On t-Derivations of BCI-Algebras

and Applied Analysis 3 Definition 2.2 see 6 . A subset S of a BCI-algebra X is called subalgebra of X if x ∗ y ∈ S whenever x, y ∈ S. For a BCI-algebra X, we denote x ∧ y y ∗ y ∗ x for all x, y ∈ X 6 . For more details we refer to 3, 5, 6 . 3. t-Derivations in a BCI-Algebra/p-Semisimple BCI-Algebra The following definitions introduce the notion of t-derivation for a BCI-algebra. Definition 3.1. Let X be a-BCI-algebra. Then for any t ∈ X, we define a self map dt : X → X by dt x x ∗ t for all x ∈ X. Definition 3.2. Let X be a BCI-algebra. Then for any t ∈ X, a self map dt : X → X is called a left-right t-derivation or l, r -t-derivation of X if it satisfies the identity dt x ∗ y dt x ∗ y ∧ x ∗ dt y for all x, y ∈ X. Similarly, we get the following. Definition 3.3. Let X be a BCI-algebra. Then for any t ∈ X, a self map dt : X → X is called a right-left t-derivation or r, l -t-derivation of X if it satisfies the identity dt x ∗ y x ∗ dt y ∧ dt x ∗ y for all x, y ∈ X. Moreover, if dt is both a l, r and a r, l -t-derivation on X, we say that dt is a tderivation on X. Example 3.4. Let X {0, 1, 2} be a BCI-algebra with the following Cayley table: ∗ 0 1 2 0 0 0 2 1 1 0 2 2 2 2 0 3.1 For any t ∈ X, define a self map dt : X → X by dt x x ∗ t for all x ∈ X. Then it is easily checked that dt is a t-derivation of X. Proposition 3.5. Let dt be a self map of an associative BCI-algebra X. Then dt is a l, r -t-derivation of X. Proof. Let X be an associative BCI-algebra, then we have dt ( x ∗ y x ∗ y ∗ t { x ∗ y ∗ t ∗ 0 by Property 6 and 2 { x ∗ y ∗ t ∗ x ∗ y ∗ t ∗ x ∗ y ∗ t by Property iii { x ∗ y ∗ t ∗ x ∗ y ∗ t ∗ x ∗ y ∗ t by Property 6 4 Abstract and Applied Analysis { x ∗ y ∗ t ∗ x ∗ y ∗ t ∗ { x ∗ t ∗ y by Property 1 ( x ∗ t ∗ y ∧ x ∗ y ∗ t ( dt x ∗ y ) ∧ x ∗ dt ( y )) . 3.2 Proposition 3.6. Let dt be a self map of an associative BCI-algebra X. Then, dt is a r, l -t-derivation of X. Proof. Let X be an associative BCI-algebra, then we have dt ( x ∗ y x ∗ y ∗ t { x ∗ t ∗ y ∗ 0 by Property 1 and 2 { x ∗ t ∗ y ∗ [{ x ∗ t ∗ y ∗ { x ∗ t ∗ y as x ∗ x 0 { x ∗ t ∗ y ∗ [{ x ∗ t ∗ y ∗ x ∗ y ∗ t by Property 1 { x ∗ t ∗ y ∗ [{ x ∗ t ∗ y ∗ x ∗ y ∗ t by Property 6 ( x ∗ y ∗ t ∧ ( x ∗ t ∗ y as y ∗ y ∗ x x ∧ y ( x ∗ dt ( y )) ∧ dt x ∗ y ) . 3.3 Combining Propositions 3.5 and 3.6, we get the following Theorem. Theorem 3.7. Let dt be a self map of an associative BCI-algebra X. Then, dt is a t-derivation of X. Definition 3.8. A self map dt of a BCI-algebra X is said to be t-regular if dt 0 0. Example 3.9. Let X {0, a, b} be a BCI-algebra with the following Cayley table: ∗ 0 a b 0 0 0 b a a 0 b b b b 0 3.4 i For any t ∈ X, define a self map dt : X → X by dt x x ∗ t { b if x 0, a 0 if x b. 3.5 Then it is easily checked that dt is l, r and r, l -t-derivations of X, which is not t-regular. Abstract and Applied Analysis 5 ii For any t ∈ X, define a self map d′ t : X → X by d′ t x x ∗ t { 0 if x 0, a b if x b. 3.6and Applied Analysis 5 ii For any t ∈ X, define a self map d′ t : X → X by d′ t x x ∗ t { 0 if x 0, a b if x b. 3.6 Then it is easily checked that d′ t is l, r and r, l -t-derivations of X, which is t-regular. Proposition 3.10. Let dt be a self map of a BCI-algebra X. Then i If dt is a l, r -t-derivation of X, then dt x dt x ∧ x for all x ∈ X. ii If dt is a r, l -t-derivation of X, then dt x x ∧ dt x for all x ∈ X if and only if dt is t-regular. Proof of (i). Let dt be a l, r -t-derivation of X, then dt x dt x ∗ 0 dt x ∗ 0 ∧ x ∗ dt 0 dt x ∧ {x ∗ dt 0 } {x ∗ dt 0 } ∗ {x ∗ dt 0 } ∗ dt x {x ∗ dt 0 } ∗ {x ∗ dt x } ∗ dt 0 ≤ x ∗ {x ∗ dt x } by Property 3 dt x ∧ x. 3.7 But dt x ∧ x ≤ dt x is trivial so i holds. Proof of (ii). Let dt be a r, l -t-derivation of X. If dt x x ∧ dt x then dt 0 0 ∧ dt 0 dt 0 ∗ {dt 0 ∗ 0} dt 0 ∗ dt 0 0 3.8 thereby implying dt is t-regular. Conversely, suppose that dt is t-regular, that is dt 0 0, then we have dt x dt x ∗ 0 x ∗ dt 0 ∧ dt x ∗ 0 x ∗ 0 ∧ dt x x ∧ dt x . 3.9 This completes the proof. 6 Abstract and Applied Analysis Theorem 3.11. Let dt be a l, r -t-derivation of a p-semisimple BCI-algebra X. Then the following hold: i dt 0 dt x ∗ x for all x ∈ X. ii dt is one-one. iii If dt is t-regular, then it is an identity map. iv if there is an element x ∈ X such that dt x x, then dt is identity map. v if x ≤ y, then dt x ≤ dt y for all x, y ∈ X. Proof of (i). Let dt be a l, r -t-derivation of a p-semisimple BCI-algebra X. Then for all x ∈ X, we have x ∗ x 0 and so dt 0 dt x ∗ x dt x ∗ x ∧ x ∗ dt x {x ∗ dt x } ∗ {x ∗ dt x } ∗ {dt x ∗ x} dt x ∗ x by property 7 . 3.10 Proof of (ii). Let dt x dt y ⇒ x ∗ t y ∗ t, then by property 12 , we have x y and so dt is one-one. Proof of (iii). Let dt be t-regular and x ∈ X. Then, 0 dt 0 so by the above part i , we have 0 dt x ∗ x and hence by property 9 , we obtain dt x x for all x ∈ X. Therefore, dt is the identity map. Proof of (iv). It is trivial and follows from the above part iii . Proof of (v). Let x ≤ y implying x ∗ y 0. Now, dt x ∗ dt ( y ) x ∗ t ∗ y ∗ t x ∗ y by property 10 0. 3.11 Therefore, dt x ≤ dt y . This completes the proof. Definition 3.12. Let dt be a t-derivation of a BCI-algebra X. Then, dt is said to be an isotone t-derivation if x ≤ y ⇒ dt x ≤ dt y for all x, y ∈ X. Example 3.13. In Example 3.9 ii , d′ t is an isotone t-derivation, while in Example 3.9 i , dt is not an isotone t-derivation. Proposition 3.14. Let X be a BCI-algebra and dt be a t-derivation on X. Then for all x, y ∈ X, the following hold: i If dt x ∧ y dt x ∧ dt y , then dt is an isotone t-derivation. ii If dt x ∗ y dt x ∗ dt y , then dt is an isotone t-derivation. Abstract and Applied Analysis 7 Proof of (i). Let dt x ∧ y dt x ∧ dt y . If x ≤ y ⇒ x ∧ y x for all x, y ∈ X. Therefore, we haveand Applied Analysis 7 Proof of (i). Let dt x ∧ y dt x ∧ dt y . If x ≤ y ⇒ x ∧ y x for all x, y ∈ X. Therefore, we have dt x dt ( x ∧ y dt x ∧ dt ( y )