Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 12 https://internationalpubls.com Endomorphisms and Their Role in Novel Derivations of Hilbert Algebras Aiyared Iampan1,*, R. Vennila2, Neelamegarajan Rajesh3, C. Arivazhagi4 1Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand. 27405 Goreway Drive, Mississauga L4T0A3, Canada. 3Department of Mathematics, Rajah Serfoji Government College (affiliated to Bharathidasan University), Thanjavur- 613005, Tamilnadu, India. 4Department of Mathematics, Government Arts and Science College, Peravurani-614804, Tamilnadu, India. E-mail: 1aiyared.ia@up.ac.th, 2vennilamaths@gmail.com, 3nrajesh_topology@yahoo.co.in, 4arivuniralya@gmail.com *Corresponding author: Aiyared Iampan Article History: Received: 10-09-2024 Revised: 16-11-2024 Accepted: 26-11-2024 Abstract: This study introduces and explores left- f -derivations, right- f -derivations, and f - derivations of type I and type II within the framework of Hilbert algebras, focusing on their fundamental properties and structural significance. We show that the kernel of an f -derivation, Ker ( )d A , forms a near filter and operates as a subalgebra of the Hilbert algebra A . These findings enhance the understanding of Hilbert algebras and offer new perspectives on the role of derivations in algebraic logic. Keywords: Hilbert algebra, near filter, subalgebra, left- f -derivation, right- f -derivation. 1. Introduction The concept of Hilbert algebras, introduced by Henkin in the early 1950s, emerged as a groundbreaking algebraic framework aimed at formalizing implications within intuitionistic and other non-classical logics [9]. Unlike classical logic, which operates on binary truth values, non-classical logic, including intuitionistic logic, requires more nuanced structures to capture their subtleties. Hilbert algebras provided a sophisticated tool for this purpose, offering an algebraic lens through which to analyze and comprehend the behaviour of implication in these alternative logical systems. By the 1960s, the fundamental importance of Hilbert algebras was further solidified through Diego's seminal work, demonstrating that Hilbert algebras constitute a locally finite variety [7]. This revelation not only underscored the structural richness of Hilbert algebras but also integrated them firmly within the broader domain of algebraic logic. Diego's results provided essential insights, laying a rigorous foundation for further explorations into their properties and applications. As a result, Hilbert algebras have become an essential structure in the study of non-classical logics, facilitating deeper inquiries into the interplay between algebra and logic. Building upon these foundational developments, subsequent research delved deeper into both the algebraic and logical aspects of Hilbert algebras, further expanding their theoretical and practical relevance. Notably, Busneag [4,5] and Jun [15] explored the critical role of filters within Hilbert algebras, establishing that these filters act as deductive systems fundamental to the underlying logical Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 13 https://internationalpubls.com structure. Their work unveiled profound connections between the algebraic properties of Hilbert algebras and their logical interpretations, offering a refined understanding of how these systems model non-classical logic. Additionally, Dudek's contributions [8] extended the versatility of Hilbert algebras through the concept of fuzzification, where the algebraic framework was adapted to accommodate fuzzy logic, characterized by truth values that exist on a continuum rather than binary distinctions. This innovation enriched the study of Hilbert algebras, enabling their application in domains that address uncertainty, partial truth, and vagueness—critical aspects of real-world reasoning systems. The inclusion of fuzzy subalgebras and deductive systems demonstrated the flexibility of Hilbert algebras and further reinforced their importance in contemporary logical and algebraic research. The study of derivations has seen remarkable progress in recent years, particularly in their application to diverse algebraic structures. In 2021, Muangkarn et al. [18] examined qf -derivations, while Bantaojai et al. [3] explored derivations induced by endomorphisms within B-algebras. This investigation revealed new insights into the interplay between derivations and algebraic morphisms, broadening the understanding of structural transformations in these systems. Continuing this trajectory, Bantaojai et al. [1,2] expanded their research in 2022 to encompass derivations on d -algebras and B- algebras, further enriching the theoretical framework and offering deeper perspectives on the behaviour of derivations in more complex algebraic systems. Simultaneously, Muangkarn et al. [17,19] focused on the structural implications of derivations induced by endomorphisms in BG-algebras and d -algebras, shedding light on the intricate relationships between these operations and the underlying algebraic properties. Additionally, Iampan et al. [10,20,21] made substantial contributions by studying derivations in UP-algebras, emphasizing the versatility and broad applicability of derivation theory across various algebraic frameworks. Building on this rich foundation, Iampan et al. [11,13] introduced and rigorously developed the concepts of ( , )l r -derivations, ( , )r l -derivations, and general derivations within the context of Hilbert algebras. Their work not only advanced the understanding of derivations in these specific algebraic structures but also illuminated their logical implications. Subsequently, their investigation into the relationship between derivations and endomorphisms opened up new pathways for exploring the structural properties of Hilbert algebras. This research has significantly enriched the algebraic theory surrounding Hilbert algebras, contributing valuable insights into the role of derivations in both classical and non-classical logical systems. Collectively, these studies have deepened our comprehension of derivations across a wide array of algebraic structures, laying a robust foundation for future research in this dynamic area of algebraic theory. This paper introduces and investigates left- f -derivations, right- f -derivations, and f -derivations of type I and type II within the framework of Hilbert algebras. We explore the fundamental properties of these derivations, highlighting their structural roles and interactions. Notably, we establish that the kernel of an f -derivation, Ker ( )d A , forms a near filter and functions as a subalgebra of the Hilbert algebra A . These results not only deepen our understanding of the algebraic structure of Hilbert algebras but also pave the way for future research into the broader implications of derivations in algebraic logic and their potential applications. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 14 https://internationalpubls.com 2. Preliminaries Before proceeding, it's important to revisit the concept of Hilbert algebras, introduced by Diego in 1966 [7]. As a key structure in algebraic logic, they capture the behaviour of implication in non- classical systems, making them central to further studies of derivations and algebraic operations. Definition 2.1. [7] A Hilbert algebra is a triplet with the formula ( , ,1)A A=  , where A is a nonempty set,  is a binary operation, and 1 is a fixed member of A that is true according to the axioms stated below: ( , )( ( ) 1)x y A x y x    = (2.1) ( , , )(( ( )) (( ) ( )) 1)x y z A x y z x y x z        = (2.2) ( , )( 1 and 1 )x y A x y y x x y   =  =  = (2.3) In [8], the following conclusion was established. Lemma 2.2. Let ( , ,1)A A=  be a Hilbert algebra. Then (1) ( )( 1)x A x x   = , (2) ( )(1 )x A x x   = , (3) ( )( 1 1)x A x   = , (4) ( , , )( ( ) ( ))x y z A x y z y x z    =   , (5) ( , , )(( ) (( ) ( )) 1)x y z A x z z y x y       = . In a Hilbert algebra ( , ,1)A A=  , the binary relation  is defined by ( , )( 1)x y A x y x y     = , which is a partial order on A with 1 as the largest element. Definition 2.3. [16] A nonempty subset D of a Hilbert algebra ( , ,1)A A=  is called a subalgebra of A if x y D  for all ,x y D . Definition 2.4. [6] A nonempty subset D of a Hilbert algebra ( , ,1)A A=  is called an ideal of A if the following conditions hold: (1) 1 D , (2) ( , )( )x y A y D x y D      , (3) 1 2 1 2 1 2( , , )( , ( ( )) )x y y A y y D y y x x D        . Definition 2.5. [12] A nonempty subset D of a Hilbert algebra ( , ,1)A A=  is called a near filter of A if the following conditions hold: (1) 1 D , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 15 https://internationalpubls.com (2) ( , )( )x y A y D x y D      . Definition 2.6. [12] A nonempty subset D of a Hilbert algebra ( , ,1)A A=  is called a filter of A if the following conditions hold: (1) 1 D , (2) ( , )( , )x y A x y x D y D      . Definition 2.7. Let ( , ,1 )AA A=  and ( , ,1 )BB B= å be Hilbert algebras. A function :f A B→ is called a homomorphism if ( ) ( ) ( )f x y f x f y = å for all ,x y A . Now, (1 ) (1 1 ) (1 ) (1 ) 1A A A A A Bf f f f=  = =å . A homomorphism :f A A→ is said to be an endomorphism. For any ,x y in a Hilbert algebra ( , ,1)A A=  , we define x y by ( )y x x  . By Lemma 2.2 (4), we can prove that x y is an upper bound of x and y . That is, ( , )( ( ) 1)x y X x x y    = , (2.4) ( , )( ( ) 1)x y X y x y    = . (2.5) A Hilbert algebra ( , ,1)A A=  is said to be  -commutative [14] if for all , , ( ) ( )x y A y x x x y y   =   , that is, x y y x =  . From [14], we know that ( )( )x X x x x   = , (2.6) ( )( 1 1 1)x X x x   =  = . (2.7) 3. Left and right- f -derivations of type I In this section, we introduce the concepts of left- f -derivations, right- f -derivations, and f - derivations of type I in Hilbert algebras, examining their fundamental properties. We then focus on analyzing the subset Ker ( )d A associated with a left (or right) f -derivation of type I , highlighting its structural significance within the algebra. Definition 3.1. Let ( , ,1)A A=  be a Hilbert algebra and f be an endomorphism of A . A self-map :d A A→ is called a left- f -derivation of type I of A if it satisfies the identity ( ) ( ( ) ( )) ( )d x y d x f y x y =    for all ,x y A . Similarly, a self-map :d A A→ is called a right- f -derivation of type I of A if it satisfies the identity ( ) ( ( ) ( )) ( )d x y f x d y x y =    for all ,x y A . Moreover, if d is a left- f -derivation of type I and a right- f -derivation of type I of A , it is called an f -derivation of type I of A . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 16 https://internationalpubls.com Example 3.2. Let {1, 2,3, 4}A = 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 1 1 4 4 1 1 3 1  Then ( , ,1)A  is a Hilbert algebra. We define an endomorphism f on A as follows: 1 2 3 4 1 1 3 4 f   =     Define a self-map 1 :d A A→ as follows: 1 1 2 3 4 1 1 3 4 d   =     Hence, 1d is a left- f -derivation of type I of A . Define a self-map 2 :d A A→ as follows: 2 1 2 3 4 1 2 1 1 d   =     Hence, 1d is a right- f -derivation of type I of A . Definition 3.3. A self-map d of a Hilbert algebra ( , ,1)A A=  is called regular if (1) 1d = . Theorem 3.4. In a Hilbert algebra ( , ,1)A A=  , the following statements hold: (1) every left- f -derivation of type I of A is regular, (2) every right- f -derivation of type I of A is regular. Proof. (1) Assume that d is a left- f -derivation of type I of A . Then (1) (1 1)d d=  [Lemma 2.2 (1)] ( (1) (1)) (1 1)d f=    ( (1) 1) 1d=   [Lemma 2.2 (1)] 1= . [(2.7)] Hence, d is regular. (2) Assume that d is a right- f -derivation of type I of A . Then (1) (1 1)d d=  [Lemma 2.2 (1)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 17 https://internationalpubls.com ( (1) (1)) (1 1)f d=    (1 (1)) 1d=   [Lemma 2.2 (1)] 1= . [(2.7)] Hence, d is regular. Corollary 3.5. Every f -derivation of type I of a Hilbert algebra A is regular. Theorem 3.6. In a Hilbert algebra ( , ,1)A A=  , the following statements hold: (1) if d is a left- f -derivation of type I of A , then ( ) ( )d x f x x=  for all x A , (2) if d is a right- f -derivation of type I of A , then ( ) ( )d x d x x=  for all x A . Proof. (1) Assume that d is a left- f -derivation of type I of A . Then, for all x A , ( ) (1 )d x d x=  [Lemma 2.2 (2)] ( (1) ( )) (1 )d f x x=    (1 ( )) (1 )f x x=    [regular] ( )f x x=  . [Lemma 2.2 (2)] (2) Assume that d is a right- f -derivation of type I of A . Then, for all x A , ( ) (1 )d x d x=  [Lemma 2.2 (2)] ( (1) ( )) (1 )f d x x=    (1 ( )) (1 )d x x=    ( )d x x=  . [Lemma 2.2 (2)] Corollary 3.7. If d is an f -derivation of type I of A , then ( ) ( ) ( )d x f x x d x x=  =  for all x A . Proposition 3.8. Let d be a left- f -derivation of type I of a Hilbert algebra ( , ,1)A A=  . Then the following properties hold: for any ,x y A , (1) ( ) ( )f x d x , (2) ( ) ( ) ( )d x f y d x y   , (3) ( ( )) ( ( )) ( ( ))d x f x f f x d d x   , (4) ( ) ( ) ( )d y x f x d x y    , (5) ( ) ( ) ( )d x d x f x=  . Proof. (1) For all x A , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 18 https://internationalpubls.com ( ) ( ) ( ) ( ( ) )f x d x f x f x x =   [Theorem 3.6 (1)] 1= . [(2.4)] Hence, ( ) ( )f x d x for all x A . (2) For all ,x y A , ( ( ) ( )) ( ) ( ( ) ( )) (( ( ) ( )) ( ))d x f y d x y d x f y d x f y x y   =      1= . [(2.4)] Hence, ( ) ( ) ( )d x f y d x y   for all ,x y A . (3) For all x A , ( ( ( )) ( ( ))) ( ( )) ( ( ( )) ( ( ))) ( ( ) )d x f x f f x d d x d x f x f f x d f x x   =     [Theorem 3.6 (1)] ( ( ( )) ( ( ))) (( ( )) ( ))d x f x f f x d x f x f x=      1= . [(2)] Hence, ( ( )) ( ( )) ( ( ))d x f x f f x d d x   for all x A . (4) For all ,x y A , ( ( ) ( )) ( ) ( ( ) ( )) (( ) )d y x f x d x y d y x f x d y x x    =      1= . [(2)] Hence, ( ) ( ) ( )d y x f x d x y    for all ,x y A . (5) For all x A , ( ) 1 ( )d x d x=  [Lemma 2.2 (2)] ( ( ) ( )) ( )f x d x d x=   [(1)] ( ) ( )d x f x=  . Proposition 3.9. Let d be a right- f -derivation of type I of a Hilbert algebra ( , ,1)A A=  . Then the following properties hold: for any ,x y A , (1) ( )x d x , (2) ( ) ( ) ( )f x d y d x y   , (3) ( ( )) ( ( )) ( ( ))f x d x d d x d d x   , (4) ( ) ( ) ( )f y x d x d x y    . Proof. (1) For all x A , ( ) ( ( ) )x d x x d x x =   [Theorem 3.6 (2)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 19 https://internationalpubls.com 1= . [(2.5)] Hence, ( )x d x for all x A . (2) For all ,x y A , ( ( ) ( )) ( ) ( ( ) ( )) (( ( ) ( )) ( ))f x d y d x y f x d y f x d y x y   =      1= . [(2.4)] Hence, ( ) ( ) ( )f x d y d x y   for all ,x y A . (3) For all x A , ( ( ( )) ( ( ))) ( ( )) ( ( ( )) ( ( ))) ( ( ) )f x d x d d x d d x f x d x d d x d d x x   =     [Theorem 3.6 (2)] ( ( ( )) ( ( ))) (( ( )) ( ))f x d x d d x d x d x d x=      1= . [(2)] Hence, ( ( )) ( ( )) ( ( ))f x d x d d x d d x   for all x A . (4) For all ,x y A , ( ( ) ( )) ( ) ( ( ) ( )) (( ) )f y x d x d x y f y x d x d y x x    =      1= . [(2)] Hence, ( ) ( ) ( )f y x d x d x y    for all ,x y A . Definition 3.10. Let d be a self-map of a Hilbert algebra ( , ,1)A A=  . We define the kernel Ker ( )d A of A as follows: Ker ( ) { : ( ) 1}d A x A d x=  = Theorem 3.11. If d is a right- f -derivation of type I of a Hilbert algebra ( , ,1)A A=  , then Ker ( )dy x A  for all Ker ( )dy A and x A . Proof. Assume that d is a right- f -derivation of A . Let Ker ( )dy A and x A . Then ( ) 1d y = . Thus, ( ) (( ) )d y x d x y y =   ( ( ) ( )) (( ) )f x y d y x y y=      ( ( ) 1) (( ) )f x y x y y=      1 (( ) )x y y=    [Lemma 2.2 (3)] 1= . [(2.7)] Hence, Ker ( )dy x A  . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 20 https://internationalpubls.com Theorem 3.12. If d is a right- f -derivation of type I of a commutative Hilbert algebra ( , ,1)A A=  and for any ,x y A is such that y x and Ker ( )dy A , then Ker ( )dx A . Proof. Assume that d is a right- f -derivation of type I of A . Let ,x y A be such that y x and Ker ( )dy A . Then 1y x = and ( ) 1d y = . Thus, ( ) (1 )d x d x=  [Lemma 2.2 (2)] (( ) )d y x x=   (( ) )d x y y=   [commutative] ( ( ) ( )) (( ) )f x y d y x y y=      ( ( ) 1) (( ) )f x y x y y=      1 (( ) )x y y=    [Lemma 2.2 (3)] 1= . [(2.7)] Hence, Ker ( )dx A . Theorem 3.13. If d is a right- f -derivation of type I of a Hilbert algebra ( , ,1)A A=  , then Ker ( )d A is a near filter (subalgebra) of A . Proof. Assume that d is a right- f -derivation of type I of A . By Theorem 3.4 (2), we have (1) 1d = and so 1 Ker ( )d A   . Let x A and Ker ( )dy A . Then ( ) 1d y = . Thus, ( ) ( ( ) ( )) ( )d x y f x d y x y =    ( ( ) 1) ( )f x x y=    1 ( )x y=   [Lemma 2.2 (3)] 1= . [(2.7)] Hence, Ker ( )dx y A  , so Ker ( )d A is a near filter of A . 4. Left and right- f -derivations of type II Building on the concepts discussed earlier, in this section, we redefine left- f -derivations, right- f - derivations, and f -derivations of type II in Hilbert algebras, examining their fundamental properties. We then analyze the subset Ker ( )d A associated with these derivations, emphasizing its structural importance within the algebra. Definition 4.1. Let ( , ,1)A A=  be a Hilbert algebra and f be an endomorphism of A . A self-map :d A A→ is called a left- f -derivation of type II of A if it satisfies the identity ( ) ( ) ( ( ) ( ))d x y x y d x f y =    for all ,x y A . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 21 https://internationalpubls.com Similarly, a self-map :d A A→ is called a right- f -derivation of type II of A if it satisfies the identity ( ) ( ) ( ( ) ( ))d x y x y f x d y =    for all ,x y A . Moreover, if d is a left- f -derivation of type II and a right- f -derivation of type II of A , it is called an f -derivation of type II of A . Example 4.2. Let {1, 2,3, 4}A = 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 1 1 1  Then ( , ,1)A  is a Hilbert algebra. We define an endomorphism f on A as follows: 1 2 3 4 1 3 2 4 f   =     Define a self-map 3 :d A A→ as follows: 3 1 2 3 4 1 1 1 4 d   =     Hence, 3d is a left- f -derivation of type II and a right- f -derivation of type II of A , so it is an f - derivation of type II of A . Theorem 4.3. In a Hilbert algebra ( , ,1)A A=  , the following statements hold: (1) every left- f -derivation of type II of A is regular, (2) every right- f -derivation of type II of A is regular. Proof. (1) Assume that d is a left- f -derivation of type II of A . Then (1) (1 1)d d=  [Lemma 2.2 (1)] (1 1) ( (1) (1))d f=    1 ( (1) 1)d=   [Lemma 2.2 (1)] 1= . [(2.7)] Hence, d is regular. (2) Assume that d is a right- f -derivation of type II of A . Then (1) (1 1)d d=  [Lemma 2.2 (1)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 22 https://internationalpubls.com (1 1) ( (1) (1))f d=    1 (1 (1))d=   [Lemma 2.2 (1)] 1= . [(2.7)] Hence, d is regular. Corollary 4.4. Every f -derivation of type II of a Hilbert algebra A is regular. Theorem 4.5. In a Hilbert algebra ( , ,1)A A=  , the following statements hold: (1) if d is a left- f -derivation of type II of A , then ( ) ( )d x x f x=  for all x A , (2) if d is a right- f -derivation of type II of A , then ( ) ( )d x x d x=  for all x A . Proof. (1) Assume that d is a left- f -derivation of type II of A . Then, for all x A , ( ) (1 )d x d x=  [Lemma 2.2 (2)] (1 ) ( (1) ( ))x d f x=    (1 ) (1 ( ))x f x=    [regular] ( )x f x=  . [Lemma 2.2 (2)] (2) Assume that d is a right- f -derivation of type II of A . Then, for all x A , ( ) (1 )d x d x=  [Lemma 2.2 (2)] (1 ) ( (1) ( ))x f d x=    (1 ) (1 ( ))x d x=    ( )x d x=  . [Lemma 2.2 (2)] Corollary 4.6. If d is an f -derivation of type II of A , then ( ) ( ) ( )d x x f x x d x=  =  for all x A . Proposition 4.7. Let d be a left- f -derivation of type II of a Hilbert algebra ( , ,1)A A=  . Then the following properties hold: for any x A , (1) ( ),x d x (2) ( ) ( )d x d x x=  . Proof. (1) For all x A , ( ) ( ( ))x d x x x f x =   [Theorem 4.5 (1)] 1= . [(2.4)] Hence, ( )x d x for all x A . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 23 https://internationalpubls.com (2) For all x A , ( ) 1 ( )d x d x=  [Lemma 2.2 (2)] ( ( )) ( )x d x d x=   [(1)] ( )d x x=  . Proposition 4.8. Let d be a right- f -derivation of type II of a Hilbert algebra ( , ,1)A A=  . Then the following properties hold: for any x A , (1) ( )x d x , (2) ( ) ( )d x d x x=  . Proof. (1) For all x A , ( ) ( ( ))x d x x x d x =   [Theorem 4.5 (2)] 1= . [(2.4)] Hence, ( )x d x for all x A . (2) For all x A , ( ) 1 ( )d x d x=  [Lemma 2.2 (2)] ( ( )) ( )x d x d x=   [(1)] ( )d x x=  . Theorem 4.9. If d is a right- f -derivation of type II of a Hilbert algebra ( , ,1)A A=  , then Ker ( )dy x A  for all Ker ( )dy A and x A . Proof. Let Ker ( )dy A and x A . Then ( ) 1d y = . Thus, ( ) (( ) )d y x d x y y =   (( ) ) ( ( ) ( ))x y y f x y d y=      ( ) ( ( ) 1)y x f x y=     ( ) 1y x=   [Lemma 2.2 (3)] 1= . [(2.7)] Hence, Ker ( )dy x A  . Theorem 4.10. If d is a right- f -derivation of type II of a commutative Hilbert algebra ( , ,1)A A=  and for any ,x y A is such that y x and Ker ( )dy A , then Ker ( )dx A . Proof. Let ,x y A be such that y x and Ker ( )dy A . Then 1y x = and ( ) 1d y = . Thus, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 24 https://internationalpubls.com ( ) (1 )d x d x=  [Lemma 2.2 (2)] (( ) )d y x x=   (( ) )d x y y=   [commutative] (( ) ) ( ( ) ( ))x y y f x y d y=      ( ) ( ( ) 1)y x f x y=     ( ) 1y x=   [Lemma 2.2 (3)] 1= . [(2.7)] Hence, Ker ( )dx A . Theorem 4.11. If d is a right- f -derivation of type II of a Hilbert algebra ( , ,1)A A=  , then Ker ( )d A is a near filter (subalgebra) of A . Proof. Assume that d is a right- f -derivation of type II of A . By Theorem 4.3 (2), we have (1) 1d = and so 1 Ker ( )d A   . Let x A and Ker ( )dy A . Then ( ) 1d y = . Thus, ( ) ( ) ( ( ) ( ))d x y x y f x d y =    ( ) ( ( ) 1)x y f x=    ( ) 1x y=   [Lemma 2.2 (3)] 1= . [(2.7)] Hence, Ker ( )dx y A  , so Ker ( )d A is a near filter of A . 5. Conclusion This study introduced and investigated left- f -derivations, right- f -derivations, and f -derivations of type I and type II within the framework of Hilbert algebras, focusing on their fundamental properties and structural roles. We demonstrated that the kernel of an f -derivation, Ker ( )d A , forms a near filter and functions as a subalgebra of the Hilbert algebra A . These results deepened the understanding of the algebraic structure of Hilbert algebras and provided new insights into the broader implications of derivations in algebraic logic. Future research could extend these concepts to other non-classical algebraic systems and explore their applications in logical frameworks and computational logic models. Acknowledgement This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567). 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