On derivations of commutative multiplicative semilattices

1 Introduction

Much of human reasoning and decision making is based on an environment of imprecision, uncertainty, incompleteness of information, partiality of truth and partiality of possibility, in short, on an environment of imperfect information. Hence how to represent and simulate human reasoning become a crucial problem in information science field. For this reason, various kinds of many valued logical algebras as the semantical systems of fuzzy logic systems have been extensively introduced, for example, MV-algebras [6], BL-algebras [8], MTL-algebras [14], residuated lattices [11] and quantales [12]. Although they are essentially different, they still have some similarities, that is, the (⊙, ∨)-reduct of the above many valued logical algebras is a commutative multiplicative semilattices, in other words, commutative multiplicative semilattices are building blocks for the above many valued logical algebras. Thus, it is meaningful to deeply study properties and algebraic structures of commutative multiplicative semilattice.

The notion of derivations, introduced from the analytic theory, is helpful for studying algebraic structures and properties in algebraic systems. In fact, the notion of derivations in ring theory is quite old and plays a significant role in algebraic geometry. In 1957, Posner [16] introduced the notion of a derivations in prime rings and give some characterizations of them. Based on this, a number of research articles have been appeared on derivations in the theory of rings and references there in [1, 19]. Inspired by derivations on rings, Ferrari et al [7, 18] introduced the notion of derivations in lattices and give some characterizations of them. Then, Jun and Zhan [15, 21] applied the notion of derivations to BCI-algebras and gave some characterizations of p-semisimple BCI-algebras. In 2010, Alshehri [2] applied the notions of (additive) derivations to MV-algebras and discussed some related properties, they also proved that an additive derivation of a linearly ordered MV-algebra is isotone. After the work of Alshehri, several authors [20] have studied derivations in MV-algebras and obtain some interesting results. Recently, authors introduced and studied new types of derivations on other logical algebras, for example, Wang [17] introduced a new type derivation in BL-algebras and give some characterization Gödel algebras by this derivation. He [13] investigated some particular derivations in residuated lattices and characterized some special types of residuated lattices in terms of derivations. As we have mentioned in the above, derivations have been studied on MV-algebras, BL-algebras, residuated lattices and lattices, etc, we have observed that although they are essentially different algebras they all are particular types of commutative multiplicative lattices. Therefore, it is meaningful to establish the derivation theory of commutative multiplicative lattices for studying the common properties of derivations in the above-mentioned algebras. This is the motivation for us to investigate derivation theory of commutative multiplicative lattices.

This paper is structured in four sections. In order to make the paper as self contained as possible, we recapitulate in Section 2 the definition of multiplicative semilattice, and review their basic properties. In Section 3, we investigate related properties of some particular derivations and give some characterizations of zero derivations in prime commutative multiplicative semilattices. Then, we prove that the set of all prefect derivations ideals on commutative multiplicative semilattices with prefect derivations can form a complete Heyting algebra. In section 4, we investigate principal derivations and prove that the structure of an idempotent commutative quantale is completely determined by its set of all principal derivations.

2 Preliminaries

In this section, we summarize some definitions and results about multiplicative semilattices, which will be used in the following sections.

Definition 2.1. [9] An algebra (L, ∨, ⊙, 0, 1) of type (2, 2, 0, 0) is called a multiplicative semilattice if it satisfies the following conditions:

(L, ∨, 0, 1) is a bounded join semilattice,

In what follows, we denote by L a multiplicative semilattice (L, ∨, ⊙, 0, 1), unless otherwise specified.

Example 2.2. (a) Let (L, ∨, ⊙, 0, 1) be a multiplicative semilattice and M ( L ) = { ( x y 0 0 ) : x , y ∈ L } be a collection of 2 × 2 matrices on L. Define operations ⊔ and ⊗ on M (L) as follows: ( x y 0 0 ) ⊔ ( z w 0 0 ) = ( x y 0 0 ) = ( x ∨ z y ∨ w 0 0 ) , ( x y 0 0 ) ⊗ ( z w 0 0 ) = ( x y 0 0 ) = ( x ⊙ z y ⊙ w 0 0 ) for any ( x y 0 0 ) , ( z w 0 0 ) ∈M (L). Then ( M ( L ) , ⊔ , ⊗ , ( 0 0 0 0 ) , ( 1 1 0 0 ) ) is a multiplicative semilattice.

(b) Let (L, ∨, 0, 1) be a bounded join semilattice. Define operation ⊙ on L as follows: x ⊙ y = { x , y ≠ 0 0 , y = 0 . Then (L, ∨, ⊙, 0, 1) is a multiplicative semilattice.

Definition 2.3. [22] Let (L, ∨, ⊙, 0, 1) be a multiplicative semilattice. Then L is called to be:

complete if every subset of it has a supremum,

A nonempty subset I of a multiplicative semilattice (L, ∨, ⊙, 0, 1) is called an ideal of L if it satisfies: (i) for all x, y ∈ I, x ∨ y ∈ I; (ii) for all x, y ∈ L, if x ∈ I and y ≤ x, then y ∈ I, (iii) for all x ∈ I, y ∈ L, x ⊙ y ∈ I and y ⊙ x ∈ I. Let I be an ideal of a multiplicative semilattice L. Define the congruence ≡ _I on L by x ≡ _F y if there exist z, w ∈ I such that x ∨ z = y ∨ w. The set of all congruence classes is denote by L/I, i.e. L/I = {[x] |x ∈ L}, where [x] = {x ∈ L|x ≡ _I y}. Then L/I becomes a multiplicative semilattice with the natural operations induced from those of L [5].

Definition 2.4. [22] Let (L₁, ∨ ₁, ⊙ ₁, 0₁, 1₁) and (L₂, ∨ ₂, ⊙ ₂, 0₂, 1₂) be two multiplicative semilattices. A mapping h : L₁ → L₂ is called a homomorphism from L₁ to L₂ if it satisfies: (1) h (x ∨ ₁y) = h (x) ∨ ₂h (y), (2) h (x ⊙ ₁y) = h (x) ⊙ ₂h (y), (3) h (1₁) =1₂, h (0₁) =0₂ for any x, y ∈ L₁. Moreover, a homomorphism is called an isomorphism if it is also a bijection.

Definition 2.5. [12] A quantale is a complete lattice L with an associative binary operation ⊙ satisfying: x ⊙ (∨ _i y _i ) = ∨ _i (x ⊙ y _i ) and (∨ _i y _i ) ⊙ x = ∨ _i (y _i ⊙ x) for all x ∈ L and {y _i } ⊆ L.

Definition 2.6. [5] A lattice (L, ∧, ∨) is called to be a Heyting algebra if for any x, y ∈ L, there exists an eloement x → y ∈ L such that z ≤ x → y if and only if x ∧ z ≤ y for all z ∈ L.

At the end of this section, we review the known main results about annihilator in a multiplicative semilattice, which is helpful for studying the relation between prefect derivations ideal and annihilator.

Definition 2.7. [10] Let (L, ∨, ⊙, 0, 1) be a multiplicative semilattice. For every nonempty subset S ⊆ L define respectively the left and right annihilator of S by Ann l ( S ) = { a ∈ L | a ⊙ s = 0 , for all s ∈ S } , Ann r ( S ) = { a ∈ L | s ⊙ a = 0 , for all s ∈ S } .

In a reduced multiplicative semilattice L, if x ⊙ y = 0 for any x, y ∈ L, then y ⊙ x = 0 and hence, there is no distinction from a left and right annihilator of S in L. In this case, we just call it by the annihilator of S in L and is denoted by Ann (S).

Proposition 2.8. Let (>L, ∨, ⊙, 0, 1) be a reduced multiplicative semilattice and S be a nonempty subset of L. Then the following hold:

Ann (S) is an ideal of L,

Proof. (1) Assume that L is a reduced multiplicative semilattice. Since 0 ∈ Ann (S), we can see that Ann (S) is a nonempty subset of L. If x, y ∈ Ann (S), then x ⊙ s = y ⊙ s = 0 for all s ∈ S. Now, (x ∨ y) ⊙ s = (x ⊙ s) ∨ (y ⊙ s) =0 ∨ 0 =0, thus x ∨ y ∈ Ann (S). Also, if y ∈ Ann (S) and x ≤ y, then x ⊙ s ≤ y ⊙ s = 0, thus x ⊙ s = 0 for all s ∈ S. Then, for any x ∈ L and y ∈ Ann (S), we have (x ⊙ y) ⊙ s = x ⊙ (y ⊙ s) =0. Thus, x ⊙ y ∈ Ann (S). Therefore, Ann (S) is an ideal of L.

(2) If x ∈ Ann (S₂), then x ⊙ s = 0 for all s ∈ S₂. This means that x annihilates all elements of S₂. From S₁ ⊆ S₂, one can see that x annihilates all elements of S₁ and hence, x ∈ Ann (S₁). Therefore, Ann (S₂) ⊆ Ann (S₁).

3 On derivations of commutative multiplicative semilattices

In this section, we introduce some particular derivations and give some characterizations of regular derivations in commutative multiplicative semilattices. Also, we give some characterizations of zero derivations in prime commutative multiplicative semilattices and prove that the set of all prefect derivations ideals on commutative multiplicative semilattices with prefect derivations can form acomplete Heyting algebra.

Definition 3.1. Let (L, ∨, ⊙, 0, 1) be a commutative multiplicative semilattice. A mapping d : L ⟶ L is called a derivation on L if it satisfies the following conditions: for any x, y ∈ L, ( 1 ) d ( x ∨ y ) = d ( x ) ∨ d ( y ) , ( 2 ) d ( x ⊙ y ) = ( d ( x ) ⊙ y ) ∨ ( x ⊙ d ( y ) ) .

The pair (L, d) is said to be a commutative multiplicative semilattice with a derivation.

Now, we present some examples for derivations on commutative multiplicative semilattices.

Example 3.2. Let L be a commutative multiplicative semilattice. Define a mapping d : L ⟶ L by d (x) =0 for all x ∈ L, then d is a derivation on L, which is called a zero derivation. Moreover, we define a mapping d : L → L by d (x) = x for all x ∈ L. Then d is a derivation on L, which is called an identity derivation.

Example 3.3. Let L be the multiplicative semilattice in Example 2.2(a). Now, we define a mapping d : M (L) → M (L) by d ( x y 0 0 ) = ( 0 y 0 0 ) for all ( x y 0 0 ) ∈ M ( L ) . Then one can check that d is a derivation on M (L).

Theorem 3.4. Let L be a commutative multiplicative semilattice, d be a mapping on L and h _d be a mapping from L to M (L) defined by h d ( x ) = ( x d ( x ) 0 x ) . Then the following statements areequivalent:

d is a derivation on L,

Proof. (1) ⇒ (2) Suppose that d is a derivation on L. For any x, y ∈ L, on one hand, if a = h _d (x ∨ y), then a = h d ( x ∨ y ) = ( x ∨ y d ( x ∨ y ) 0 x ∨ y ) . From Definition 3.1, we have a = h d ( x ∨ y ) = ( x ∨ y d ( x ) ∨ d ( y ) 0 x ∨ y ) . On the other hand, we have h d ( x ) ∨ h d ( y ) = ( x d ( x ) 0 x ) ∨ ( y d ( y ) 0 y ) = ( x ∨ y d ( x ) ∨ d ( y ) 0 x ∨ y ) . Thus, h _d (x ∨ y) = h _d (x) ∨ h _d (y) for any x, y ∈ L. Moreover, h d ( x ⊙ y ) = ( x ⊙ y d ( x ⊙ y ) 0 x ⊙ y ) and h d ( x ) ⊙ h d ( y ) = ( x ⊙ y ( d ( x ) ⊙ y ) ∨ ( x ⊙ d ( y ) ) 0 x ⊙ y ) . From Definition 3.1, we have h _d (x ⊙ y) = h _d (x) ⊙ h _d (y) for any x, y ∈ L. Therefore, h _d is a homomorphism from L to M (L).

(2) ⇒ (1) Assume that h _d is a homomorphism from L to M (L). For any x, y ∈ L, on one hand, from Definition 2.4(1), we have ( x ∨ y d ( x ∨ y ) 0 x ∨ y ) = h d ( x ∨ y ) = h d ( x ) ∨ h d ( y ) = ( x d ( x ) 0 x ) ∨ ( y d ( y ) 0 y ) = ( x ∨ y d ( x ) ∨ d ( y ) 0 x ∨ y ) . Thus, d (x ∨ y) = d (x) ∨ d (y) for any x, y ∈ L. On the other hand, from Definition 2.4(2), we have ( x ⊙ y d ( x ⊙ y ) 0 x ⊙ y ) = h d ( x ⊙ y ) = h d ( x ) ⊙ h d ( y ) = ( x d ( x ) 0 x ) ⊙ ( y d ( y ) 0 y ) = ( x ⊙ y ( d ( x ) ⊙ y ) ∨ ( x ⊙ d ( y ) ) 0 x ⊙ y ) . Thus, d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y)), for any x, y ∈ L. Therefore, d is a derivation on L.

Then ({0, a, b, c, 1}, ∨, ⊙, 0, 1) is a commutative multiplicative semilattice. Define a mapping d : L ⟶ L by d (0) =0, d (a) = a, d (b) = b, d (c) = d (1) = c. One can check that d is a derivation on L.

Proposition 3.6. Let (>L, ∨, ⊙, 0, 1) be a commutative multiplicative semilattice and d be a derivation on L. Then we have: for any x, y ∈ L,

d (0) =0,

Proof. (1) From the definition of derivations, we have d (0) = d (0 ⊙0) = (d (0) ⊙0) ∨ (0 ⊙ d (0)) =0, that is, d (0) =0.

(2) For all x ∈ L, we have d (x) = d (x ⊙ 1) = (d (x) ⊙1) ∨ (x ⊙ d (1)) = d (x) ∨ (x ⊙ d (1)). Thus, d (x) = d (x) ∨ (x ⊙ d (1)), for all x ∈ L.

(3) From Definition 3.1, we deduce that d (x²) = d (x ⊙ x) = (x ⊙ d (x)) ∨ (d (x) ⊙ x) = x ⊙ d (x) for all x ∈ L. By induction, we can obtain that d (x ⁿ ) = x^n-1 ⊙ d (x) for n ≥ 1.

(4) If x ≤ y, from Definition 3.1, we have d (y) = d (x ∨ y) = d (x) ∨ d (y). Thus, d (x) ≤ d (y).

(5) For all x, y ∈ L, we have d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y)) ≤ d (x) ∨ d (y). Thus, d (x ⊙ y) ≤ d (x) ∨ d (y).

In what follows, we introduce prefect derivations in a commutative multiplicative semilattice and investigate some related properties of them.

Definition 3.7. Let (L, ∨, ⊙, 0, 1) be a commutative multiplicative semilattice and d be a derivation on L.

d is called a regular derivation provided that d (1) =1,

Then ({0, a, 1}, ∨, ⊙, 0, 1) is a commutative multiplicative semilattice. Define a mapping d : L ⟶ L by d (0) =0, d (a) = d (1) =1. One can check that d is a prefect derivation on L.

Example 3.9. Considering the derivation d in Example 3.3, one can see that d is neither an idempotent derivation nor a regular derivation on L.

Theorem 3.10. Let L be a commutative multiplicative semilattice and d be a derivation on L. Then the following statements are equivalent:

d is a regular derivation on L,

Proof. (1) ⇒ (2) If d is a regular derivation on L, then d (x) = x ∨ d (x) from Proposition 3.6(2). Thus, we have x ≤ d (x).

(2) ⇒ (1) Since 1 ≤ d (1) and hence d (1) =1. Thus, d is a regular derivation on L.

Proposition 3.11. Let L be a commutative multiplicative semilattice and d be a prefect derivation on L. Then we have: for any x, y ∈ L,

d (x) ⊙ d (x) = d (x),

Proof. (1) For all x ∈ L, we have d (x) ⊙ d (x) ≤ d (x) ⊙1 = d (x). On the other hand, since d is a prefect derivation on L, we have d (x) = d (x ⊙ x) = (d (x) ⊙ x) ∨ (x ⊙ d (x)) ≤ d (x) ⊙ d (x). Thus, d (x) ⊙ d (x) = d (x).

(2) For all x ∈ L, from (1), we have d (d (x)) = d (d (x) ⊙ d (x)) = (d (d (x)) ⊙ d (x)) ∨ (d (x) ⊙ d (d (x))) ≤ d (x) ∨ d (x) = d (x). On the other hand, since d is a regular derivation on L, we have d (x) ≤ d (d (x)). Thus, dd (x) = d (x).

(3) Let y ∈ d (L). So there exists x ∈ L such that y = d (x) and hence d (y) = dd (x) = d (x) = y, that is, y ∈ Fix _d (L). Conversely, if y ∈ Fix _d (L), then we have y ∈ d (L). Therefore, Fix _d (L) = d (L).

(4) For any x ∈ L, we have x = d (x₀) for some x₀ ∈ L. From (3), we have d (x) = dd (x₀) = d (x₀) = x. Thu, d = id _L . Conversely, suppose that d = id _L , we have d (L) = L.

(5) For all x, y ∈ L, assume that x ≤ d (y), we have d (x) ≤ d (d (y)). By the statement (2), we get d (d (y)) = d (y). Thus d (x) ≤ d (y). Conversely, suppose that d (x) ≤ d (y), we have x ≤ d (x) ≤ d (y) for all x, y ∈ L.

(6) For all x ∈ L, from Proposition 3.6(3) and d is a prefect derivation on L, we have d (x) = d (x ⊙ x) = x ⊙ d (x). Thus, d (x) = x ⊙ d (x).

(7) An immediate consequence of regular derivations is that x ⊙ d (y) ≤ d (x) ⊙ d (y) and d (x) ⊙ y ≤ d (x) ⊙ d (y) for all x, y ∈ L. It follows that d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y) ≤ d (x) ⊙ d (y) for all x, y ∈ L. Thus, d (x ⊙ y) ≤ d (x) ⊙ d (y).

(8) From Proposition 3.6(1), we have d (0) =0, that is, 0∈ Ker(d). If x, y∈ Ker(d), then d (x) = d (y) =0 and hence d (x ∨ y) = d (x) ∨ d (y) =0, that is, x∨ y ∈ Ker(d). Also, if x ≤ y and y∈ Ker(d), then d (x) ≤ d (y) =0 and hence d (x) =0, that is, x∈ Ker(d). Finally, for all x ∈ L, y∈ Ker(d), from (6), we have d (x ⊙ y) ≤ d (x) ⊙ d (y) =0 ⊙ d (y) =0 and hence x⊙ y ∈ Ker(d), in the similar way, one can prove that y⊙ x ∈ Ker(d). Thus, Ker(d) is an ideal of L.

(9) By Definition 3.1, Proposition 3.6(1), Definition 3.7(1) and Proposition 3.11(7), we have Fix _d (L) is closed under the operations ∨, ⊙ and 0, 1. Therefore, Fix _d (L) is a subalgebra of L.

Remark 3.12. The statement (9) in the above proposition shows that the fixed point set Fix _d (L) of prefect derivations in a commutative multiplicative semilattice L has the same structure as L, which reveals the essence for the fixed point set of prefect derivations.

Theorem 3.13. Let L be a commutative multiplicative semilattice and d a prefect derivation on L. Then we have the following properties:

d : L ⟶ Fix _d (L) is a surjective homomorphism,

Proof. (1) From Proposition 3.11(2), we can prove that d is well defined. Now, we show that d (x ⊙ y) = d (x) ⊙ d (y) for all x, y ∈ L. Since d is a prefect derivation on L, we have d (x ⊙ y) = d (d (x ⊙ y)) = d ((d (x) ⊙ y) ∨ (x ⊙ d (y))) = d (d (x) ⊙ y) ∨ d (x ⊙ d (y)) ≥ d (d (x) ⊙ y) ∨ (d (x) ⊙ d (y)) ≥ d (x) ⊙ d (y). On the other hand, d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y)) ≤ d (x) ⊙ d (y). Thus, d (x ⊙ y) = d (x) ⊙ d (y). Moreover, d preserves joint, we know that d is a homomorphism from L to Fix _d (L).

(2) From Proposition 3.11(8), we obtain that Ker (d) is an ideal of L. Let x ≡ _Ker(d)y. Then there exists z, w ∈ Ker (d) such that x ∨ z = y ∨ w, which implies d (x) ∨ d (z) = d (x ∨ z) = d (y ∨ w) = d (y) ∨ d (w), that is, d (x) = d (y). Thus, d ¯ is well defined. Moreover, it follows from (1) that d ¯ : L / Ker ( d ) ⟶ Fix d ( L ) is an isomorphism.

In what follows, we give some characterizations of zero derivations in prime commutative multiplicative semilattices.

Theorem 3.14. Let L be a prime commutative multiplicative semilattice and d be a derivation on L. The the following statements are equivalent:

if there exists an element u ∈ L such that u ⊙ d (x) =0 for all x ∈ L,

Proof. (1) ⇒ (2) For any x, y ∈ L, assume that there exists an element u such that u ⊙ d (x) =0, then 0 = u ⊙ d (x ⊙ y) = u ⊙ ((d (x) ⊙ y) ∨ (x ⊙ d (y))) = (u ⊙ d (x) ⊙ y) ∨ (u ⊙ x ⊙ d (y)) = u ⊙ x ⊙ d (y). Thus, u ⊙ x ⊙ d (y) =0. Since L is a prime multiplicative semilattice, we have u = 0 or d (y) =0. If u ≠ 0, then d (y) =0 for all y ∈ L, that is, d = 0.

(2) ⇒ (1) It is clear.

Theorem 3.15. Let L be a prime commutative multiplicative semilattice and d be a derivation on L. Then the following statements are equivalent:

d² (x) =0 for all x ∈ L,

Proof. (1) ⇒ (2) Assume that d ≠ 0, then there exists an element x such that d (x) ≠0. If d² = 0, then for any y ∈ L, we have 0 = d² (x ⊙ y) = d (d (x ⊙ y)) = d ((d (x) ⊙ y) ∨ (x ⊙ d (y)) = d (d (x) ⊙ y) ∨ d (x ⊙ d (y)) = d² (x) ⊙ y ∨ d (x) ⊙ d (y) ∨ d (x) ⊙ d (y) ∨ x ⊙ d² (y) = d (x) ⊙ d (y) ∨ d (x) ⊙ d (y) = d (x) ⊙ d (y), that is, d (x) ⊙ d (y) =0. Since L is a prime multiplicative semilattice, by the Theorem 3.14, we have d (y) =0 for any y ∈ L, which means that d = 0, that is a contradiction to the assumption. Thus, d = 0.

(2) ⇒ (1) It is clear.

Theorem 3.16. Let L be a prime commutative multiplicative semilattice and d₁, d₂ be two derivations on L. Then the following statements are equivalent:

d₁d₂ (x) =0 for all x ∈ L,

Proof. (1) ⇒ (2) For any x, y ∈ L, we have 0 = d₁d₂ (x ⊙ y) = d₁ (d₂ (x ⊙ y)) = d₁ ((d₂ (x) ⊙ y) ∨ (x ⊙ d₂ (y))) = d₁ (d₂ (x) ⊙ y) ∨ d₁ (x ⊙ d₂ (y)) = d₁d₂ (x) ⊙ y ∨ d₂ (x) ⊙ d₁ (y) ∨ d₁ (x) ⊙ d₂ (y) ∨ x _odotd1 (x) ⊙ d₂ (y) = d₂ (x) d₁ (y) ∨ d₁ (x) ⊙ d₂ (y). Repl-ace x by d₂ (x), we get d 2 ( d 2 ( x ) ) ⊙ d 1 ( y ) ∨ d 1 ( d 2 ( x ) ) ⊙ d 2 ( y ) = d 2 2 ( x ) d 1 ( y ) . Now, by the Theorem 3.14, we have d₁ (x) =0 or d 2 2 ( x ) = 0 for all x ∈ L, further by Theorem 3.15, we have d₂ (x) =0 for all x ∈ L.

(2) ⇒ (1) It is clear.

In what follows, we introduce the corresponding ideals of multiplicative semilattices with prefect derivations, which are called prefect derivation ideals. Moreover, we prove that the set of all prefect derivations ideals on commutative multiplicative semilattices with prefect derivations can form a complete Heyting algebra.

Definition 3.17. Let (L, d) be a commutative multiplicative semilattice with a prefect derivation and I be an ideal of L. Then I is called a prefect derivation ideal of (L, d) if x ∈ I implies d (x) ∈ I for allx ∈ L.

We will denote the set of all prefect derivation ideal of a commutative multiplicative semilattice with prefect derivation (L, d) by DI [L, d].

Example 3.18. Considering the commutative multiplicative semilattice with a prefect derivation (L, d) in Example 3.8, one can easily check that the prefect derivation ideals of (L, d) are {0},{a, 1} and L. Moreover, one can see that {0, a} is an ideal of L, but it is not a prefect derivation ideal of (L, d).

In the following, we will show that the annihilator of nonempty subset is a prefect derivation ideal in a reduced commutative multiplicative semilattice with a with prefect derivation.

Proposition 3.19. Let (>L, d) be a reduced commutative multiplicative semilattice with prefect derivations. Then for any nonempty subset S of L, Ann (S) is a prefect derivation ideal of (L, d).

Proof. From Proposition 2.8(1), one can see that Ann (S) is an ideal of L. Now, we will prove that Ann (S) is a prefect derivation ideal of (L, d). If x ∈ Ann (S), then 0 = d (x ⊙ s) = (d (x) ⊙ s) ∨ (x ⊙ d (s)) for any s ∈ S. Multiplying by s from right, we have 0 = d (x ⊙ s) ⊙ s = (d (x) ⊙ s ⊙ s) ∨ (x ⊙ d (s) ⊙ s). Since x ⊙ s = 0 and s ⊙ x = 0 and hence s ⊙ d (x) ⊙ s = 0. Also, multiplying by d (x) from right, we get s ⊙ d (x) ⊙ s ⊙ d (x) =0, that is, d (x) ⊙ x = 0. This means that d (x) ∈ Ann (S). Thus, Ann (S) is a prefect derivation ideal of (L, d).

Open problem. For any prefect derivation ideal I of a reduced commutative multiplicative semilattice with prefect derivation (L, d), whether there exists a nonempty subset X of L such that Ann (X) = I.

Theorem 3.20. Let (>L, d) be a commutative multiplicative semilattice with prefect derivations. Then (DI [L, d], ∧, ∨, →) is a complete Heyting algebra, where for all I₁, I₂ ∈ DI [L, d], ∧_i∈ΛI _i = ∩ _i∈ΛI _i , ∨_i∈II _i = {x ∈ L|x ≤ x _{i 1} ∨ x _{i 2} ∨ ⋯ ∨ x _{i m} , i _j ∈ I _i , 1 ≤ j ≤ m}, I₁ → I₂ = {x ∈ L|a ⊙ d (x) ∈ I₂, for all a ∈ I₁}.

Proof. Suppose that {I _i } _i∈Λ is a family of prefect derivation ideals of (L, d). It is easy to check that the infimum of {I _i } _i∈Λ is ∧_i∈ΛI _i = ∩ _i∈ΛI _i and the supremum is ∨_i∈II _i = {x ∈ L|x ≤ x _{i 1} ∨ x _{i 2} ∨ ⋯ ∨ x _{i m} , i _j ∈ I _i , 1 ≤ j ≤ m}. Therefore, (DI [L, d], ∧, ∨) is a complete lattice. Next, we define I₁ → I₂ = {x ∈ L|a ⊙ d (x) ∈ I₂, for all a ∈ I₁} for any I₁, I₂ ∈ DI [L, d]. And we shall prove that I₁ ∩ I ⊆ I₂ if and only if I ⊆ I₁ → I₂ for all I₁, I₂ ∈ DI [L, d], that is, (DI [L, d], ∧, ∨, →) is a complete Heyting algebra. In order to do this, we first show that I₁ → I₂ is a prefect derivation ideal of (L, d).

Now, let x, y ∈ I₁ → I₂. Then a ⊙ d (x) ∈ I₁ and a ⊙ d (y) ∈ I₁ for all a ∈ I₁. Hence (a ⊙ d (x)) ∨ (a ⊙ d (y)) ∈ I₁. Hence a ⊙ d (x ∨ y) = a ⊙ (d (x) ∨ d (y)) = (a ⊙ d (x)) ∨ (a ⊙ d (y)) ∈ I₁ for all a ∈ I₂, we have a ⊙ d (x ∨ y) ∈ I₁ for all a ∈ I₁. Thus, x ∨ y ∈ I₁ → I₂. On other hand, let y ∈ I₁ → I₂ and y ∈ L such that x ≤ y. Then d (x) ≤ d (y). It follows that a ⊙ d (x) ≤ a ⊙ d (y) ∈ I₂ for all a ∈ I₁, that is, x ∈ I₁ → I₂. Moreover, for all x ∈ I₁ → I₂ and y ∈ L, we have a ⊙ d (x ⊙ y) ≤ a ⊙ d (x) ∈ I₂ and a ⊙ d (y ⊙ x) ≤ a ⊙ d (x) ∈ I₂ for all a ∈ L, that is, x ⊙ y, y ⊙ x ∈ I₁ → I₂. Finally, for all x ∈ I₁ → I₂, we have a ⊙ dd (x) = a ⊙ d (x) for all a ∈ L, that is, d (x) ∈ I₁ → I₂. Therefore, I₁ → I₂ is a prefect derivation ideal of (L, d).

Finally, we prove that I₁ ∩ I₂ ⊆ I if and only if I₁ ⊆ I₂ → I for any I₁, I₂ ∈ di [L, d]. First, suppose that x ∈ I₁, then d (x) ∈ I₁. For any a ∈ I₂, we have a ⊙ d (x) ≤ d (x), a ⊙ d (x) ≤ a and hence a ⊙ d (x) ∈ I₁ ∩ I₂ ⊆ I, that is, x ∈ I₂ → I. Thus, I₁ ⊆ I₂ → I. Conversely, assume that I₁ ⊆ I₂ → I and x ∈ I₁ ∩ I₂, then x ∈ I₂ → I. For any a ∈ F₂, we have a ⊙ d (x) ∈ I. Taking a = x ∈ I₂, we have d (x) = x ⊙ d (x) ∈ I and hence x ∈ I. Thus, I₁ ⊆ I₂ → I.

Therefore, (DI [L, d], ∧, ∨, →) is a complete Heyting algebra.

Let (L, d) be a commutative multiplicative semilattice with a derivation and I be a derivation ideal. We define a mapping d _I : L/I → L/I such that d _I ([x]) = [dx], for any x ∈ L.

Proposition 3.21. Let (>L, d) be a commutative multiplicative semilattice with a derivation and I be a derivation ideal of (L, d). Then (L/I, d _I ) is a multiplicative semilattice with a derivation.

Proof. Let (L, d) be a commutative multiplicative semilattice with a derivation and I be a derivation ideal of (L, d). Define a mapping d _I : L/I → L/I by d _I ([x]) = [d (x)] for any [x] ∈ L/I.

First, we show that d _I is well defined. In fact, if x, y ∈ L such that [x] = [y], then x ∈ [y] and hence d (x) ∈ [d (y)] by d (I) ⊆ I, that is, [d (x)] = [d (y)]. Thus, d _I (x) = d _I (y). Therefore, d _I is well defined.

Next, let [x], [y] ∈ L/I. Then d _I ([x] ∨ [y]) = d _I ([x ∨ y]) = [d (x ∨ y)] = [d (x) ∨ d (y)] = [d (x)] ∨ [d (y)] = d _I (x) ∨ d _I (y). Also, we have (d _I ([x]) ⊙ [y]) ∨ ([x] ⊙ d _I [y]) = [(d (x) ⊙ y) ∨ (x ⊙ d (y))], on the other hand, we have d _I ([x] ⊙ [y]) = [d (x ⊙ y)]. Thus, d _I ([x] ⊙ [y]) = d _I ([x]) ⊙ [y]) ∨ ([x] ⊙ d _I [y]). One can easy to check that d _I is a prefect derivation on L. Therefore, (L/I, d _I ) is a commutative multiplicative semilattice with a derivation.

Theorem 3.22. Let (>L, d) be a commutative multiplicative semilattice with a derivation. Then there exist a one to one correspondence between DI [L, d] and DI [L/I, d _I ].

Proof. Considering the following map: u : DI [ L , d ] ⟶ DI [ L / I , d I ] , u ( I ) = J / I

Since J is a prefect derivation ideal of (L, d), so J/I is a prefect derivation ideal of (L/I, d _I ). Hence the map u is well defined.

Let M, N ∈ DI [L, d] such that u (M) = u (N). So M/I = N/I. For any x ∈ M, we have [x] ∈ M/I = N/I, so [x] = [y] for some y ∈ N. That is x ∈ [y] ⊆ N. Hence M ⊆ N. The proof of N ⊆ M is similar to that of M ⊆ N. Hence M = N. That is, the function u is an epic function. It is clear to see that u is a monomorphic function. Therefore u is a bijective map between DI [L, d] and DI [L/I, d _I ]

Corollary 3.23. Let (>L, d) be a commutative multiplicative semilattice with a prefect derivation and I be a prefect derivation ideal. Then (d _I ) (L/I) is a subalgebra of L/I.

Proof. It follows from Proposition 3.21.

Proposition 3.24. Let (>L, d) be a commutative multiplicative semilattice with a prefect derivation and I be a prefect derivation ideal. Then I ∩ d (L) = d (I).

Proof. It is clear from d (I) ⊆ I and d (I) ⊆ I ∩ d (L). Conversely, suppose that a ∈ I ∩ d (L). There exists x ∈ L, such that a = d (x) ∈ I. It follows that a = d (x) = dd (x) = d (a) ∈ d (I) and hence I ∩ d (L) = d (I).

Theorem 3.25. Let (>L, d) be a commutative multiplicative semilattice with a prefect derivation and I be a prefect derivation ideal. Then (d _I ) (L/I) is isomorphic to d (L)/d (I).

Proof. Let α : d (L) → (d _I ) (L/I) defined by α (d (x)) = d _I ([x]) = [d (x)]. It is clear that α is well defined and surjective. Moreover, it is easy to show that α is a homomorphism from d (L) and (d _I ) (L/I). It follows from the homomorphism theorem that d (L)/Ker (α) ≅ (d _I ) (L/I). For the kernel Ker(α) of α, we get that d (x) ∈ Ker (α) ⇔ α (d (x)) =0/I and d (x) ∈ d (L) ⇔ d (x)/I = 0/I and d (x) ∈ d (L) ⇔ d (x) ∈ I and d (x) ∈ d (L) ⇔ d (x) ∈ I ∩ d (L) ⇔ d (x) ∈ d (I).

4 Principal derivations and their applications

In this section, we investigate principal derivations in commutative multiplicative semilattices. Then, we show that the structure of an idempotent commutative quantale is completely determined by its set of all principal derivations.

In what follows, let L be a commutative multiplicative semilattice and a ∈ L, we define a map d _a : L → L as follows: d _a (x) = a ⊙ x for all x ∈ L.

Proposition 4.1. Let L be a commutative multiplicative semilattice and a ∈ L. Then the map d _a is an derivation on L, which is called a principal derivation on L.

Proof. For all x, y ∈ L, we have d _a (x ∨ y) = a ⊙ (x ∨ y) = (a ⊙ x) ∨ (a ⊙ y) = d _a (x) ∨ d _a (y). Moreover, d _a (x ⊙ y) = (a ⊙ x ⊙ y) ∨ (a ⊙ x ⊙ y) = (d _a (x) ⊙ y) ∨ (x ⊙ d _a (y). Thus, d _a is a derivation on L.

In what follows, we focus on algebraic structure the set of all principal derivations. We denote by D (L) = {d _a |a ∈ L} be the set of all principal derivations of L.

Theorem 4.2. Let L be an idempotent commutative multiplicative semilattice. Then (D (L), ⊔, ⊗, d₀, d₁) is an idempotent commutative multiplicative semilattice, where (d _a ⊔ d _b ) x = (d _a x) ∨ (d _b x), (d _a ⊗ d _b ) x = (d _a x) ⊙ (d _b x), for any d _a , d _b ∈ D (L).

Proof. Now, we show that (D (L), ⊔, ⊗, d₀, d₁) is a commutative multiplicative semilattice. For all d _a , d _b ∈ D (L) and x ∈ L, we have (d _a ⊔ d _b ) (x) = (d _a (x) ∨ d _b (x)) = (a ⊙ x) ∨ (b ⊙ x) = (a ∨ b) odotx = d_a∨b (x), that is, (d _a ⊔ d _b ) = d_a∨b and hence d _a ⊔ d _b ∈ D (L). Moreover, for all d _a ∈ D (L) and x ∈ L, we have (d _a ⊔ d₀) (x) = d _a (x) ∨ d₀ (x) = a ⊙ x = d _a (x), (d _a ⊔ d₁ (x) = d _a (x) ∨ d₁ (x) = x = d₁ (x). Thus, (D (L), ⊔, d₀, d₁) is a joint semilattice. Also, we have (d _a ⊗ d _b ) (x) = (d _a (x)) ⊙ (d _b (x)) = (a ⊙ x) ⊙ (b ⊙ x) = a ⊙ b ⊙ x ⊙ x = (a ⊙ b) ⊙ x = d_a⊙b (x), that is, (d _a ⊗ d _b ) = d_a⊙b and hence d _a ⊗ d _b ∈ D (L). Furthermore, d _a ⊗ d₁) (x) = d _a (x) ⊙ d₁ (x) = a ⊙ x = d _a (x) and (d _a ⊗ d₀) (x) = d _a (x) ⊙ d₀ (x) =0 = d₀ (x). Thus, (D (L), ⊗, d₁) is a monoid with d₀ as a bilaterally absorbing element.

Next, we prove that ⊗ is distributive with respect to the ⊔. For all d _a , d _b , d _c ∈ D (L) and x ∈ L, we have (d _a ⊗ (d _b ⊔ d _c )) (x) = d _a (x) ⊙ (d _b (x) ∨ d _c (x)) = (d _a (x) ⊙ d _b (x)) ∨ (d _a (x) ⊙ d _c (x)) = (d _a ⊗ d _b ) (x) ∨ (d _a ⊗ d _c ) (x) = ((d _a ⊗ d _b ) ⊔ (d _a ⊗ d _c )) (x), that is, d _a ⊗ (d _b ⊔ d _c ) = (d _a ⊗ d _b ) ⊔ (d _a ⊗ d _c ). In the simil-arity way, we can prove that (d _b ⊔ d _c ) ⊗ d _a = (d _b ⊗ d _a ) ⊔ (d _c ⊗ d _a ).

Finally, it is clear that (D (L), ⊗) is an idempotent commutative monoid.

Therefore, (D (L), ⊔, ⊗, d₀, d₁) is an idempotent multiplicative semilattice.

Corollary 4.3. Let L be an idempotent commutative quantale. Then (D (L), ⊔, ⊗, d₀, d₁) is an idempotent commutative quantale, where (d _a ⊔ d _b ) x = (d _a x) ∨ (d _b x), (d _a ⊗ d _b ) x = (d _a x) ⊙ (d _b x), for any d _a , d _b ∈ D (L).

Proof. For any {d _i } ⊆ D (L) and x ∈ L, we have ⊔ _i d _{a i} (x) = (∨ _i (d _{a i} )) (x) = ∨ _i (a _i ⊙ x) = (∨ _i a _i ⊙ x) = d_{∨ a i} (x). Also, d₀ is a smallest principal derivation on L, then (D (L), ∨, d₀, d₁) is a complete lattice. Finally, from Theorem 4.2, we obtain that (D (L), ⊔, ⊗, d₀, d₁) is an idempotent quantale.

Theorem 4.4. Let L be an idempotent commutative quantale. Then idempotent commutative quantale (L, ∨, ⊙, 0, 1) and (D (L), ⊔, ⊗, d₀, d₁) are isomorphic.

Proof. For all a ∈ L, let φ : L ⟶ D (L) be defined by φ (a) = d _a for all a ∈ L. Clearly, φ is a map from L to D (L), that is, φ is well defined. One can easily see that φ is one to one and onto.

Furthermore, for any a, b ∈ L, we have φ (a ⊙ b) = d_a⊙b = d _a ⊗ d _b = φ (a) ⊗ φ (b), φ (a ∨ b) = d_a∨b = d _a ⊔ d _b = φ (a) ⊔ φ (b).

Combining them, we obtain that φ is an isomorphism from L to D (L). Therefore, idempotent multiplicative semilattice. (L, ∨, ⊙, 0, 1) and (D (L), ⊔, ⊗, d₀, d₁) are isomorphic.

Corollary 4.5. Let L be an idempotent commutative complete multiplicative semilattice. Then idempotent quantale (L, ∨, ⊙, 0, 1) and (D (L), ⊔, ⊗, d₀, d₁) are isomorphic.

Proof. The proof is similar to that of Theorem 4.4.

5 Conclusions

Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. The theory of derivations of algebraic structures is a direct descendant of the development of classical Galois theory and the theory of invariants. In the paper, some useful properties of particular derivations in commutative multiplicative semilattices are discussed. Also, we obtain that the fixed point set of prefect derivations is still a commutative multiplicative semilattices. Besides, we prove that the set of all prefect derivations ideals on commutative multiplicative semilattices with prefect derivations can form a complete Heyting algebra. Finally, we show that the structure of an idempotent commutative quantale is completely determined by its set of all principal derivations.

In our future work, the following topics should be considered:

to find an isomorphism between the set of all prefect derivation ideals on commutative multiplicative semilattices with prefect derivations and the lattice of all ideals in the fixed point set for prefect derivations in commutative multiplicative semilattices.