t-derivations on complicated subtraction algebras

Journal of Discrete Mathematical Sciences and Cryptography

ISSN: 0972-0529 (Print) 2169-0065 (Online) Journal homepage: http://www.tandfonline.com/loi/tdmc20

t-derivations on complicated subtraction algebras Chiranjibe Jana, Tapan Senapati & Madhumangal Pal To cite this article: Chiranjibe Jana, Tapan Senapati & Madhumangal Pal (2017) t-derivations on complicated subtraction algebras, Journal of Discrete Mathematical Sciences and Cryptography, 20:8, 1583-1595, DOI: 10.1080/09720529.2017.1308663 To link to this article: https://doi.org/10.1080/09720529.2017.1308663

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Journal of Discrete Mathematical Sciences & Cryptography ISSN 0972-0529 (Print), ISSN 2169-0065 (Online) Vol. 20 (2017), No. 8, pp. 1583–1595 DOI : 10.1080/09720529.2017.1308663

t-derivations on complicated subtraction algebras Chiranjibe Jana * Department of Applied Mathematics with Oceanology & Computer Programming Vidyasagar University Midnapore 721102 West Bengal India Tapan Senapati † Department of Mathematics Padima Janakalyan Banipith Kukurakhupi 721517 West Bengal India Madhumangal Pal § Department of Applied Mathematics with Oceanology & Computer Programming Vidyasagar University Midnapore 721102 West Bengal India Abstract In this paper, the notion of t-derivation, generalized t-derivation are introduced on complicated subtraction (c-subtraction) algebra, which is a generalization of subtraction algebra and investigated their properties in details. Subject Classification: 06F35, 03G25 Keywords: Subtraction algebra, c-subtraction algebra, derivation, t-derivation, isotone derivation. *E-mail: [email protected] † E-mail: [email protected] (Corresponding Author) § E-mail: [email protected]

©

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C. JANA, T. SENAPATI AND M. PAL

1. Introduction Schein [16] cogitated the system of the form (F, o, \), where F is a set of functions closed under the composition “o” of functions (and hence (F, o) is a functions of semigroup), and set theoretic subtraction “\” (and hence (F, \) is a subtraction algebra in the sense of Abbott [1]). He suggested a problem concerning the structure of multiplication in a subtraction semigroup. It was explained by Zelinka [26], and he had solved the problem for subtraction algebras of a special type known as the “atomic subtraction algebras”. Kim et al. [12] showed that a subtraction algebra is equivalent to an implicative BCK-algebra, and a subtraction semigroup is a special case of a BCI-semigroup which is a generalization of a ring. Senapati together with colleagues [2, 4, 5, 17-25] has done lot of works on BCK/BCI-algebras and related algebraic systems. Jun et al. [6-8] introduced the notion of ideals, prime ideals and irreducible ideals in subtraction algebras and investigated their characterizations. Again, Jun et al. [6-8] introduced the notion of complicated subtraction algebras and investigated some of its properties. Ceven and Öztürk [3] introduced the notion of additional concepts on subtraction algebras, so called subalgebra, bounded subtraction algebra, union of subtraction algebras and investigated some of its related properties. Prince Willams et al. [15] established fuzzy soft ideals in subtraction algebras. Öztürk and Ceven [14] introduced the notion of derivation on subtraction algebras. Recently, Khan et al. [10] have studied (∈, ∈ ∨ qk ) -intuitionistic fuzzy (soft) ideals of subtraction algebras and have discussed their properties in details. To the best of our knowledge no works are available on t-derivation on c-subtraction algebras. For this reason we are motivated to develop the theories on t-derivation on c-subtraction algebras. In this paper, the notion of t-derivations, which is a generalization of derivation in c-subtraction algebras is introduced and studied related properties of t-derivation on c-subtraction algebras. Also, generalized t-derivations are given and discussed their related properties. 2. Subtraction Algebra In this section we recall some of the basic concepts of subtraction algebra which will be very helpful in further study of the paper. Throughout the paper X denotes the subtraction algebra unless otherwise specified. By a subtraction algebra we mean an algebra (X, –) with a single binary operation “–” that satisfies the following identities, for any x, y, z ŒX,

COMPLICATED SUBTRACTION ALGEBRAS

(S1) (S2) (S3)

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x − ( y − x) = x x − (x − y ) = y − ( y − x) ( x − y ) − z = ( x − z ) − y.

The last identity permits us to omit parenthesis in expressions of the form (x – y) – z. The subtraction determines an order relation on X : a ≤ b ⇔ a − b = 0, where 0 = a – a is an element that does not depend on the choice of a Œ X. The ordered set (X, £) is a semi-Boolean algebra in the sense of [1], that is, it is a meet semilattice with zero (0) in which every interval [0, a] is a Boolean algebra with respect to the induced order. Here a ∧ b = a − (a − b), the complement of an element b Œ [0, a] is a – b. In a subtraction algebra, for any x, y, z Œ X, the following are true [7, 8] ( p1) ( p2) ( p3) ( p 4) ( p5) ( p6) ( p7) ( p8) ( p9) ( p10) ( p11) ( p12)

(x − y ) − y = x − y x − 0 = x and 0 − x = 0 (x − y ) − x = 0 x − (x − y ) ≤ y (x − y ) − ( y − x) = x − y x − (x − (x − y )) = x − y (x − y ) − ( z − y ) ≤ x − z x ≤ y if and only if x = y − w for some w ∈ X x ≤ y implies x − z ≤ y − z and z − y ≤ z − x x , y ≤ z implies x − y = x ∧ ( z − y ) (x ∧ y ) − (x ∧ z) ≤ x ∧ ( y − z) (x − y ) − z = (x − z) − ( y − z).

A non-empty subset S of a subtraction algebra X is called a subalgebra of X if x – y Œ S for all x, y Œ X. A mapping f from a subtraction algebra X to a subtraction algebra Y is called a homomorphism of X if f(x – y) = f(x) – f(y) for all x, y Œ X. A homomorphism f from a subtraction algebra X to itself is called an endomorphism of X. If x £ y implies f(x) £ f(y), then f is called an isotone map. Definition 2.1: [7] A nonempty subset S of a subtraction algebra X is called an ideal of X if it satisfies (a) 0 Œ S (b) for all x Œ X, y Œ S and x – y Œ S implies x Œ S.

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For an ideal S of a subtraction algebra X, we have that x £ y and y Œ S imply that x Œ S for any x, y Œ X [8]. Lemma 2.2: [13] Let X be a subtraction algebra. Then (a) x ∧ y = y ∧ x for any z Œ X (b) x – y £ x for any x, y Œ X. Definition 2.3: [9] Let X be a subtraction algebra. For any a, b Œ X, let G(a, b= ) {x|x − a ≤ b}. Then X is said to be complicated subtraction algebra (c-subtraction algebra) if for any a, b Œ X, the set G(a, b) has the greatest element. Note that 0, a, b Œ G(a, b). The greatest element of G(a, b) is denoted by a + b. Proposition 2.4 [9] If X is a c-subtraction algebra, then for all x, y, z Œ X, the followings are true (a) x £ x + y and y £ x + y (b) x + 0 = 0 = 0 + x (c) x + y = y + x (d) x £ y ﬁ x + z £ y + z (e) x £ y ﬁ x + y = y (f) x + y is the least upper bound of x and y. Definition 2.5 [3] Let X be a subtraction algebra. Then X is called bounded subtraction algebra if there is an element 1 of X satisfying the condition x £ 1 for all x Œ X. 3. t-derivation of c-subtraction algebras The following definitions introduce the notion of t-derivation of c-subtraction algebras. Definition 3.1: [4] Let X be a c-subtraction algebra and d is a selfmap on X. Then d is called a derivation of X if it satisfies the identity d(x ∧ y= ) (d(x ) ∧ y ) + (x ∧ d( y )) for all x, y Œ X.


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Example 3.2: [9] Let X = {0, a, b, c} be a c-subtraction algebra with the following Caley table: –

0

a

b

c

0

0

0

0

0

a

a

0

a

0

b

b

b

0

0

c

c

b

a

0

In a complicated subtraction algebra X, 0 + 0 = 0, 0 + a = a, 0 + b = b, 0 + c = c, a + a = a, a + b = c, a + c = c, b + b = b, b + c = c, and c + c = c. Define a mapping d : X Æ X by

0, if x = 0, a d(x ) =  b, if x = b, c. Then it is easy to verify that d is a derivation on X. Now, we define the t-derivation on c-subtraction algebras X.

Definition 3.3: Let X be a c-subtraction algebra. Then for any t Œ X, we define a self-map dt : X Æ X by dt (x ) = t ∧ x for all x Œ X. Definition 3.4: Let X be a c-subtraction algebra. Then for any t Œ X, a self-map dt : X Æ X is called t-derivation of X if it satisfy the condition dt (x ∧ y= ) (dt (x ) ∧ y ) + (x ∧ dt ( y )) for all x, y Œ X. Example 3.5: For t Œ X, define a self-map dt : X Æ X of a subtraction algebra X in Example 3.2. Define the mapping dt as follows, for t = 0, dt(x) = 0 for all x Œ X for t = a, dt(0) = 0, dt(a) = a, dt(b) = 0, dt(c) = a for t = b, dt(0) = 0, dt(a) = 0, dt(b) = b, dt(c) = b for t = c, dt(0) = 0, dt(a) = a, dt(b) = b, dt(c) = c. Hence, for each t Œ X, dt is a t-derivation of X. Example 3.6: In Example 3.2, for any t Œ X, defining a self-map dt : X Æ X by

0, if x = 0, b dt (x ) =t ∧ x = c , if x = a, c.

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Then, it is routine to checked that dt is not a t-derivation on X since dt (a ∧ b) = dt (0) = 0 ≠ (dt (a) ∧ b) + (a ∧ dt (b)) = (c ∧ b) + (a ∧ 0) = b + 0 = b. Theorem 3.7: Let X be a c-subtraction algebra and dt be a t-derivation on X. Then the following conditions hold ) dt (x ) ∧ x ≤ x for all x Œ X (a) dt (x= (b) If x £ X then dt(x + y) = dt(y) for all x, y Œ X (c) dt (G(a, b)) ⊆ G(a, b) (d) G(dt (a), dt (b)) ⊆ G(a, b). Proof: (a) Since x Ÿ x = x, dt(x) = dt(x Ÿ x) = dt(x) Ÿ x + x Ÿ dt(x) = dt(x) Ÿ x = dt(x) – (dt(x) – x) = x – (x – dt(x)) £ x by (S2) and (p4). (b) We have x + y = y, so we get dt (x + y ) = dt ( y ). (c) For all x Œ G(a, b), we have x – a £ b. From Theorem 3.7 (a), we have dt(x) £ x and dt (x ) − a ≤ x − a ≤ b by (p9). Therefore, dt (x ) ∈ G(a, b). (d) For all x ∈ G(dt (a), dt (b)), we have x − dt (a) ≤ dt (b) ≤ b. Then, x – b £ dt(a) £ a, so x – a £ b. Hence x Œ G(a, b).  Corollary 3.8: Let X be a c-subtraction algebra and dt be a t-derivation on X. Then the following properties hold (a) dt (a + b) ≤ a + b

(b) dt (a) + dt (b) ≤ a + b. Proof: (a) It is trivial from Theorem 3.7(a). (b) Since, the greatest element of G(dt (a), dt (b)) is dt (a) + dt (b) and the greatest of G(a, b) is a + b, then we get dt (a) + dt (b) ≤ a + b, by using the Theorem (c).  Theorem 3.9: Let dt be a t-derivation on a c-subtraction algebra X and I is an ideal of X. Then dt ( I ) ⊆ I . Proof: Let x be a c-subtraction algebra and dt be a t-derivation on X. Then for all x Œ X, dt(x) £ x. Therefore, dt (x ) − x = 0 ∈ I . By definition of an ideal of X, we have dt(x) Œx. 


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Theorem 3.10: Let X be a c-subtraction algebra and dt be a t-derivation on X. Then (i) d2t = dt, for all t Œ X, (ii) dt(0) = 0, (iii) dt(t) = t, for all t Œ X, (iv) for t = 0, d0 (x ∧ y ) = 0, for all x, y Œ X,

(v) dt (x ) ∧ dt ( y ) ≤ dt (x ∧ y ) ≤ dt (x ) + dt ( y ), for all x, y Œ X. Proof: Let dt be a t-derivation on c-subtraction algebra X. 2 (x ) dt (dt (= x )) dt (dt (x ) ∧= x ) (dt2 (x ) (i) Since dt2 (x ) ≤ dt (x ) ≤ x , we get dt= 2 ∧ x ) + (dt (x ) ∧ dt (x= )) dt2 (x ) + dt (x= ) dt (x ). Hence, dt (x ) = dt (x ).

(ii) Since 0 ∧ 0 = 0, then for any t Œ X, we have dt (0 ∧ 0) = (dt (0) ∧ 0) + (0 ∧ dt (0)) =((t ∧ 0) ∧ 0) + (0 ∧ (t ∧ 0)) =(0 ∧ 0) + (0 ∧ 0) =0 + 0 =0

t) (dt (t) ∧ t) + (t ∧ dt (t)) (iii) Since t ∧ t = t , then for all t Œ X, we get dt (t ∧ = =((t ∧ t) ∧ t) + (t ∧ (t ∧ t)) =(t ∧ t) + (t ∧ t) =t + t =t. Hence, dt(t) = t for all t Œ X. (d0 (x ) (iv) Let x, y Œ X. Then, by applying (p2) for t = 0, we get d0 (x ∧ y ) = ∧ y ) + (x ∧ d0 ( y )) =((0 ∧ x ) ∧ y ) + (x ∧ (0 ∧ y )) =(0 ∧ y ) + (x ∧ 0) =0 + 0 =0. (v) Since dt(y) £ y, for any t Œ X, we have dt (x ) − y ≤ dt (x ) − dt ( y ) and dt (x ) − (dt (x ) − dt ( y )) ≤ dt (x ) − (dt (x ) − y ), hence, we have dt (x ) ∧ dt ( y ) ≤ dt (x ) ∧ y. Similarly, since dt(x) £ x, we have dt ( y ) − x ≤ dt ( y ) − dt (x ) and dt ( y ) − (dt ( y ) − dt (x )) ≤ dt ( y ) − (dt ( y ) − x ), that is, dt ( y ) ∧ dt (x ) ≤ dt (x ) ∧ dt ( y ) ≤ d( y ) ∧ x. Hence, we get dt (x ) ∧ dt ( y ) ≤ dt (x ) ∧ y + x ∧ dt = ( y ) dt (x ∧ y ). Also, since dt (x ) ∧ y ≤ dt (x ) and dt ( y ) ∧ x ≤ dt ( y ) by (p4), hence, we get dt (x ∧ y ) = dt (x ) ∧ y + x ∧ dt ( y ) ≤ dt (x ) + x ∧ dt ( y ) ≤ dt (x ) + dt ( y ). This completes the proof.  Theorem 3.11: Let X be a c-subtraction algebra and dt be a t-derivation on X. Let 1 be the greatest element of X. Then for all z Œ X the following conditions hold (a) If x £ dt(1) then dt(x) = x (b) If x ≥ dt(1), then dt(1) £ dt(x) (c) If x £ y and dt(y) = y, then dt(x) = x. Proof: Let X be a c-subtraction algebra and dt be a t-derivation on X. Then for any t Œ X, we have since dt (x )= dt (x ∧ 1)= (dt (x ) ∧ 1) + (x ∧ dt (1)) = dt (x ) + (x ∧ dt (1)), we have x ∧ d(1) ≤ dt (x ). Then

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(a) if x £ dt(1) since x - dt(1) = 0, we have x =x − (x − dt (1)) =x ∧ dt (1) ≤ dt (x ) and dt(x) = x by Theorem 3.7 (a). (b) if x ≥ dt(1), since dt(1) – x = 0, we get dt (1) − (dt (1) − x= ) dt (1) ∧ x= x ∧ dt (1) ≤ dt (x ).

(c) if x £ y, since x = x ∧ y , we get dt (x )= dt (x ∧ y )= (dt (x ) ∧ y ) + (x ∧ dt ( y ))= dt (x ) + (x ∧ y= ) dt (x ) + = x x.  4. Isotone derivation In physics, two nuclides are isotones if they have the very same neutron number N, but different proton number Z. In biochemistry, an isotonic solution is one in which its effective osmole concentration is the same as the solute concentration of a cell. In calculus, a function f define on a subset of real numbers with real values is called monotonic if it is either entirely non-increasing or non-decreasing. It is called monotonically increasing if for all real x, y such that x £ y one has f(x) £ f(y). Likewise a function f is called monotonically decreasing if whenever x £ y, then f(x) ≥ f(y). But, isotone in derivational approach in algebraic system defined below. Definition 4.1: Let X be a c-subtraction algebra and d be a derivation on X. If x £ y implies d(x) £ d(y), then d is called an isotone derivation. Theorem 4.2: Let X be a c-subtraction algebra and dt be a t-derivation on X. Then for any t Œ X and for all x, y Œ X, then followings hold, (a) If dt (x ∧ y ) = dt (x ) ∧ dt ( y ), then dt is an isotone t-derivation (b) If dt (x + y ) = dt (x ) + dt ( y ), then dt is an isotone t-derivation. Proof: Let X be a c-subtraction algebra and dt be a t-derivation on X. Then for any t Œ X, we have (a) Let x £ y. Then by (p4), dt (x )= dt (x ∧ y )= dt (x ) ∧ dt ( y ) ≤ dt ( y ). (b) Let x £ y. Then since x + y = y from Proposition 2.4 (e), dt(y) = dt(x + y) = dt(x) + dt(y). Hence, we get dt(x) £ dt(y).  Theorem 4.3: Let X be a c-subtraction algebra with greatest element 1 and dt be a t-derivation on X. Then dt(1) = 1 if and only if dt is an identity t-derivation. Proof: Let dt be a t-derivation on c-subtraction algebra X with greatest element 1. We assume that dt(1) = 1. Now, by using the Theorem 3.11


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(c), for all x Œ X, since x £ 1, then x ∧ 1 = x. We get dt (x )= dt (x ∧ 1)= (dt (x ) ∧ 1) + (x ∧ dt (1))= dt (x ) + (x ∧ 1)= dt (x ) + x= x. This completes the proof.  Theorem 4.4: Let dt be a t-derivation on subtraction algebra X with greatest element 1. Then the followings are equivalent (a) dt is an isotone t-derivation (b) dt (x) = x ∧ dt (1) (c) dt (x ∧ y) = dt (x) ∧ dt (y) (d) dt (x) + dt (y) ≤ dt (x + y). Proof: Let dt be a t-derivation on subtraction algebra X. (a) ⇒ (b). Since dt is isotone t-derivation then dt (x ) ≤ dt (1). Also, dt(x) £ x, then dt (x ) ≤ x ∧ dt (1). By Theorem 3.11, dt (x ) = dt (x ) + (x ∧ dt (1)) = x ∧ dt (1). (b) ⇒ (c). Assume that (b) holds, then dt (x ) ∧ dt ( y ) = (x ∧ dt (1)) ∧ ( y ∧ dt (1)) = x ∧ y ∧ dt (1) = dt (x ∧ y ).

(c) ⇒ (a). Assume that (c) holds. Let x £ y, then x= x ∧ y so we get dt (x ) = dt (x ∧ y ) = dt (x ) ∧ dt ( y ) by (c), therefore, dt (x ) = dt (x ) ∧ dt ( y ) that is dt is an isotone t-derivation. (a) ⇒ (d). Assume that (a) holds. We have dt (x ) ≤ dt (x + y ) and dt ( y ) ≤ dt (x + y ), so we get dt (x ) + dt ( y ) ≤ dt (x + y ).

(d) ⇒ (a). Let x Œ y, then dt (x ) + dt ( y ) ≤ dt (x + y ) = dt ( y ). Then dt(x) + dt(y) = dt(y) which implies that dt(x) £ dt(y).  Theorem 4.5: Let dt be a t-derivation on a c-subtraction algebra X. Then the followings hold (a) dt is the identity derivation (b) dt(x + y) = (dt(x) + y)) Ÿ(x + dt(y)) (c) dt is one-to-one t-derivation (d) dt is onto t-derivation. Proof: Let dt be a t-derivation on c-subtraction algebra. (a) ⇒ (b) is clear. (a) ⇒ (c) is straight forward. (a) ⇒ (d) is also trivial.

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(b) ⇒ (a). Let y = x, by (b) we get dt (x + x ) = (dt (x ) + x ) ∧ (x + dt (x )). Since dt(x) £ x, we have dt (x ) = x ∧ x = x. (c) ⇒ (a). Let dt be a one-to-one t-derivation. If there exists x Œ X such that dt(x) π x, then dt(x) £ x. Let it be dt(x) = x1. Then x1 Œ x so, we get dt (x1 ) = dt (x1 ∧ x ) = (dt (x1 ) ∧ x ) + (x1 ∧ dt (x )) = dt (x1 ) + x1 = x1 = dt (x ). Since x1 π x which contradicts that dt is one-to-one t-derivation. (d) ⇒ (a). We assume that dt is onto t-derivation i.e. dt(X) = X. Then, for every x Œ X, there exists y Œ X such that x = dt(y). By Theorem 3.10(i), we 2 (x ) dt (dt= ( y )) d= ( y ) d= ( y ) x , which imply that dt is an identity get d= t t t t-derivation.  Definition 4.6 [14] Let X be a c-subtraction algebra. A self-map D : X Æ X is called generalized derivation if there exists a derivation determined (D(x ) by self-map d : X Æ X if it satisfies the following identity D(x ∧ y ) = ∧ y ) + (x ∧ d( y )) for all x, y Œ X. Example 4.7: In Example 3.2, defining a mapping D : X Æ X by

0, if x = 0  = D(x ) = a, if x a b, if x = b, c.  Also, defining a mapping dt : X Æ X by

0, if x = 0, c  = d(x ) = a, if x a b, if x = b. 

Then it is easy to check that D is a generalized derivation on X determined by the derivation d. Definition 4.8 Let X be a c-subtraction algebra. Then for any t Œ X, define a self-map Dt : X Æ X such that Dt(x) = x Ÿ t for all x Œ X. Definition 4.9: Let X be a c-subtraction algebra. A self-map Dt : X Æ X is called a generalized t-derivation on X if there exists a t-derivation determined by self-map dt : X Æ X if it satisfies the following identity Dt (x ∧ = y ) (Dt (x ) ∧ y ) + (x ∧ dt ( y )), where for all x, y Œ X and for any t Œ X. Example 4.10: In Example 3.2, for any t Œ X, defining a self-map Dt : X Æ X


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0, if x = 0  Dt (x ) = x ∧ t = a, if x = a c , if x = b, c. 

Also, for any t Œ X, defining a self-map dt : X Æ X by

0, if x = 0, c  dt (x ) =t ∧ x = a, if x = a b, if x = b. 

We can easily checked that dt is a t-derivation on X. But, Dt (b ∧ a) = Dt (0) = 0 ≠ Dt (b) ∧ a + b ∧ dt (a) = (c ∧ a) + (b ∧ a) = a + 0 = a, so it not a generalized t-derivation determined by t-derivation dt. Proposition 4.11: Let X be a c-subtraction algebra and Dt be a generalized t-derivation determined by dt. Then we have Dt (dt (x )) ≤ dt (x ) ≤ x for all x Œ X. Proof: Let X be a c-subtraction algebra. Since dt2 (x ) ≤ dt (x ) ≤ x and by S2 and (P 4) Dt (dt (x )) − (Dt (dt (x )) − dt (x )) ≤ dt (x ). We get, D= (dt (x )) Dt (dt (x ) ∧ t dt (= x )) (Dt (dt (x )) ∧ dt (x )) + (dt (x ) ∧ dt2 (x )) ≤ dt (x ) + dt = (x ) dt (x ). Hence, Dt(dt (x)) £ dt(x) £ x for all x Œ X by Theorem 3.7 (a). 5. Conclusions and future work In this paper, we have considered the notion of t-derivations on complicated (c-subtraction) subtraction algebras and investigated some useful properties of t-derivations on c-subtraction algebras. In our opinion, these result can be similarly extended to the other algebraic structure such as B/BG/BF/MV/Q-algebras, d-algebras, KU-algebras, and Lie algebras. The study of t-derivations on different algebraic structures may have a lot of applications in different branches of theoretical physics, engineering and computer science etc. It is our hope that this work would serve as a foundation for further study in the theory of derivations of subtraction algebras. References [1] J.C. Abbott, Sets, Lattices and Boolean Algebras, Allyn and Bacon, Boston, 1969. 137 (1987), 31-35.

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Received January, 2015