ISSN: 2322-1666 print/2251-8436 online
RIGHT DERIVATIONS ON ORDERED SEMIGROUPS
M. MURALI KRISHNA RAO
Abstract.
Over the last few decades, several authors have
investi-gated the relationship between the commutativity of ring R and the
existence of certain specified derivations of
R.
In this paper, we
in-troduce the concept of right derivation on semigroups and we study
some of the properties of right derivation of semigroups. We prove
that if
d
be a non-zero right derivation of a cancellative ordered
semigroup
M,
then
M
is a commutative ordered semigroup.
Key Words: Ordered Semigroup, Right Derivation, Derivation, Negatively Ordered. 2010 Mathematics Subject Classification:Primary: 20M17, 06F05.
1. Introduction
Semigroup, as the basic algebraic structure was used in the areas of theoretical computer science as well as in the solutions of graph theory, optimization theory and in particular for studying automata, coding theory and formal languages. The first result in this deriva-tion is due to Posner [15] in 1957. In the year 1990, Bresar and Vukman [1] established that a prime ring must be commutative if it admits a nonzero left derivation. Kim [2,3,4 ] studied right derivation and generalized derivation of incline algebra. The notion of deriva-tion of algebraic structures is useful for characterizaderiva-tion of algebraic structures. The noderiva-tion of derivation has also been generalized in various directions such as right derivation, left derivation, f-derivation, reverse derivation, orthogonal derivation, (f, g)-derivation, general-ized right derivation, etc. Murali Krishna Rao and Venkateswarlu[7,8 ] introduced the notion of generalized right derivation of Γ−incline and right derivation of ordered Γ−semiring. Murali Krishna Rao [9-14 ] studied ideals of various aigebraic structures. In this paper, we introduce the concept of right derivation on semigroups and we study some of the properties of right derivation of semigroups.
Received: 22 December 2018, Accepted: 1 September 2019. Communicated by Ali Taghavi
∗Address correspondence to M. Murali Krishna Rao; E-mail: [email protected]. c
⃝2019 University of Mohaghegh Ardabili
.
1042. Preliminaries
In this section, we will recall some of the fundamental concepts and definitions necessary for this paper.
Definition 2.1. A semigroup is an algebraic system (S, .) consisting of a non-empty setS together with an associative binary operation′′·′′.
Definition 2.2. A non-empty subsetAof a semigroupMis called (i) a subsemigroup ofMifAA⊆A.
(ii) a quasi ideal ofM ifAisAM∩M A⊆A.
(iii) a bi-ideal ofM ifAis a subsemigroup ofMandAM A⊆A. (iv) an interior ideal ofM ifAis a subsemigroup ofMandM AM⊆A.
(v) a left (right) ideal ofM ifM A⊆A(AM⊆A). (vi) an ideal ifAM⊆AandM A⊆A.
(vii) a bi-quasi-interior ideal ofMifAis a subsemigroup ofM andAM AM A⊆A. ((viii) a bi-interior ideal ofMifAis a subsemigroup ofMandM AM∩AM A⊆A.
((ix) a left bi-quasi ideal (right bi-quasi ideal) ofM ifAis a subsemigroup andM A∩ AM A⊆A(AM∩AM A⊆A).
((x) a left quasi-interior ideal (right quasi-interior ideal) ofMifAis a subsemigroup of (M,+) andM AM A⊆A(AM AM⊆A).
Definition 2.3. LetMbe a semigroup. If there exists 1∈Msuch thata·1 = 1·a=a,for alla∈M,is called an unity element ofMthenM is said to be semigroup with unity. Definition 2.4. An elementaof a semigroupSis called a regular element if there exists an elementbofSsuch thata=aba.
Definition 2.5. A semigroup S is called a regular semigroup if every element of S is a regular element.
Definition 2.6. An elementaof a semigroupSis called an idempotent element ifaa=a. Definition 2.7. An elementbof a semigroupM is called an inverse element ofaofM if ab=ba= 1.
Definition 2.8. A non-empty subsetAof a semigroupMis called (i) a subsemigroup ofMifAA⊆A.
(ii) a left(right) ideal ofMifAis an subsemigroup ofM andM A⊆A(AM⊆A). (iii) an ideal if M A⊆AandAM⊆A.
Definition 2.9. A semigroupM is called a group if for each non-zero element ofM has multiplication inverse.
Definition 2.10. A semigroupM is called an ordered semigroup if it admits a compatible relation≤. i.e. ≤is a partial ordering onM satisfies the following conditions. Ifa≤band c≤dthen
(i) ac≤bd
(ii) ca≤db,for alla, b, c, d∈M
Definition 2.11. An ordered semigroupM is said to have zero element if there exists an element 0∈M such that 0x=x0 = 0, for allx∈M.
An ordered semigroupMis said to be commutative semigroup ifxy=yx,for allx, y∈M Definition 2.12. A non zero element ain an ordered semigroup M is said to be a zero divisor if there exists non zero elementb∈M,such thatab=ba= 0.
Definition 2.13. An ordered semigroup M with unity 1 and zero element 0 is called an integral ordered semigroup if it has no zero divisors.
Definition 2.14. An ordered semigroupM is said to be totally ordered semigroup if any two elements ofM are comparable.
Definition 2.15. In an ordered semigroupM
(i) the semigroupMis said to be positively ordered, ifa≤abandb≤ab,for alla, b∈ M.
(ii) the semigroupMis said to be negatively ordered ifab≤aandab≤b,for alla, b∈ M.
Definition 2.16. A non-empty subsetAof an ordered semigroupMis called a subsemigroup Mifab∈Afor alla, b∈A.
Definition 2.17. LetM be an ordered semigroup. A non-empty subsetIofM is called a left (right) ideal of an ordered semigroupM ifM I⊆I(IM⊆I) and if for anya∈M, b∈ I, a≤b⇒a∈I. Iis called an ideal ofM if it is both a left ideal and a right ideal ofM. Definition 2.18. LetM andN be ordered semigroups. A mappingf:M→N is called a homomorphism iff(ab) =f(a)f(b),for alla, b∈M.
Definition 2.19. LetM be an ordered semigroup. A mappingf :M →M is called an endomorphism if
(i) f is an onto ,
(ii) f(ab) =f(a)f(b),for alla, b∈M.
3. Right derivation of ordered semigroups
In this section, we introduce the concept of right derivation on semigroups and we study some of the properties of right derivation of semigroups
Definition 3.1. LetM be an ordered semigroup. If a mappingd:M →M satisfies the following conditions
(i) d(xy) =d(x)yd(y)x
(ii) Ifx≤ythend(x)≤d(y),for allx, y∈M. thendis called a right derivation ofM.
Example3.2. LetNbe a the set of all natural numbers. LetM =
{( a b c 0
)
|a, b, c∈N∪ {0} }
be additive semigroup with respect to usual multiplication of matrices. Define [aij]≤ [bij] if and only if aij ≤bij,for alli, j,where aij, bij∈M.ThenM is an ordered semigroup.
Defined:M→M byd (
c a b 0
) =
( 0 a b o
)
.Thendis a right derivation ofM. Theorem 3.3. LetM be a commutative idempotent ordered semigroup. Then for a fixed elementa∈M, the mapping da :M →M given by da(x) =xa,for all x∈ M is a right derivation ofM.
Proof. LetM be a commutative ordered semigroup anda∈M.Supposex, y∈M. da(xy) = (xy)a
= (xy)a(xy)a = (xa)y(ya)x =da(x)yda(y)x. Supposex≤y⇒xa≤ya
⇒da(x)≤da(y).
Hencedais a right derivation ofM. □
Theorem 3.4. Letdbe a right derivation of an ordered semigroupM.Thend(0) = 0 Proof. Letdbe a right derivation of the ordered semigroupM.Then
d(0) =d(00) =d(0)0d(0)0 = 00 = 0. Therefored(0) = 0
□ Theorem 3.5. Letdbe a right derivation of an idempotent negatively ordered semigroup M. Thend(x)≤x,for allx∈M.
Proof. Letdbe a right derivation of the idempotent negatively ordered semigroupM. Then d(x) =d(xx)
=d(x)xd(x)x
≤d(x)x
⇒d(x)≤d(x)x
⇒d(x)≤x.
Hence the theorem. □
Theorem 3.6. LetM be a negatively ordered semigroup . Thend(xy)≤d(x)d(y)). Proof. LetM be a negatively ordered semigroup. Supposex, y∈M.
Thend(x)y≤d(x) andd(y)x≤d(y).Therefore d(xy) =d(x)yd(y)x
≤d(x)d(y)
Hence the theorem. □
Theorem 3.7. Let M be an idempotent negatively ordered semigroup. Thend(xd(x)) ≤ d(x),for allx∈M.
Proof. LetMbe an idempotent negatively ordered semigroup. Then d(xd(x)) =d(x)d(x)d(d(x))x
≤d(x)d(x)
≤d(x). Therefored(xd(x))≤d(x).
□ Letdbe a right derivation of an ordered semigroupM.Definekerd={x∈M|d(x) = 0} Theorem 3.8. Let d be a right derivation of an idempotent negatively ordered integral semigroupM.Thenkerdis an ideal of an ordered semigroupM.
Proof. Letdbe a right derivation of the idempotent negatively ordered integral semigroup M.Suppose x, y ∈ kerd. Thend(x) = 0 andd(y) = 0 ⇒ d(xy) = 0. ⇒ xy ∈ kerd. Thereforekerdis a subsemigroup ofM.
Letx∈M andy∈kerdsuch thatx≤y.Thend(y) = 0.
⇒d(xy) =d(x)yd(y)x =d(x)y0x = 0 sincex≤y⇒xx≤xy
⇒d(xx)≤d(xy)
⇒d(xx)≤0.
⇒d(xx) = 0. Therefored(x) = 0
Hencex∈kerd.Thenkerdis an ideal of the ordered semigroupM. □ Theorem 3.9. Let M be a negatively ordered semigroup. Then d(xy) ≤ d(x), for all x, y∈M andd(xy)≤d(y).
Proof. LetMbe a negatively ordered semigroup. d(xy) =d(x)yd(y)x
≤d(x)d(y)
≤d(x).
Similarly we can proved(xy)≤d(y). □
Theorem 3.10. Letdbe a right derivation of a commutative idempotent -negatively ordered semigroupM.Define a setF ixd(M) ={x∈M|d(x) =x}.ThenF ixd(M)is an ideal of M.
Proof. Letdbe a right derivation of the commutative idempotent -negatively ordered semi-groupM.Supposex, y∈F ixd(M).Then
d(x) =x, d(y) =y d(xy) =d(x)yd(y)x
=xyyx
=xyxy=xy.Thereforexy∈F ixd(M)
ThereforeF ixd(M) is a subsemigroup ofM.Supposex≤yandy∈F ixd(M). x≤y
⇒xx≤xy
⇒x≤xy Thereforexy=x
⇒d(xy) =d(x)
⇒=d(x)yd(y)x=d(x)
⇒d(x)yyx=d(x)
⇒d(x)yx=d(x). Thereforex≤d(x)
By Theorem 3.5d(x)≤x..Therefored(x) =x.HenceF ixd(M) is an ideal ofM. □ Theorem 3.11. LetM be a negatively ordered semigroup with unity1 and dbe a right derivation ofM.Then
(i) d(1)x≥d(x)
(ii) Ifd(1) = 1thenx≥d(x).
Proof. LetM be a negatively ordered semigroup with unity 1, dbe a right derivation ofM andx∈M.Thenx1 =xand 1x=x.
d(x) =d(x1) =d(x)1d(1)x
⇒d(x) =d(x)1d(1)x≤d(1)x
Supposed(1) = 1 thend(1)x≥d(x).Therefore 1x≥d(x)⇒x≥d(x).Hence the theorem. □ Theorem 3.12. LetM be an idempotent negatively ordered semigroup with unity anddbe a right derivation ofM.Thend(1) = 1if and only ifd(x) =x.
Proof. LetM be a negatively idempotent ordered semigroup with unity anddbe a right derivation. Supposex∈M.
d(x) =d(xx) =d(x)xd(x)x =d(x)x≤x d(x)≤x.
Supposed(1) = 1.By Theorem [3.11 ], we havex≤d(x).Therefored(x) =x.Converse is
obvious. □
Theorem 3.13. LetMbe a cancellative ordered semgroup anddbe a non-zero right deriva-tion ofM.ThenM is a commutative ordered semigroup.
Proof. LetM be a cancellative ordered semgroup andd : M → M be a non-zero right derivation ofM.Supposea, b∈M.
d(aba) =d(a)bd(ba)a
=d(a)bd(b)ad(a)ba· · · (1) d(aba) =d(ab)ad(a)ab
=d(a)bd(b)ad(a)ab· · · (2). From (1) and (2) ,
d(a)bd(b)ad(a)ba=d(a)bd(b)ad(a)ab
⇒ ba=ab, for all a, b∈M.
HenceM is a commutative ordered semigroup. □
4. Conclusion
Over the last few decades, several authors have investigated the relationship between the commutativity of ring R and the existence of certain specified derivations ofR.In this paper, we introduced the concept of right derivation on semigroups, we studied some of the properties of right derivation of semigroups. We proved that ifdis a non-zero right derivation of a cancellative ordered semigroupM,thenM is a commutative ordered semigroup.
References
