IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation90306310.1155/2010/903063903063Research ArticleOn Regular Elements in an InclineMeenakshiA. R.AnbalaganS.KriegAloysDepartment of MathematicsKarpagam UniversityCoimbatore 641 021Indiakarpagamuniversity.ac.in2010172201020101708200931122009280120102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Inclines are additively idempotent semirings in which products are less than (or) equal to either factor. Necessary and sufficient conditions for an element in an incline to be regular are obtained. It is proved that every regular incline is a distributive lattice. The existence of the Moore-Penrose inverse of an element in an incline with involution is discussed. Characterizations of the set of all generalized inverses are presented as a generalization and development of regular elements in a -regular ring.

1. Introduction

The notion of inclines and their applications are described comprehensively in Cao et al. . Recently, Kim and Roush have surveyed and outlined algebraic properties of inclines and of matrices over inclines . Multiplicative semigroups unlike matrices over a field are not regular; that is, it is not always possible to solve the regularity equation axa  =  a. If there exists x, x is called a g-inverse of a and the element a is said to be regular. This concept of regularity of elements in a ring goes back to Neumann . If every element in a ring is regular, then it is called a regular ring. Regular rings are important in many branches of mathematics, especially in matrix theory, since the regularity condition is a linear condition that solves linear equations and takes the place of canonical decomposition.

In , Hartwig has studied on existence and construction of various g-inverses associated with an element in a *-regular ring, that is, regular ring with an anti-automorphism and developed a technique for computing g-inverses mainly by using star cancellation law. In semirings one of the most important aspects of structure is a collection of equivalence relations called Green’s relations and the corresponding equivalence classes. In , it is stated that an element is regular if and only if the equivalence 𝔇 class contains an idempotent.

In this paper, we exhibit that Green’s equivalence relations on a pair of elements in an incline reduce to the equality of elements. This leads to the characterization of regular element in an incline that is, an element in an incline is regular if and only if it is idempotent and structure of set of all g-inverses of an element in an incline with involution. In Section 2, we present the basic definitions, notations, and required results on inclines. In Section 3, some characterization of regular elements in an incline are obtained as a generalization of regular elements in a *-regular ring studied by Hartwig and as a development of results available in a Fuzzy algebra. The invariance of the product ba-c  for elements a,b,c in a regular incline and a g-inverse a- of a is discussed. For elements in a regular incline it is proved that equality of right ideals coincides with equality of left ideals. In Section 4, equivalent conditions for the existence of the Moore-Penrose inverse of an element in an incline with involution-T are determined.

Green’s equivalence relation reduces to equality of elements. We conclude that the proofs are purely based on incline property without using star cancellation law as in the work of Hartwig .

2. Preliminaries

In this section, we give some definitions and notations.

Definition 2.1.

An incline is a nonempty set R with binary operations addition and multiplication denoted as +, · defined on R  ·  R    R  such that for all x, y, z  Rx+y=y+x,x+(y+z)=(x+y)+z,x(y+z)=xy+xz,(y+z)x=yx+zx,x(yz)=(xy)z,x+x=x,x+xy=x,y+xy=y.

Definition 2.2.

An incline R is said to be commutative if xy=yx for all x,yR.

Definition 2.3.

(R,) is an incline with order relation “” defined on R such that for x,yR, xy if and only if x+y=y. If xy, then y is said to dominate x.

Property 2.4.

For x,y in an incline R, x+yx  and x+yy.

For x+y=(x+x)+y=x+(x+y), and x+y=x+(y+y)=(x+y)+y

Thus x+yx and x+yy.

Property 2.5.

For x, y in an incline R, xyx and xyy.

Throughout let R denote an incline with order relation . For an element aR, aR={ax/xR} is the right ideal of a and Ra={xa/xR} is the left ideal of a.

Definition 2.6 (Green’s relation [<xref ref-type="bibr" rid="B2">5</xref>]).

For any two elements a, b in a semigroup S.

ab if there exist x,yS such that xa=b and yb=a.

ab if there exist x,yS such that ax=b and by=a.

a𝒥b if there exist w,x,y,z such that wax=b, ybz=a.

ab if ab and ab.

a𝔇b if there exists cS such that ac and cb.

3. Regular Elements in an Incline

In this section, equivalent conditions for regularity of an element in an incline are obtained and it is proved that a regular commutative incline is a distributive lattice. The equality of right (left) ideals of a pair of elements in a regular incline reduces to the equality of elements. This leads to the invariance of the product ba-c for all choice a- of a and a,b,c in a regular incline. Characterization of the set of all g-inverses of an element in terms of a particular g-inverse is determined.

Just for sake of completeness we will introduce g-inverses of an element in an incline.

Definition 3.1.

aR is said to be regular if there exists an element xR such that axa=a. Then x is called a generalized inverse, in short g-inverse or 1-inverse of a and is denoted as a-. Let a{1} denotes the set of all 1-inverses of a.

Definition 3.2.

An element aR is called antiregular, if there exists an element xR such that xax  =  x. Then x is called the 2-inverse of a. a{2} denotes the set of all 2-inverses of a.

Definition 3.3.

For aR if there exists xR such that axa=a, xax=x, and ax=xa, then x is called the Group inverse of a. The Group inverse of a is a commuting 1-2 inverse of a.

An incline R is said to be regular if every element of R is regular.

Example 3.4.

The Fuzzy algebra with support [0,1] under the max. min. operation is an incline . Each element in is regular as well as idempotent [6, page 212]. Thus is a regular incline.

Example 3.5.

Let D={a,b,c} and R=(𝒫(D),,), where (𝒫(D)) the power set of D is an incline. Here for each element x𝒫(D), x2=xx=x. Hence x is idempotent and x is regular (refer Proposition 3.7). Thus R is a regular incline.

Lemma 3.6.

Let aR be regular. Then a=ax=xa for all xa{1}.

Proof.

If a is regular, then by Property 2.5a=axaaxa.

Therefore ax=a.

Similarly, from axaa, it follows that a=xa. Thus, a=xa=ax for all xa  {1}.

Proposition 3.7.

For aR, ais regular if and only if a is idempotent.

Proof.

Let aR be regular. Then by Lemma 3.6,  a=ax=xa for all xa{1}. a=axa=(ax)a=a·a=a2. Thus a is idempotent.

Converse is trivial.

Example 3.8.

Let us consider the example 72=([0,1],sup(x,y),xy) of an incline given in . Here xy is usual multiplication of real numbers. Hence for each x72, x2x and x, is not idempotent. Therefore by Proposition 3.7, 72 is not a regular incline.

Proposition 3.9.

If a is regular, then a is the smallest g-inverse of a, that is, ax for all xa{1}.

Proof.

Let a be regular, then by Proposition 3.7, aa{1}. By Lemma 3.6  a=ax for all xa{1}. Hence by Property 2.5  ax. Thus a is the smallest g-inverse of a.

It is well known that  every distributive lattice is an incline, but an incline need not be a distributive lattice. Now we shall show that regular commutative incline is a distributive lattice in the following.

Proposition 3.10 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

A commutative incline is a distributive lattice as (semiring) if and only if x2=x for all xX.

Lemma 3.11 (see [<xref ref-type="bibr" rid="B4">7</xref>]).

DL is a distributive lattice. (DL is the set of all idempotent elements in an incline L.)

Proposition 3.12.

Let  R be a commutative incline, R is regular R is a distributive lattice.

Proof.

Let R is commutative incline.

R is regular: every element in R is idempotent (by Proposition 3.7),

DR  =  R, where DR is the set of all idempotent elements of R,

R=DR is distributive lattice (by [7, Lemma  2.1]).

Conversely, if R is a distributive lattice then by Proposition 111 in  every element of R is idempotent, again by Proposition 3.7R is a regular incline.

Next we shall see some characterization of regular elements in an incline.

Theorem 3.13.

For aR, the following are equivalent:

a is regular,

a is idempotent,

a{1,2}={a},

group inverse of a exists and coincides with a,

a=va2 for some va{1},

a=a2u for some ua{1}.

In either case v,u,vau are all g-inverses of a and vau is invariant for all choice of u,va{1}. vau is the smallest g-inverse of a.

Proof.

(i)(ii) This is precisely Proposition 3.7.

To prove the theorem it is enough to prove the following implications:

(ii)(iii)(iv)(i); (i)(v)(ii) and (i)(vi)(ii).

(ii)(iii) If a is idempotent, then aa{1}. For any xa{1,2} we have x=xax and by

Lemma 3.6 we get x=(xa)x=ax=a. Therefore a{1,2}={a}.

Thus (iii) holds.

(iii)(iv) If a{1,2}={a} then a is the only commuting 1-2 inverse of a.

Therefore by Definition 3.3 the Group inverse of a exists and coincides with a.

(iv)(i) This is trivial.

(i)(v) Let a be regular, then by Lemma 3.6, for some va{1},a=(av)a=(va)a=va2.

Thus (v) holds.

(v)(ii) Let a=va2 for some va{1}. By Property 2.5,a=va2vaaa=va=va2a=va2=(va)a=a2.

Therefore a is idempotent

Thus (ii) holds.

(i)(vi)(ii) can be proved along the same lines and hence omitted.

Now if a=va2 holds then we can show that va{1}a=va2vaa. Therefore a=va2=vaa2=a·a=va2=a, and ava=a2=a

Thus va{1}.

In a similar manner we can show ua{1}.

Now consider, x=vau, where u,va{1}. It can be verified thatx=vaua{1,2}={a}.

Hence, a=vau for all v,u  a{1}.

Thus vau is invariant for all choice of g-inverse of a. By Proposition 3.9, a=vau is the smallest g-inverse of a.

Remark 3.14.

If a is regular, then (i) aR=a2R and (ii) Ra=Ra2 automatically holds. The converse holds for an incline with unit.

Remark 3.15.

Let us illustrate the relation between various inverses associated with an element in an incline in the following.

Let R={0,a,b,c,d,1} be a lattice ordered by the following Hasse graph. Define ·: R×RR by x·y=d for all x,y{1,b,c,d} and 0 otherwise. Then (R,, ·) is an incline which is not a distributive lattice.

In this incline R, the only two elements 0, d are regular which satisfies the Theorem 3.13.

d·x·d=d for each x{b,c,d,1}.

Hence d{1}={b,c,d,1} and 0{1}=R.

Since dR, x·d·x=x for x=0, and x=d.

Hence d is antiregular d{2}={0,d}.

d{1}d{2} and d{1,2}=d{1}d{2}  =  d.

Theorem 3.16.

Let R be a regular incline. For a,b,cR the following hold:

b=yab=baRbRa,c=axc=accRaR.

x is a 1-inverse of a

(ax)2=ax,axR=aR,(xa)2=xa,Rxa=Ra.

x is a 2-inverse of a(xa)2=xa,xaR=xR,(ax)2=ax,Rax=Rx.

qaRqapR and RaRqa implies pq=a-.

If c=ax and b=ya then ba-c is invariant under all choice of 1-inverse of a.

pwq=a-{qapwR=qaR  and  w=(qap)-,Rqa=Ra.

Proof.

Whenever two symmetric results are involved we shall prove the first leaving the second.

Let b=ya, since a is regular. ba=ba-a=yaa-a=ya=b. Thus b=yaba=b.

Let ba=b then for zRbz=xb  for some  xR=  (xb)aRa. Thus b=baRbRa.

Since b is regular, by Lemma 3.6,  b=xbRb, since RbRa, b=xb=ya. Thus RbRab=ya.

Hence (i) holds.

Let xa{1} then by Lemma 3.6 and Proposition 3.7 we have (ax)2=ax=  (xa)=a. Hence, axR=aR and Rxa=Ra.

Conversely, let axR=aR and (ax)2=ax.

Then, aRaxRaxa=a (by (i))xa{1}.

Interchange x and a in (ii) then (iii) holds.

Let qaRqapR and RaRqa

qaRqapRqapqa=qa  (by  (i))

RaRqaaqa=a, that is, a is regular with qa{1}.

Now,qapqa=qa,(aqa)pqa=aqa,apqa=a. Therefore pqa{1}.

Thus (iv) holds.

Let c=ax and b=ya for some x,yRba-c=y(aa-a)x=yax. Which is independent of a- and ba-c is invariant for all choice of a- of a.

Let pwq=a-a=a(pwq)a (By Definition 3.1).

From the statement (ii), we have

Ra=RpwqaRqaRaRa=Rqa.

Therefore qa{1} (by (ii)).

Now,apwqa=aqapwaqa=qa,qaRqapwRqaR,qaR=qapwR,qapwqa=qa,(aqa)pwqa=aqa,a(pwq)a=a,pwqa{1}.

Thus (vi) holds.

Corollary 3.17.

For a, b in a regular incline one has the following: Ra=RbaR=bRa=b.

Proof.

Since Ra=Rb,RaRb,and  RbRa.

By Theorem 3.16(i) we have RaRba=abab(by  Property  2.5)RbRab=baba(by  Property  2.5).

Therefore a=b. In a similar manner we can show aR=bRa=b.

On the other hand a=b automatically implies Ra=Rb and aR=bR.

Remark 3.18.

We note that Corollary 3.17 fails for regular matrices over an incline.

Let us consider B=(1110) and A=(1011)=  PB, where P=(0110).

Since P2=I2,PA=B. Here A and B are regular.

By Proposition 2.4 in [8, page 297], R(A)R(B) and R(B)R(A).

Hence R(A)=R(B) but AB.

It is well known that [9, page 26] if a- is a particular g-inverse of a in a ring with unit, then the general solution of the equation axa=a is given by a-+h-a-ahaa-, where h is arbitrary. Here we shall generalize this for incline.

Theorem 3.19.

Let aR and a- be any particular 1-inverse of a then ag-{1}={a-+h/h  is  arbitrary  element  in  R} is the set of all g-inverses of a dominating a-. Furthermore, a{1}=ag-{1}, union over all g-inverses of a.

Proof.

Let 𝒜 denote the set {a-+h/h  is  arbitrary  element  in  R}. Suppose that x is arbitrary element of ag-{1} then xa- which implies x+ka-+k for kR and by Property 2.4 we have a-+ka-.

Therefore x+ka-+ka-

Pre- and postmultiplication by a we getaxa+akaa(a-+k)aaa-a,a+akaa(a-+k)aa (By Definition 3.1).

By Property 2.5  akaa, hence a+aka=a.aa(a-+k)aa,a(a-+k)a=a. Therefore (a-+k)a{1}.

Thus for each xag-{1} there exists an element in 𝒜. Hence ag-{1}𝒜.

On the other hand for any y𝒜,y=a-+ha- by Property 2.4.

From Definition 3.1 and Property 2.5, we getaya=a(a-+h)a=a+aha=a. Hence yag-{1}, which implies 𝒜  ag-{1}. Therefore 𝒜  =ag-{1}a{1}=  set  of  all  g-inverses  of  a=  ag-{1},union  over  all  g-inverses  of  a.

4. Projection on an Incline with Involution-<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M384"><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>

In this section, the existence of the Moore-Penrose inverse of an element in an incline with involution-T is discussed as a generalization of that for elements is a *-regular ring and for elements in a Fuzzy algebra studied by Hartwig , Kim and Roush  and Meenakshi , respectively. Characterization of the set of all {1,3}, {1,4} inverses and a formula for Moore-Penrose are obtained analogous to those of the result established for fuzzy matrices in [6, 8].

An involution-T of an incline R is an involutary anti-automorphism, that is, (aT)T=a,(a+b)T=aT+bT,(ab)T=bTaT,aT=0 if and only if a=0 for all a,bR.

Definition 4.1.

An element aR is said to be a projection if aT=a=a2, that is a is symmetric and idempotent.

Definition 4.2.

For a in an incline R with involution-T, we say that xR is a 3-inverse of a if (ax)T=ax, and we say that yR is a 4-inverse of a if (ya)T=ya.

Definition 4.3.

An element xR is said to be Moore-Penrose inverse of a, if x satisfies the following: (i) axa=a, (ii) xax=x, (iii) (ax)T=ax, and (iv) (xa)T=xa, denoted as a.

In  it is stated that for an element a in an incline with involution-T, a exists if and only if aaaTa. Here we derive equivalent condition for the existence of a in terms of the weaker relation aaaTa.

First we shall show that Green’s equivalence relation on an incline R reduces to equality of elements in R.

Lemma 4.4.

For a,bR the following hold:

(i)aba=b, (ii)aba=b.

Converse holds for elements in a regular incline or incline with unit.

Proof.

(i) If ab then by Definition 2.6 there exist x,yR such that xa=b and yb=a. By Property 2.5 we have xa=bba and yb=aab.

Therefore, aba=b.

(ii) This can be proved in a similar manner and hence omitted.

The converse holds for a regular incline. For, if a,b are regular, then by Lemma 3.6  a=b=yb=by and b=a=xa=ax for some x,ya{1}. Hence a=bab and ab. ab and ab trivially hold for incline with unit.

Theorem 4.5.

Let R be an incline with involution-T. For aR the following are equivalent:

a is a projection,

a has 1-3 inverse,

a has 1-4 inverse,

a exists and equals a,

aTax = aT has a solution in R,

xaaT = aT has a solution in R,

a is regular and aTa{1},

aaaTa,

aaaTa,

Proof.

Let a be a projection, by Definition 4.1  a is symmetric idempotent. a is regular follows from Proposition 3.7. Thus a has 1-inverse x (say) and by Lemma 3.6  a=ax=xa. Since a is symmetric, a=aT. Therefore x is a 1-3 inverse of a.

Thus a has 1–3 inverses. Coverersly if a has 1–3 inverses, then again by Lemma 3.6 there exists xR, such that a=ax=xa and ax=(ax)T. Hence a is symmetric idempotent. Thus (i) holds.

This can be proved along the same lines as that of (i)(iii), hence omitted.

This equivalence can be proved directly by verifying that a satisfy the four equations in Definition 4.3.

Let a has 1–3 inverses, x (say) then

aTax=aT(ax)=aT(ax)T=aTxTaT=(axa)T=aT,aTax=aT, (by Definition 4.2).

Conversely, if aTax=aT, then aTaTaaTaT=aTa and therefore a=aTa and a is symmetric. Hence the given condition aTax=aT reduces to a x=aT

Now, axa=(ax)a=aTa=a and ax=aT=a=(ax)T

xa{1,3}. Thus a has 1–3 inverses.

This can be proved in the same manner and hence omitted.

a is regular and aTa{1}

a is regular and a=aaTa

a is idempotent and a=aTa=aaT (by Proposition 3.7 and Lemma 3.6)

a is symmetric and idempotent

a is a projection.

Remark 4.6.

It is well known that  for an element a in a *-regular ring if a exists then a=a(1,4)aa(1,3). We observe that for an element a in an incline with involution-T if a is regular, then by Lemma 3.6 it follows that a{1,3}=a{1,4}. If a exists it is unique and given by a=a(1,3)aa(1,3).

Remark 4.7.

Let us consider the incline R in Remark 3.15 under the identity involution-T on R. Here each element in R is symmetric and the 3-inverse of the element d is R and 4-inverse also the same.

Hence 0,d are the only projections in Rd=d=d{1,2}d{1,3}=d{1,4}=d{1}={b,c,d,1},b·d·c=d=db,cd{1,4}.

Theorem 4.8.

Let R be an incline with involution-T. For any element aR and xa{1,3} given, then ag{1,3}={x+h/h  is  arbitrary  element  in  R} is the set of all {1,3} inverses of a  dominating x.

Proof.

This can be proved along the same lines as that of Theorem 3.19 and hence omitted.

5. Conclusion

The main results in the present paper are the generalization of the available results shown in the reference for elements in a *-regular ring  and for elements in a Fuzzy algebra . We have proved the results by using Property 2.5 without using star cancellation law.

In  it is stated that an element is regular if and only if 𝔇 class contains an idempotent. By Lemma 4.4 the 𝔇 class {b/b𝔇a}={a} and by Proposition 3.1  a is regular if and only if a is idempotent.

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