Int. J. Anal. Appl. (2023), 21:39 Derivations of Hilbert Algebras Aiyared Iampan1,∗, R. Alayakkaniamuthu2, P. Gomathi Sundari2, N. Rajesh2 1Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2Department of Mathematics, Rajah Serfoji Government College (Affiliated to Bharathidasan University), Thanjavur 613005, Tamilnadu, India ∗Corresponding author: aiyared.ia@up.ac.th Abstract. In this paper, we introduce the notions of (l, r)-derivations, (r, l)-derivations, and derivations of Hilbert algebras and investigate some related properties. In addition, we define two subsets for a derivation d of a Hilbert algebra X, Kerd(X) and Fixd(X), and we also take a look at some of their characteristics. 1. Introduction and Preliminaries Logic algebras are a significant class of algebras among several other algebraic structures. The concept of Hilbert algebras was introduced in early 50-ties by Henkin [9] for some investigations of implication in intuitionistic and other non-classical logics. In 60-ties, these algebras were studied especially by Diego [7] from algebraic point of view. Diego [7] proved that Hilbert algebras form a variety which is locally finite. Hilbert algebras were treated by Busneag [4,5] and Jun [12] and some of their filters forming deductive systems were recognized. The study of derivations has continued, for example, in 2021, Muangkarn et al. [14] studied fq- derivations, and Bantaojai et al. [3] studied derivations induced by an endomorphism of B-algebras. In 2022, Bantaojai et al. [1, 2] studied derivations on d-algebras and B-algebras, and Muangkarn et al. [13,15] studied derivations induced by an endomorphism of BG-algebras and d-algebras. Iampan et al. [10,16,17] studied derivations on UP-algebras. Received: Feb. 21, 2023. 2020 Mathematics Subject Classification. 03G25. Key words and phrases. Hilbert algebra; (l, r)-derivation; (r, l)-derivation; derivation. https://doi.org/10.28924/2291-8639-21-2023-39 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-39 2 Int. J. Anal. Appl. (2023), 21:39 The concepts of (l, r)-derivations, (r, l)-derivations, and derivations of Hilbert algebras are intro- duced in this work along with several related features. In addition, we define two subsets for a derivation d of a Hilbert algebra X, Kerd(X) and Fixd(X), and we also take a look at some of their characteristics. Let’s go through the idea of Hilbert algebras as it was introduced by Diego [7] in 1966 before we start. Definition 1.1. [7] A Hilbert algebra is a triplet with the formula X =(X, ·,1), where X is a nonempty set, · is a binary operation, and 1 is a fixed member of X that is true according to the axioms stated below: (1) (∀x,y ∈X)(x · (y ·x)=1), (2) (∀x,y,z ∈X)((x · (y ·z)) · ((x ·y) · (x ·z))=1), (3) (∀x,y ∈X)(x ·y =1,y ·x =1⇒ x = y). In [8], the following conclusion was established. Lemma 1.1. Let X =(X, ·,1) be a Hilbert algebra. Then (1) (∀x ∈X)(x ·x =1), (2) (∀x ∈X)(1 ·x = x), (3) (∀x ∈X)(x ·1=1), (4) (∀x,y,z ∈X)(x · (y ·z)= y · (x ·z)). In a Hilbert algebra X =(X, ·,1), the binary relation ≤ is defined by (∀x,y ∈X)(x ≤ y ⇔ x ·y =1), which is a partial order on X with 1 as the largest element. Definition 1.2. [18] A nonempty subset D of a Hilbert algebra X = (X, ·,1) is called a subalgebra of X if x ·y ∈D for all x,y ∈D. Definition 1.3. [6] A nonempty subset D of a Hilbert algebra X =(X, ·,1) is called an ideal of X if the following conditions hold: (1) 1∈D, (2) (∀x,y ∈X)(y ∈D ⇒ x ·y ∈D), (3) (∀x,y1,y2 ∈X)(y1,y2 ∈D ⇒ (y1 · (y2 ·x)) ·x ∈D). For any x,y in a Hilbert algebra X =(X, ·,1), we define x ∨y by (y ·x) ·x. Note that x ∨y is an upper bound of x and y for all x,y ∈X. A Hilbert algebra X =(X, ·,1) is said to be commutative [11] Int. J. Anal. Appl. (2023), 21:39 3 if for all x,y ∈X, (y ·x) ·x =(x ·y) ·y, that is, x ∨y = y ∨x. From [11], we know that (∀x ∈X)(x ∨x = x), (∀x ∈X)(x ∨1=1∨x =1). 2. Main Results In this section, we introduce the notions of an (l, r)-derivation, an (r, l)-derivation and a derivation of a Hilbert algebra and study some of their basic properties. Finally, we define two subsets Kerd(X) and Fixd(X) for a derivation d of a Hilbert algebra X, and we consider some properties of these as well. Definition 2.1. Let X = (X, ·,1) be a Hilbert algebra. A self-map d : X → X is called an (l, r)- derivation of X if it satisfies the identity d(x ·y)= (d(x)·y)∨(x ·d(y)) for all x,y ∈X. Similarly, a self- map d :X →X is called an (r, l)-derivation of X if it satisfies the identity d(x·y)= (x·d(y))∨(d(x)·y) for all x,y ∈X. Moreover, if d is both an (l, r)-derivation and an (r, l)-derivation of X, it is called a derivation of X. Example 2.1. Let X = {1,2,3,4} be a Hilbert algebra with a fixed element 1 and a binary operation · defined by the following Cayley table: · 1 2 3 4 1 1 2 3 4 2 1 1 3 4 3 1 2 1 4 4 1 2 3 1 Define a self-map d :X →X by for any x ∈X, d(x)= { 1 if x 6=2 2 if x =2. Then d is a derivation of X. Definition 2.2. An (l, r)-derivation (resp., (r, l)-derivation, derivation) d of a Hilbert algebra X = (X, ·,1) is said to be regular if d(1)=1. Theorem 2.1. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) every (l, r)-derivation of X is regular, (2) every (r, l)-derivation of X is regular. Proof. (1) Assume that d is an (l, r)-derivation of X. Then d(1)= d(1 ·1)= (d(1) ·1)∨(1 ·d(1))= 1∨d(1)=1. Hence d is regular. 4 Int. J. Anal. Appl. (2023), 21:39 (2) Assume that d is an (r, l)-derivation of X. Then d(1) = d(1 ·1) = (1 ·d(1))∨ (d(1) ·1) = d(1)∨1=1. Hence d is regular. � Corollary 2.1. Every derivation of a Hilbert algebra X =(X, ·,1) is regular. Theorem 2.2. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X, then d(x)= x ∨d(x) for all x ∈X, (2) if d is an (r, l)-derivation of X, then d(x)= d(x)∨x for all x ∈X. Proof. (1) Assume that d is an (l, r)-derivation of X. Then for all x ∈ X, d(x) = d(1 · x) = (d(1) ·x)∨ (1 ·d(x))= (1 ·x)∨d(x)= x ∨d(x). (2) Assume that d is an (r, l)-derivation of X. Then for all x ∈X, d(x)= d(1 ·x)= (1 ·d(x))∨ (d(1) ·x)= d(x)∨ (1 ·x)= d(x)∨x. � Corollary 2.2. If d is a derivation of a Hilbert algebra X =(X, ·,1), then d(x)∨x = x ∨d(x) for all x ∈X. Definition 2.3. Let d be an (l, r)-derivation (resp., (r, l)-derivation, derivation) of a Hilbert algebra X =(X, ·,1). We define a subset Kerd(X) of X by Kerd(X)= {x ∈X : d(x)=1}. Proposition 2.1. Let d be an (l, r)-derivation of a Hilbert algebra X = (X, ·,1). Then the following properties hold: for any x,y ∈X, (1) x ≤ d(x), (2) d(x) ·y ≤ d(x ·y), (3) d(x ·d(x))=1, (4) d(d(x) ·x)=1, (5) x ≤ d(d(x)). Proof. (1) For all x ∈X, x ·d(x)= x · (x ∨d(x))= x · ((d(x) ·x) ·x)=1. Hence x ≤ d(x). (2) For all x,y ∈X, (d(x) ·y) ·d(x ·y)= (d(x) ·y) · ((d(x) ·y)∨ (x ·d(y)))= (d(x) ·y) · (((x · d(y)) · (d(x) ·y)) · (d(x) ·y))=1. Hence d(x) ·y ≤ d(x ·y). (3) For all x ∈X, d(x ·d(x))= (d(x) ·d(x))∨ (x ·d(d(x)))=1∨ (x ·d(d(x)))=1. (4) For all x ∈X, d(d(x) ·x)= (d(d(x)) ·x)∨ (d(x) ·d(x))= (d(d(x)) ·x)∨1=1. (5) For all x ∈ X, d(d(x)) = d(x ∨ d(x)) = d((d(x) · x) · x) = (d(d(x) · x) · x)∨ ((d(x) · x) · d(x)) = (1 · x)∨ ((d(x) · x) · d(x)) = x ∨ ((d(x) · x) · d(x)) = (((d(x) · x) · d(x)) · x) · x. Thus x ·d(d(x))= x · ((((d(x) ·x) ·d(x)) ·x) ·x)=1. Hence x ≤ d(d(x)). � Proposition 2.2. Let d be an (r, l)-derivation of a Hilbert algebra X = (X, ·,1). Then the following properties hold: for any x,y ∈X, (1) x ·d(y)≤ d(x ·y), (2) d(x ·d(x))=1, Int. J. Anal. Appl. (2023), 21:39 5 (3) d(d(x) ·x)=1. Proof. (1) For all x,y ∈ X, (x ·d(y)) ·d(x · y) = (x ·d(y)) · ((x ·d(y))∨ (d(x) · y)) = (x ·d(y)) · (((d(x) ·y) · (x ·d(y))) · (x ·d(y)))=1. Hence x ·d(y)≤ d(x ·y). (2) For all x ∈X, d(x ·d(x))= (x ·d(d(x)))∨ (d(x) ·d(x))= (x ·d(d(x)))∨1=1. (3) For all x ∈X, d(d(x) ·x)= (d(x) ·d(x))∨ (d(d(x)) ·x)=1∨ (d(d(x)) ·x)=1. � Theorem 2.3. Let d1,d2, . . . ,dn be (l, r)-derivations of a Hilbert algebra X =(X, ·,1) for all n ∈N. Then x ≤ dn(dn−1(. . .(d2(d1(x))) . . .)) for all x ∈ X. In particular, if d is an (l, r)-derivation of X, then x ≤ dn(x) for all n ∈N and x ∈X. Proof. For n = 1, it follows from Proposition 2.1 (1) that x ≤ d1(x) for all x ∈ X. Let n ∈ N and assume that x ≤ dn(dn−1(. . .(d2(d1(x))) . . .)) for all x ∈X. Let Dn = dn(dn−1(. . .(d2(d1(x))) . . .)). Then dn+1(Dn)= dn+1(1 ·Dn) = (dn+1(1) ·Dn)∨ (1 ·dn+1(Dn)) = (1 ·Dn)∨ (1 ·dn+1(Dn)) =Dn ∨dn+1(Dn) = (dn+1(Dn) ·Dn) ·Dn. Thus Dn ·dn+1(Dn)=Dn · ((dn+1(Dn) ·Dn) ·Dn)=1. Therefore, Dn ≤ dn+1(Dn). By assumption, we get x ≤Dn ≤ dn+1(Dn)= dn+1(dn(dn−1(. . .(d2(d1(x))) . . .))) for all x ∈ X. Hence x ≤ dn(dn−1(. . .(d2(d1(x))) . . .)) for all n ∈ N and x ∈ X. In particular, put d = dn for all n ∈N. Hence x ≤ dn(dn−1(. . .(d2(d1(x))) . . .))= dn(x) for all n ∈N and x ∈X. � Definition 2.4. An ideal D of a Hilbert algebra X = (X, ·,1) is said to be invariant (with respect to an (l, r)-derivation (resp., (r, l)-derivation, derivation) d of X) if d(D)⊆D. Theorem 2.4. Every ideal of a Hilbert algebra X = (X, ·,1) is invariant with respect to any (l, r)- derivation of X. Proof. Let D be an ideal of X and d an (l, r)-derivation of X. Let y ∈ d(D). Then y = d(x) for some x ∈D. It follows that y ·x = d(x) ·x =1∈D, which implies y ∈D. Thus d(D)⊆D. Hence D is invariant with respect to an (l, r)-derivation d of X. � Corollary 2.3. Every ideal of a Hilbert algebra X =(X, ·,1) is invariant with respect to any derivation of X. 6 Int. J. Anal. Appl. (2023), 21:39 Theorem 2.5. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X, then y ∨x ∈Kerd(X) for all y ∈Kerd(X) and x ∈X, (2) if d is an (r, l)-derivation of X, then y ∨x ∈Kerd(X) for all y ∈Kerd(X) and x ∈X. Proof. (1) Assume that d is an (l, r)-derivation of X. Let y ∈Kerd(X) and x ∈X. Then d(y)=1. Thusd(y∨x)= d((x·y)·y)= (d(x·y)·y)∨((x·y)·d(y))= (d(x·y)·y)∨((x·y)·1)= (d(x·y)·y)∨1=1. Hence y ∨x ∈Kerd(X). (2) Assume that d is an (r, l)-derivation of X. Let y ∈Kerd(X) and x ∈X. Then d(y)=1. Thus d(y∨x)= d((x ·y)·y)= ((x ·y)·d(y))∨(d(x ·y)·y)= ((x ·y)·1)∨(d(x ·y)·y)=1∨(d(x ·y)·y)=1. Hence y ∨x ∈Kerd(X). � Corollary 2.4. If d is a derivation of a Hilbert algebra X = (X, ·,1), then y ∨ x ∈ Kerd(X) for all y ∈Kerd(X) and x ∈X. Theorem 2.6. In a commutative Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X and for any x,y ∈ X is such that y ≤ x and y ∈ Kerd(X), then x ∈Kerd(X), (2) if d is an (r, l)-derivation of X and for any x,y ∈ X is such that y ≤ x and y ∈ Kerd(X), then x ∈Kerd(X). Proof. (1) Assume that d is an (l, r)-derivation of X. Let x,y ∈ X be such that y ≤ x and y ∈Kerd(X). Then y ·x =1 and d(y)=1. Thus d(x)= d(1 ·x)= d((y ·x) ·x)= d((x ·y) ·y)= (d(x ·y)·y)∨((x ·y)·d(y))= (d(x ·y)·y)∨((x ·y)·1)= (d(x ·y)·y)∨1=1. Hence x ∈Kerd(X). (2) Assume that d is an (r, l)-derivation of X. Let x,y ∈X be such that y ≤ x and y ∈Kerd(X). Then y ·x =1 and d(y)=1. Thus d(x)= d(1 ·x)= d((y ·x) ·x)= d((x ·y) ·y)= ((x ·y) ·d(y))∨ (d(x ·y) ·y)= ((x ·y) ·1)∨ (d(x ·y) ·y)=1∨ (d(x ·y) ·y)=1. Hence x ∈Kerd(X). � Corollary 2.5. If d is a derivation of a commutative Hilbert algebra X =(X, ·,1) and for any x,y ∈X is such that y ≤ x and y ∈Kerd(X), then x ∈Kerd(X). Theorem 2.7. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X, then y ·x ∈Kerd(X) for all x ∈Kerd(X) and y ∈X, (2) if d is an (r, l)-derivation of X, then y ·x ∈Kerd(X) for all x ∈Kerd(X) and y ∈X. Proof. (1) Assume that d is an (l, r)-derivation of X. Let x ∈Kerd(X) and y ∈X. Then d(x)=1. Thus d(y ·x)= (d(y)·x)∨(y ·d(x))= (d(y)·x)∨(y ·1)= (d(y)·x)∨1=1. Hence y ·x ∈Kerd(X). (2) Assume that d is an (r, l)-derivation of X. Let x ∈Kerd(X) and y ∈X. Then d(x)=1. Thus d(y ·x)= (y ·d(x))∨(d(y)·x)= (y ·1)∨(d(y)·x)=1∨(d(y)·x)=1. Hence y ·x ∈Kerd(X). � Corollary 2.6. If d is a derivation of a Hilbert algebra X = (X, ·,1), then y · x ∈ Kerd(X) for all x ∈Kerd(X) and y ∈X. Int. J. Anal. Appl. (2023), 21:39 7 Theorem 2.8. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X, then Kerd(X) is a subalgebra of X, (2) if d is an (r, l)-derivation of X, then Kerd(X) is a subalgebra of X. Proof. (1) Assume that d is an (l, r)-derivation of X. By Theorem 2.1 (1), we have d(1) = 1 and so 1 ∈ Kerd(X) 6= ∅. Let x,y ∈ Kerd(X). Then d(x) = 1 and d(y) = 1. Thus d(x · y) = (d(x) · y)∨ (x · d(y)) = (1 · y)∨ (x · 1) = y ∨ 1 = 1. Hence x · y ∈ Kerd(X), so Kerd(X) is a subalgebra of X. (2) Assume that d is an (r, l)-derivation of X. By Theorem 2.1 (2), we have d(1) = 1 and so 1 ∈ Kerd(X) 6= ∅. Let x,y ∈ Kerd(X). Then d(x) = 1 and d(y) = 1. Thus d(x · y) = (x · d(y))∨ (d(x) · y) = (x · 1)∨ (1 · y) = 1∨ y = 1. Hence x · y ∈ Kerd(X), so Kerd(X) is a subalgebra of X. � Corollary 2.7. If d is a derivation of a Hilbert algebra X =(X, ·,1), then Kerd(X) is a subalgebra of X. Definition 2.5. Let d be an (l, r)-derivation (resp., (r, l)-derivation, derivation) of a Hilbert algebra X =(X, ·,1). We define a subset Fixd(X) of X by Fixd(X)= {x ∈X : d(x)= x}. Theorem 2.9. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X, then Fixd(X) is a subalgebra of X, (2) if d is an (r, l)-derivation of X, then Fixd(X) is a subalgebra of X. Proof. (1) Assume that d is an (l, r)-derivation of X. By Theorem 2.1 (1), we have d(1) = 1 and so 1 ∈ Fixd(X) 6= ∅. Let x,y ∈ Fixd(X). Then d(x) = x and d(y) = y. Thus d(x · y) = (d(x) ·y)∨ (x ·d(y))= (x ·y)∨ (x ·y)= x ·y. Hence x ·y ∈Fixd(X), so Fixd(X) is a subalgebra of X. (2) Assume that d is an (r, l)-derivation of X. By Theorem 2.1 (2), we have d(1) = 1 and so 1 ∈ Fixd(X) 6= ∅. Let x,y ∈ Fixd(X). Then d(x) = x and d(y) = y. Thus d(x · y) = (x ·d(y))∨ (d(x) ·y)= (x ·y)∨ (x ·y)= x ·y. Hence x ·y ∈Fixd(X), so Fixd(X) is a subalgebra of X. � Corollary 2.8. If d is a derivation of a Hilbert algebra X =(X, ·,1), then Fixd(X) is a subalgebra of X. Theorem 2.10. In a Hilbert algebra X =(X, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of X, then x ∨y ∈Fixd(X) for all x,y ∈Fixd(X), (2) if d is an (r, l)-derivation of X, then x ∨y ∈Fixd(X) for all x,y ∈Fixd(X). Proof. (1) Assume that d is an (l, r)-derivation of X. Let x,y ∈ Fixd(X). Then d(x) = x and d(y) = y. By Theorem 2.9 (1), we get d(y · x) = y · x. Thus d(x ∨ y) = d((y · x) · x) = (d(y ·x) ·x)∨((y ·x) ·d(x))= ((y ·x) ·x)∨((y ·x) ·x)= (y ·x) ·x = x∨y. Hence x∨y ∈Fixd(X). 8 Int. J. Anal. Appl. (2023), 21:39 (2) Assume that d is an (r, l)-derivation of X. Let x,y ∈Fixd(X). Then d(x)= x and d(y)= y. By Theorem 2.9 (2), we get d(y ·x)= y ·x. Thus d(x∨y)= d((y ·x)·x)= ((y·x)·d(x))∨(d(y·x)·x)= ((y ·x) ·x)∨ ((y ·x) ·x)= (y ·x) ·x = x ∨y. Hence x ∨y ∈Fixd(X). � Corollary 2.9. If d is a derivation of a Hilbert algebra X = (X, ·,1), then x ∨ y ∈ Fixd(X) for all x,y ∈Fixd(X). 3. Conclusion In this article, we introduced the ideas of (l, r)-derivations, (r, l)-derivations, and derivations of Hilbert algebras, and deduced their significant features. Additionally, two subsets Kerd(X) and Fixd(X) for a derivation d of a Hilbert algebra X are defined. As a result, we have found that Kerd(X) and Fixd(X) are subalgebras of X. Acknowledgment: This research project was supported by the Thailand Science Research and Inno- vation Fund and the University of Phayao (Grant No. FF66-UoE017). Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] T. Bantaojai, C. Suanoom, J. Phuto, A. Iampan, A Bi-endomorphism Induces a New Type of Derivations on B-Algebras, Italian J. Pure Appl. Math. 48 (2022), 336-348. [2] T. Bantaojai, C. Suanoom, J. Phuto, A. 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