ii)𝑥∗𝑦∈𝐼 and 𝑦∈𝐼 imply 𝑥∈𝐼.

Definition 6. Let 𝑋 be a 𝐵𝐶𝐶-algebra. A map 𝑑∶𝑋→𝑋 is a left-right derivation (briefly,(𝑙,𝑟)-derivation) of 𝑋, if it satisfies the identity 𝑑𝑥∗𝑦=𝑑(𝑥)∗𝑦∧𝑥∗𝑑𝑦∀𝑥,𝑦∈𝑋. (1)
If 𝑑 satisfies the identity 𝑑𝑥∗𝑦=𝑥∗𝑑𝑦∧𝑑(𝑥)∗𝑦∀𝑥,𝑦∈𝑋, (2)
then 𝑑 is a right-left derivation (briefly, (𝑟,𝑙)-derivation) of 𝑋. Moreover, if 𝑑 is both (𝑙,𝑟)-derivation and (𝑟,𝑙)-derivation, then 𝑑 is a derivation of 𝑋.

Definition 7. A self map 𝑑 of a 𝐵𝐶𝐶-algebra 𝑋 is said to be regular if 𝑑(0)=0. (3)

Corollary 8. A derivation 𝑑 of a 𝐵𝐶𝐶-algebra 𝑋is regular.

Definition 9. Let 𝑑 be a derivation of a 𝐵𝐶𝐶-algebra 𝑋. An ideal 𝐴 of 𝑋 is said to be 𝑑-invariant if 𝑑(𝐴)⊆𝐴, where 𝑑(𝐴)={𝑑(𝑥)∣𝑥∈𝐴}.

Proposition 10. Let 𝑋 be a 𝐵𝐶𝐶-algebra with partial order ≤, and let 𝑑 be a derivation of 𝑋. Then the following hold for all 𝑥,𝑦∈𝑋:(i)𝑑(𝑥)≤𝑥,(ii)𝑑(𝑥∗𝑦)≤𝑑(𝑥)∗𝑦, (iii)𝑑(𝑥∗𝑦)≤𝑥∗𝑑(𝑦), (iv)𝑑(𝑑(𝑥))≤𝑥, (v)𝑑(𝑥∗𝑑(𝑥))=0,(vi)𝑑⁻¹(0)={𝑥∈𝑋∶𝑑(𝑥)=0} is a subalgebra of 𝑋.

3. Generalized Derivations of BCC-Algebras

Definition 11. Let 𝑋 be a 𝐵𝐶𝐶-algebra. A mapping 𝐷∶𝑋→𝑋 is called a generalized (𝑙,𝑟)-derivation if there exists an (𝑙,𝑟)-derivation 𝑑∶𝑋→𝑋 such that 𝐷(𝑥∗𝑦)=(𝐷(𝑥)∗𝑦)∧(𝑥∗𝑑(𝑦)) for all 𝑥,𝑦∈𝑋; if there exists an (𝑟,𝑙)-derivation 𝑑∶𝑋→𝑋 such that 𝐷(𝑥∗𝑦)=(𝑥∗𝐷(𝑦))∧(𝑑(𝑥)∗𝑦) for all 𝑥,𝑦∈𝑋, the mapping 𝐷∶𝑋→𝑋 is called a generalized (𝑟,𝑙)-derivation. Moreover, if 𝐷 is both a generalized (𝑙,𝑟)- and (𝑟,𝑙)-derivation, we say that 𝐷 is a generalized derivation.

Example 12. Let 𝑋={0,1,2,3} be a 𝐵𝐶𝐶-algebra with Cayley table (see Table 1).
Define a map 𝑑∶𝑋→𝑋 by 𝑑(𝑥)=0if𝑥=0,1,32if𝑥=2. (4) Then 𝑑 is a derivation of 𝑋. Now we define a map 𝐷∶𝑋→𝑋 by 𝐷(𝑥)=0if𝑥=0,2𝑥if𝑥=1,3. (5) It is easy to verify that 𝐷 is a generalized (𝑟,𝑙)-derivation of 𝑋.

Table 1 

Proposition 13. A generalized derivation 𝐷 of a 𝐵𝐶𝐶-algebra 𝑋 is regular.

Proof. Directly from Corollary 8, we have that 𝐷(0)=0.

Proposition 14. Let 𝐷 be a self-map of a 𝐵𝐶𝐶-algebra 𝑋. Then,(i)if 𝐷 is a generalized (𝑙,𝑟)-derivation of 𝑋, then 𝐷(𝑥)=𝐷(𝑥)∧𝑥 for all 𝑥∈𝑋;(ii)if 𝐷 is a generalized (𝑟,𝑙)-derivation of 𝑋, then 𝐷(𝑥)=𝑥∧𝑑(𝑥) for all 𝑥∈𝑋.

Proof. (i) If 𝐷 is a generalized (𝑙,𝑟)-derivation of 𝑋, then there exists an (𝑙,𝑟)-derivation 𝑑 such that 𝐷(𝑥∗𝑦)=(𝐷(𝑥)∗𝑦)∧(𝑥∗𝑑(𝑦)) for all 𝑥,𝑦∈𝑋. Hence, we get 𝐷(𝑥)()=(𝐷(𝑥))∧((0(𝑥)∧((0=((0∗(0(𝑥=()∗)(𝑥((𝑥(𝑥)=𝐷𝑥∗0∗0𝑥∗𝑑))=𝐷𝑥∗𝑑))𝑥∗𝑑))((𝑥∗𝑑))∗𝐷))𝑥∗0((𝑥∗0∗𝐷))=𝑥∗𝑥∗𝐷))=𝐷∧𝑥. (6)
(ii) If 𝐷 is a generalized (𝑟,𝑙)-derivation of 𝑋, then there exists an (𝑟,𝑙)-derivation 𝑑 such that 𝐷(𝑥∗𝑦)=(𝑥∗𝐷(𝑦))∧(𝑑(𝑥)∗𝑦) for all 𝑥,𝑦∈𝑋. Hence, we get 𝐷(𝑥)()=((0∧(𝑑(𝑥))=()(𝑥)(𝑥).=𝐷𝑥∗0𝑥∗𝐷))∗0𝑥∗0∧𝑑=𝑥∧𝑑 (7)

Proposition 15. Let 𝑋 be a 𝐵𝐶𝐶-algebra with partial order ≤, and let 𝐷 be a generalized derivation of 𝑋. Then the following hold for all 𝑥,𝑦∈𝑋:(1)𝐷(𝑥)≤𝑑(𝑥)≤𝑥, (2)𝐷(𝑥∗𝑦)≤𝑥∗𝑑(𝑦), (3)𝐷(𝑥∗𝑦)≤𝑑(𝑥)∗𝑦, (4)𝐷(𝑥∗𝐷(𝑥))=0,(5)𝐷(𝐷(𝑥)∗𝑥)=0,(6)𝐷(𝑑(𝑥)∗𝑥)=0,(7)𝐷(𝑥∗𝑑(𝑥))=0,(8)𝐷(𝐷(𝑥))≤𝑥.

Proof. Consider (1) 𝐷(𝑥)=𝐷(𝑥∗0)=(𝑥∗𝐷(0))∧(𝑑(𝑥)∗0)=(𝑥∗0)∧𝑑(𝑥)=𝑥∧𝑑(𝑥)≤𝑑(𝑥).
  From Proposition 10,   𝑑(𝑥)≤𝑥.
(2) 𝐷(𝑥∗𝑦)=(𝐷(𝑥)∗𝑦)∧(𝑥∗𝑑(𝑦))≤𝑥∗𝑑(𝑦).
(3) 𝐷(𝑥∗𝑦)=(𝑥∗𝐷(𝑦))∧(𝑑(𝑥)∗𝑦)≤𝑑(𝑥)∗𝑦.
(4) 𝐷(𝑥∗𝐷(𝑥))=(𝐷(𝑥)∗𝐷(𝑥))∧(𝑥∗𝑑(𝐷(𝑥)))=0∧(𝑥∗𝑑(𝐷(𝑥)))=0.
(5) 𝐷(𝐷(𝑥)∗𝑥)=(𝐷(𝑥)∗𝐷(𝑥))∧(𝑑(𝐷(𝑥))∗𝑥)=0∧(𝑑(𝐷(𝑥))∗𝑥)=0.
(6) 𝐷(𝑑(𝑥)∗𝑥)=(𝐷(𝑑(𝑥))∗𝑥)∧(𝑑(𝑥)∗𝑑(𝑥))=(𝐷(𝑑(𝑥))∗𝑥)∧0=0∗(0∗(𝐷(𝑑(𝑥))∗𝑥))=0.
(7) 𝐷(𝑥∗𝑑(𝑥))=(𝑥∗𝐷(𝑑(𝑥)))∧(𝑑(𝑥)∗𝑑(𝑥))=(𝑥∗𝐷(𝑑(𝑥)))∧0=0∗(0∗(𝑥∗𝐷(𝑑(𝑥))))=0.
(8) From Proposition 10, we have 𝐷(𝐷(𝑥((𝑥(𝑑(𝑥)∗(𝑑(𝑥)=(𝑑(𝑥)(𝑑(𝑥)(𝑑(𝑥)∗(𝑑(𝑥)=(𝑑(𝑥)(0(𝑑(𝑥))=(𝑑(𝑥))(𝑑(𝑥(𝑥)(𝑑(𝑥(𝑑(𝑥))=𝐷𝑥∧𝑑))=𝐷∗𝑥))∗𝐷∗𝑥))∧𝑑∗𝑥))∗𝐷))∧𝑑∗0∗0∧𝑑))=𝑑∧𝑑))≤𝑑))≤𝑥. (8)

Proposition 16. Let 𝑋 be a 𝐵𝐶𝐶-algebra. Then 𝐷_𝑛(𝐷_𝑛−1(…(𝐷₂(𝐷₁(𝑥)))…))≤𝑥 for 𝑛∈𝑁, where 𝐷₁,𝐷₂,…,𝐷_𝑛 are generalized derivations of 𝑋.

Proof. Trivial.

Definition 17. Let 𝐷 be a generalized derivation of a 𝐵𝐶𝐶-algebra 𝑋. An ideal 𝐴 of 𝑋 is said to be 𝐷-invariant if 𝐷(𝐴)⊆𝐴, where 𝐷(𝐴)={𝐷(𝑥)∣𝑥∈𝐴}. (9)

Theorem 18. Let 𝐷 be a generalized derivation of a 𝐵𝐶𝐶-algebra 𝑋. Then every ideal 𝐴 of 𝑋 is 𝐷-invariant.

Proof. Let 𝐴 be an ideal of a 𝐵𝐶𝐶-algebra 𝑋. Let 𝑦∈𝐷(𝐴). Then 𝑦=𝐷(𝑥) for some 𝑥∈𝐴. It follows that 𝑦∗𝑥=𝐷(𝑥)∗𝑥=0∈𝐴, which implies that 𝑦∈𝐴. Thus 𝐷(𝐴)⊆𝐴. Hence, 𝐴 is 𝐷-invariant.

Definition 19. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation. Define a Ker𝐷 by Ker𝐷={𝑥∈𝑋∶𝐷(𝑥)=0}.

Theorem 20. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be generalized derivation. If 𝑦∈Ker(𝐷) and 𝑥∈𝑋, then 𝑥∧𝑦∈Ker(𝐷).

Proof. Let 𝑦∈Ker(𝐷). Then, we get 𝐷(𝑦)=0, and so 𝐷=𝐷𝑦∗∧=∧𝑥∧𝑦=𝐷𝑦∗𝑦∗𝑥𝑦∗𝑥𝑦∗𝑑𝑦∗𝑥0∗𝑦∗𝑥𝑦∗𝑑𝑦∗𝑥=0∧𝑦∗𝑑𝑦∗𝑥=0. (10)
Hence, we have 𝑥∧𝑦∈Ker(𝐷).

Theorem 21. Let 𝑋 be a commutative 𝐵𝐶𝐶-algebra and 𝐷 be a generalized derivation. If 𝑥≤𝑦 and 𝑦∈Ker(𝐷), then 𝑥∈Ker(𝐷).

Proof. Let 𝑥≤𝑦 and 𝑦∈Ker(𝐷). Then we get 𝑥∗𝑦=0 and 𝐷(𝑦)=0, and so 𝐷(𝑥)()=𝐷𝑦∗∧=∧=𝐷𝑥∗0=𝐷𝑥∗𝑥∗𝑦=𝐷𝑦∗𝑦∗𝑥𝑦∗𝑥𝑦∗𝑑𝑦∗𝑥0∗𝑦∗𝑥𝑦∗𝑑𝑦∗𝑥=0∧𝑦∗𝑑𝑦∗𝑥=0. (11)
Hence we have 𝑥∈Ker(𝐷).

Theorem 22. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation. If 𝑥∈Ker(𝐷), one has 𝑥∗𝑦∈Ker(𝐷) for all 𝑦∈𝑋.

Proof. Let 𝑥∈Ker(𝐷). Then, 𝐷(𝑥)=0. Thus, we have 𝐷=𝐷(𝑥)∧𝑦=∧𝑦𝑦𝑥∗𝑦∗𝑦𝑥∗𝑑0∗𝑦𝑥∗𝑑=0∧𝑥∗𝑑=0. (12)
Hence, 𝑥∗𝑦∈Ker(𝐷).

Theorem 23. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation.Then Ker(𝐷) is a subalgebra of 𝑋.

Proof. Directly from Theorem 22.

Definition 24. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation on 𝑋. Denote Fix_𝐷(𝑋)={𝑥∈𝑋∶𝐷(𝑥)=𝑥}.

Proposition 25. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation on 𝑋. If 𝑥∈Fix_𝐷(𝑋), then 𝑑(𝑥)=𝑥.

Proof. Since 𝑥∈Fix_𝐷(𝑋), then 𝐷(𝑥)=𝑥. From Proposition 15, we have 𝐷(𝑥)≤𝑑(𝑥)≤𝑥 that implies that 𝑥≤𝑑(𝑥)≤𝑥. Hence, 𝑑(𝑥)=𝑥.

Proposition 26. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation on 𝑋. Then Fix_𝐷(𝑋) is a subalgebra of 𝑋.

Proof. If 𝑥,𝑦∈Fix_𝐷(𝑋), we get 𝐷(𝑥)=𝑥 and 𝑑(𝑦)=𝑦, and so 𝐷(𝑥∗𝑦)=(𝐷(𝑥)∗𝑦)∧(𝑥∗𝑑(𝑦))=(𝑥∗𝑦)∧(𝑥∗𝑦)=𝑥∗𝑦. Hence, 𝑥∗𝑦∈Fix_𝐷(𝑋).

Proposition 27. Let 𝑋 be a 𝐵𝐶𝐶-algebra and let 𝐷 be a generalized derivation on 𝑋. If 𝑥,𝑦∈Fix_𝐷(𝑋), we have 𝑥∧𝑦∈Fix_𝐷(𝑋).

Proof. Let 𝑥,𝑦∈Fix_𝐷(𝑋), Then 𝐷(𝑥)=𝑥 and 𝐷(𝑦)=𝑦.
From Proposition 26, we have 𝑦∗𝑥∈Fix_𝐷(𝑋), and so 𝑑(𝑦∗𝑥)=𝑦∗𝑥. Hence, we have 𝐷=𝐷𝑦∗∧=∧𝑥∧𝑦=𝐷𝑦∗𝑦∗𝑥𝑦∗𝑥𝑦∗𝑑𝑦∗𝑥𝑦∗𝑦∗𝑥𝑦∗𝑦∗𝑥=𝑦∗𝑦∗𝑥=𝑥∧𝑦. (13)
Hence, 𝑥∧𝑦∈Fix_𝐷(𝑋).