On t-derivations of PMS-algebras

Definition 2.1.

A PMS-algebra is an algebra (X; ⋆, 0) of type (2, 0) with a constant “0” and a binary operation “⋆” satisfying the following axioms:

In X, we define a binary relation “≤” by: a ≤ b if and only if a ⋆ b = 0.

Proposition 2.2.

In a PMS-algebra (X; ⋆, 0) the following properties hold for all a, b, c ∈ X.

Definition 2.3.

Let S be a nonempty subset of a PMS-algebra (X; ⋆, 0). Then S is called a PMS-subalgebra of X if a ∗ b ∈ S for all a, b ∈ S.

Definition 2.4.

Let X be a PMS-algebra. The set G (X) = {a ∈ X: a⋆ 0 = a} is called the G-part of X.

Definition 2.5.

Let X be a PMS-algebra. The set B (X) = {a ∈ X: a⋆ 0 = 0} is called the p-radical of X.

Remark 3.1.

Let (X; ⋆, 0) be a PMS-algebra.

Proof.

Let a, b, c ∈ X.

Definition 3.2.

A PMS-algebra X is said to be associative if (x ⋆ y) ⋆ z = x ⋆ (y ⋆ z) for all x, y, z ∈ X.

Example 3.3.

Let X be the PMS-algebra with the following Cayley table.

Then (X; ⋆, 0) is an associative PMS-algebra.

Definition 3.4.

Let X be a PMS-algebra. Then, for t ∈ X, define a self-map d_t: X→ X by d_t(x) = t ⋆ x for all x ∈ X.

Definition 3.5.

Let X be a PMS-algebra. Then, for t ∈ X, a self-map d_t: X→ X is said to be a (left, right)-t-derivation of X, denoted by, (l, r)-t-derivation if d_t(x ⋆ y) = (d_t(x) ⋆ y) ∧ (x ⋆ d_t(y)) for all x, y ∈ X.

Remark 3.6.

Let d_t be a (l, r)-t-derivation on a PMS-algebra X. Then d_t(x ⋆ y) = d_t(x) ⋆ y for all x, y ∈ X.

Example 3.7.

Let X be an associative PMS-algebra with the following Cayley Table.

Define, when t = 0, d_t(x) = x, ∀x ∈ X. when t = 1, d_t(0) = 1, d_t(1) = 0, d_t(2) = 1. when t = 2, d_t(0) = 2, d_t(1) = 1, d_t(2) = 0. Therefore, for each t ∈ X, d_t is a (l, r)-t-derivation on X.

Definition 3.8.

Let X be a PMS-algebra. Then, for t ∈ X, a self-map d_t: X→ X is said to be a (l, r)-t-derivation of X, denoted by, (r, l)-t-derivation if d_t(x ⋆ y) = (x ⋆ d_t(y)) ∧ (d_t(x) ⋆ y) for all x, y ∈ X.

Remark 3.9.

Let d_t be a (r, l)-t-derivation on a PMS-algebra X. Then d_t(x ⋆ y) = x ⋆ d_t(y) for all x, y ∈ X.

Example 3.10.

Let X be a non-associative PMS-algebra with the following Cayley Table.

Define, when t = 0, d_t(x) = x, ∀x ∈ X. when t = 1, d_t(0) = 2, d_t(1) = 0, d_t(2) = 1.

when t = 2, d_t(0) = 1, d_t(1) = 2, d_t(2) = 0.

Therefore, for each t ∈ X, d_t is a (r, l)-t-derivation on X.

Proposition 3.11.

Let d_t be a self-map on a PMS-algebra X. Then d_t is a (r, l)-t-derivation on X.

Proof.

Let d_t: X→ X defined by d_t(x) = t ⋆ x. Now for x, y ∈ X,

Hence d_t is a (r, l)-t-derivation on X. ■

Definition 3.12.

Let X be a PMS-algebra. Then, for t ∈ X, a self-map d_t: X→ X is called the t-derivation on X if d_t is both a (l, r)-t-derivation and a (r, l)-t-derivation on X.

Example 3.13.

Consider the PMS-algebra X in Example 3.3. Define the mapping d_t as follows:

When t = 0, d_t(x) = x, ∀x ∈ X.

When t = 1, d_t(0) = 1, d_t(1) = 0, d_t(2) = 3, d_t(3) = 2.

When t = 2, d_t(0) = 2, d_t(1) = 3, d_t(2) = 0, d_t(3) = 1.

When t = 3, d_t(0) = 3, d_t(1) = 2, d_t(2) = 1, d_t(3) = 0. Therefore, for each t ∈ X, d_t is a t-derivation of X.

Proposition 3.14.

Let d_t be a self-map on an associative PMS-algebra X. Then d_t is a (l, r)-t-derivation on X.

Proof.

Let X be an associative PMS-algebra and d_t be a self-map on X.

Hence d_t is (r, l)-t-derivation on X.

By combining Proposition 3.11 and Proposition 3.14, we get the following theorem. ■

Theorem 3.15.

Let X be an associative PMS-algebra. For t ∈ X, a self-map d_t is a t-derivation on X.

Definition 3.16.

A self-map d_t on a PMS-algebra X is said to be t-regular if d_t(0) = 0.

Example 3.17.

In Example 3.13, d_t is a regular t-derivation on X where t = 0. However, for t = 1, t = 2, or t = 3, d_t is not a regular t-derivation on X.

Proposition 3.18.

Let X be the p-radical of a PMS-algebra and d_t be a self-map on X. Then d_t is t-regular.

Proof.

Let X be a p-radical PMS-algebra.

Let x ∈ X. Then x ⋆ 0 = 0. Now d_t(0) = t ⋆ 0 = 0. Hence d_t is t-regular. ■

Theorem 3.19.

Let d_t be a self-map on a PMS-algebra X. Then

Proof.

(1) Let d_t is a (l, r)-t-derivation on X. Suppose that d_t(x) = x ∧ d_t(x).

d_t(0) = 0 ∧ d_t(0) = (0 ⋆ d_t(0)) ⋆ d_t(0) = (d_t(0) ⋆ d_t(0)) ⋆ 0 = 0 ⋆ 0 = 0.

d_t is regualr on X.

Conversely, assume that d_t is regular.

Now d_t(x) = d_t(0 ⋆ x) = (d_t(0) ⋆ x) ∧ (0 ⋆ d_t(x)) = (0 ⋆ x) ∧ d_t(x) = x ∧ d_t(x). Hence d_t(x) = x ∧ d_t(x)

(2) Let d_t is a (r, l)-t-derivation on X.

Suppose d_t(0) = 0.

Now d_t(x) = d_t(0 ⋆ x) = (0 ⋆ d_t(x) ∧ (d_t(0) ⋆ x) = d_t(x) ∧ (0 ⋆ x) = d_t(x) ∧ x ■

Theorem 3.20.

Let d_t be a (r, l)-t-derivation on a PMS-algebra. Then the following holds true.

Proof.

Let d_t be a (r, l)-t-derivation on a PMS-algebra.

Theorem 3.21.

Let X be a PMS-algebra and d_t be a t-derivation on X. If y ≤ x and d_t(x ⋆ y) = d_t(x) ⋆ d_t(y) for all x, y ∈ X,t hen d_t(x) = d_t(y).

Proof.

Let x ≤ y and d_t(x ⋆ y) = d_t(x) ⋆ d_t(y).

Now for x ∈ X, d_t(x) = d_t(0⋆x) = d_t((y⋆x)⋆x) = d_t(y⋆x)⋆d_t(x) = (d_t(y)⋆d_t(x))⋆d_t(x) = (d_t(x) ⋆ d_t(x)) ⋆ d_t(y) = 0 ⋆ d_t(y) = d_t(y).

Hence d_t(x) = d_t(y).■

Theorem 3.22.

Let d_t be a t-regular (l, r)-t-derivation on a PMS-algebra X. Then the following hold.

Proof.

Let d_t be a t-regular (l, r)-t-derivation on a PMS-algebra X.

Theorem 3.23.

Let X be a PMS-algebra. Then Ker(d_t) = {0} if and only if d_t is t-regular.

Proof.

Suppose Ker(d_t) = {0}. Then d_t(x) = 0 ∀x ∈ X. In particular, d_t(0) = 0. Hence d_t is regular.

Conversely, assume d_t is regular. Let x ∈ Ker(d_t). Then d_t(x) = 0. Now

Thus x = 0 implies that Ker(d_t) = {0} ■

Definition 3.24.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be self-maps on X. Then define a mapping d_t ◦ ${\mathrm{d}}_t^{\prime }$ : X → X by (d_t ◦ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t( ${\mathrm{d}}_t^{\prime }$ (x)) for all x ∈ X.

Theorem 3.25.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be a (l, r)-t-derivation on X. Then (d_t ◦ ${\mathrm{d}}_t^{\prime }$ ) is a (l, r)-t-derivation on X.

Proof.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be a (l, r)-t-derivation on X.

For x, y ∈ X, we have

Hence (d_t ◦ ${\mathrm{d}}_t^{\prime }$ ) is a (l, r)- t-derivation of X. ■

Theorem 3.26.

Let X be a PMS algebra and d_t d^'_t be a (r, l)- t-derivation of X. Then (d_t ◦ ${\mathrm{d}}_t^{\prime }$ ) is a (r, l)-t-derivation on X.

Proof.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be a (r, l)-t-derivation on X.

For x, y ∈ X, we have

Hence (d_t ◦ ${\mathrm{d}}_t^{\prime }$ ) is a (r, l)-t-derivation of X.

Combining Theorem 3.25 and Theorem 3.26, we get the following theorem. ■

Theorem 3.27.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be t-derivation of X. Then (d_t ◦ ${\mathrm{d}}_t^{\prime }$ ) is a t-derivation on X.

Theorem 3.28.

Let X be a PMS-algebra. Let d_t be a (r, l)-t-derivation on X and ${\mathrm{d}}_t^{\prime }$ be a (l, r)-t- derivation on X. Then d_t ◦ ${\mathrm{d}}_t^{\prime }$ = ${\mathrm{d}}_t^{\prime }$ ◦ d_t.

Proof.

Let d_t be a (r, l)-t-derivation on X and ${\mathrm{d}}_t^{\prime }$ be a (l, r)-t-derivation on X. Now

From (3.1) and (3.2) we have, (d_t ◦ ${\mathrm{d}}_t^{\prime }$ )(x) = ( ${\mathrm{d}}_t^{\prime }$ ◦ d_t)(x), ∀x ∈ X. Thus d_t ◦ ${\mathrm{d}}_t^{\prime }$ = ${\mathrm{d}}_t^{\prime }$ ◦ d_t. ■

Theorem 3.29.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be two t-derivations on X, then d_t ◦ ${\mathrm{d}}_t^{\prime }$ = ${\mathrm{d}}_t^{\prime }$ ◦ d_t.

Definition 3.30.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be self-maps on X. Then define d_t ⋆ ${\mathrm{d}}_t^{\prime }$ : X→ X by (d_t ⋆ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t(x) ⋆ ${\mathrm{d}}_t^{\prime }$ (x) for all x ∈ X.

Theorem 3.31.

Let X be a PMS-algebra and d_t ${\mathrm{d}}_t^{\prime }$ be t-derivations on X. Then d_t ⋆ ${\mathrm{d}}_t^{\prime }$ = ${\mathrm{d}}_t^{\prime }$ ⋆ d_t.

Definition 3.32.

Let LDer_t(X) be the set of all (l, r)-t-derivations on a PMS-algebra X. Then define a binary operation “∧” on LDer_t(X) by (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t (x) ∧ ${\mathrm{d}}_t^{\prime }$ (x), ∀x ∈ X, where d_t ${\mathrm{d}}_t^{\prime }$ ∈ LDer_t(X).

Lemma 3.33.

If d_t and d^'_t are (l, r)-t-derivations on X. Then (d_t ∧ ${\mathrm{d}}_t^{\prime }$ ) is a (l, r)-t-derivation on X.

Proof.

From (3.3) and (3.4), we get (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x ⋆ y) = (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x) ⋆ y. ■

Lemma 3.34.

The binary operation ∧ defined on L_tDer(X) is associative.

Proof.

Let d_t ${\mathrm{d}}_t^{\prime }$ ${\mathrm{d}}_t^{″}$ ∈ LDer_t(X). Now for x ∈ X,

From (3.5) and (3.6), we have ((d_t ∧ ${\mathrm{d}}_t^{\prime }$ ) ∧ ${\mathrm{d}}_t^{″}$ )(x) = (d_t ∧ ( ${\mathrm{d}}_t^{\prime }$ ∧ ${\mathrm{d}}_t^{″}$ ))(x), ∀x ∈ X.

Thus “∧” is associative. ■

Theorem 3.35.

L_tDer(X) is a semigroup under the binary operation ∧ defined by (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t(x) ∧ ${\mathrm{d}}_t^{\prime }$ (x) for all x ∈ X, and d_t ${\mathrm{d}}_t^{\prime }$ ∈ L_tDer(X).

Definition 3.36.

Let R_tDer(X) be the set of all (r, l)-t-derivations on a PMS-algebra X. Then define a binary operation ∧ on R_tDer(X) by (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t(x) ∧ ${\mathrm{d}}_t^{\prime }$ (x), ∀x ∈ X, where d_t ${\mathrm{d}}_t^{\prime }$ ∈ R_tDer(X).

Lemma 3.37.

If d_t and d^'_t are (r, l)-t-derivations on X. Then (d_t ∧ ${\mathrm{d}}_t^{\prime }$ ) is also a (r, l)-t-derivation on X.

Proof.

Let d_t ${\mathrm{d}}_t^{\prime }$ ∈ RDer_t(X). For

From (3.7) and (3.8), we have, (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x ⋆ y) = x ⋆ (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(y). ■

Lemma 3.38.

The binary operation “∧” defined on RDer_t(X) is associative.

Proof.

Let d_t ${\mathrm{d}}_t^{\prime }$ ${\mathrm{d}}_t^{″}$ ∈ RDer_t(X). Now for, x ∈ X.

From (3.9) and (3.10), we have ((d_t ∧ ${\mathrm{d}}_t^{\prime }$ ) ∧ ${\mathrm{d}}_t^{″}$ )(x) = (d_t ∧ ( ${\mathrm{d}}_t^{\prime }$ ∧ ${\mathrm{d}}_t^{″}$ ))(x), ∀x ∈ X. Thus the binary operation “∧” is associative on RDer_t(X). ■

Theorem 3.39.

RDer_t(X) is a semigroup under the binary operation “∧” defined by (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t(x) ∧ ${\mathrm{d}}_t^{\prime }$ (x) for all x ∈ X, where d_t ${\mathrm{d}}_t^{\prime }$ ∈ R_tDer(X).

Combining Theorem (3.35) and Theorem (3.39), we get the following theorem.

Theorem 3.40.

If Der_t(X) denotes the set of all t-derivations on X. Then it is a semigroup under the binary operation “∧” defined by (d_t ∧ ${\mathrm{d}}_t^{\prime }$ )(x) = d_t(x) ∧ ${\mathrm{d}}_t^{\prime }$ (x) for all x ∈ X, where d_t ${\mathrm{d}}_t^{\prime }$ ∈ Der_t(X).