Volume 2010, Article ID 903063,12pages doi:10.1155/2010/903063
Research Article
On Regular Elements in an Incline
A. R. Meenakshi and S. Anbalagan
Department of Mathematics, Karpagam University, Coimbatore 641 021, India
Correspondence should be addressed to A. R. Meenakshi,arm meenakshi@yahoo.co.in
Received 17 August 2009; Revised 31 December 2009; Accepted 28 January 2010
Academic Editor: Aloys Krieg
Copyrightq2010 A. R. Meenakshi and S. Anbalagan. This 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 thanorequal 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.1.
Recently, Kim and Roush have surveyed and outlined algebraic properties of inclines and of matrices over inclines2. 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 existsx,xis called ag-inverse of a and the element a is said to be regular. This concept of regularity of elements in a ring goes back to Neumann3. 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 4, 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 computingg-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. In2, it is stated
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 allg-inverses of an element in an incline with involution. InSection 2, we present the basic definitions, notations, and required results on inclines. InSection 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 ag-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. InSection 4, equivalent conditions for the existence of the Moore-Penrose inverse of an element in an incline with involution-Tare 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 Hartwig4.
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 onR · R → R such that for allx,y,z ∈R
xyyx, xyzxyz,
xyzxyxz, yzxyxzx,
xyzxyz, xxx, xxyx, yxyy.
2.1
Definition 2.2. An inclineRis said to be commutative ifxyyxfor allx, y∈R.
Definition 2.3. R,≤is an incline with order relation “≤” defined onRsuch that forx, y∈R,
x≤yif and only ifxyy. Ifx≤y, thenyis said to dominatex.
Property 2.4. Forx, yin an inclineR,xy≥xandxy≥y.
Forxy xx yx xy, andxyx yy xy y
Thusxy≥xandxy≥y.
Property 2.5. Forx,yin an inclineR,xy≤xandxy≤y.
Throughout letRdenote an incline with order relation≤. For an elementa∈R,aR {ax/x∈R}is the right ideal of a andRa{xa/x∈R}is the left ideal of a.
Definition 2.6Green’s relation5. For any two elementsa,bin a semigroupS. iaLbif there existx, y∈Ssuch thatxabandyba.
iiaRbif there existx, y∈Ssuch thataxbandbya. iiiaJbif there existw, x, y, zsuch thatwaxb,ybza. ivaHbifaRbandaLb.
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 rightleftideals 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 introduceg-inverses of an element in an incline.
Definition 3.1. a ∈Ris said to be regular if there exists an elementx ∈Rsuch that axa a. Thenxis called a generalized inverse, in shortg-inverse or 1-inverse of a and is denoted as
a−.Leta{1}denotes the set of all 1-inverses of a.
Definition 3.2. An element a ∈ Ris called antiregular, if there exists an elementx ∈ Rsuch that xax x.Thenxis called the 2-inverse of a. a{2}denotes the set of all 2-inverses of a.
Definition 3.3. Fora∈Rif there existsx∈Rsuch that axaa, xaxx, and axxa, thenxis 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 algebraFwith support 0,1under the max. min. operation is an incline 2. Each element inF is regular as well as idempotent 6, page 212. Thus F is a regular incline.
Example 3.5. LetD {a, b, c}andR PD,∪,∩,wherePDthe power set ofDis an incline. Here for each elementx ∈ PD,x2 x∩x x. Hencexis idempotent andxis
regularreferProposition 3.7. ThusRis a regular incline.
Lemma 3.6. Let a∈Rbe regular. Thenaaxxa for allx∈a{1}. Proof. If a is regular, then byProperty 2.5
aaxa≤ax≤a. 3.1
Thereforeaxa.
Similarly, from a ≤ xa ≤ a, it follows that a xa. Thus, a xa ax for all
x∈a{1}.
Proposition 3.7. Fora∈R, ais regular if and only if a is idempotent.
Proof. Leta ∈ R be regular. Then byLemma 3.6,a ax xafor allx ∈ a{1}.a axa
axaa·aa2. Thus a is idempotent.
Converse is trivial.
Example 3.8. Let us consider the example 72 0,1,supx, y, xyof an incline given in2.
Herexyis usual multiplication of real numbers. Hence for eachx ∈72,x2 ≤xandx,is not
idempotent. Therefore byProposition 3.7, 72is not a regular incline.
Proof. Let a be regular, then byProposition 3.7,a ∈ a{1}. ByLemma 3.6a ax for allx ∈ a{1}. Hence byProperty 2.5a≤x. Thus a is the smallestg-inverse of a.
It is well known that7every 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 3.10see1. A commutative incline is a distributive lattice as (semiring) if and only
ifx2xfor allx∈X.
Lemma 3.11see7. DL is a distributive lattice. (DL is the set of all idempotent elements in an
incline L.)
Proposition 3.12. LetRbe a commutative incline,Ris regular⇔Ris a distributive lattice. Proof. LetRis commutative incline.
Ris regular:⇔every element inRis idempotentbyProposition 3.7,
⇔DR R, whereDRis the set of all idempotent elements ofR, ⇒RDRis distributive latticeby7, Lemma 2.1.
Conversely, ifRis a distributive lattice then by Proposition 111 in1every element of
Ris idempotent, again byProposition 3.7Ris a regular incline.
Next we shall see some characterization of regular elements in an incline.
Theorem 3.13. For a∈R, the following are equivalent:
ia is regular,
iia is idempotent,
iiia{1,2}{a},
ivgroup inverse of a exists and coincides with a,
vava2for somev∈a{1},
viaa2ufor someu∈a{1}.
In either casev, u, vauare allg-inverses of a andvauis invariant for all choice ofu, v∈a{1}.vau is the smallestg-inverse of a.
Proof. i⇔iiThis is preciselyProposition 3.7.
To prove the theorem it is enough to prove the following implications: ii⇒iii⇒iv⇒i;i⇒v⇒iiandi⇒vi⇒ii.
ii⇒iiiIf a is idempotent, thena∈a{1}. For anyx∈a{1,2}we havex xaxand by
Lemma 3.6we getx xaxaxa. Thereforea{1,2}{a}.
Thusiiiholds.
iii⇒ivIfa{1,2}{a}then a is the only commuting 1-2 inverse of a. Therefore byDefinition 3.3the Group inverse of a exists and coincides with a. iv⇒iThis is trivial.
i⇒vLet a be regular, then byLemma 3.6, for somev∈a{1},
Thusvholds.
v⇒iiLetava2for somev∈a{1}. ByProperty 2.5,
ava2≤va≤a
⇒avava2
ava2 vaaa2.
3.3
Therefore a is idempotent Thusiiholds.
i⇒vi⇒iican be proved along the same lines and hence omitted. Now ifava2holds then we can show thatv∈a{1}
ava2≤va≤a. 3.4
Thereforeava2va
a2a·ava2a, 3.5
andavaa2a
Thusv∈a{1}.
In a similar manner we can showu∈a{1}.
Now consider,xvau, whereu, v∈a{1}. It can be verified that
xvau∈a{1,2}{a}. 3.6
Hence,avaufor allv, u ∈a{1}.
Thusvauis invariant for all choice ofg-inverse of a. ByProposition 3.9,avauis the smallestg-inverse of a.
Remark 3.14. If a is regular, theniaR a2R and iiRa 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×R → Rbyx·y dfor allx, y ∈ {1, b, c, d}and 0 otherwise. ThenR,∨,·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.
1d·x·ddfor eachx∈ {b, c, d,1}.
Henced{1}{b, c, d,1}and 0{1}R.
a b c
d
1
0
Figure 1
Hencedis antiregular
d{2}{0, d}. 3.7
3d{1}/d{2}andd{1,2}d{1} ∩d{2} d.
Theorem 3.16. LetRbe a regular incline. Fora, b, c∈Rthe following hold:
i
bya⇐⇒bba⇐⇒Rb⊆Ra,
cax⇐⇒cac⇐⇒cR⊆aR. 3.8
iixis a 1-inverse of a
⇐⇒ax2ax, axRaR,
⇐⇒xa2xa, RxaRa. 3.9
iiixis a 2-inverse of a
⇐⇒xa2 xa, xaRxR,
⇐⇒ax2 ax, RaxRx. 3.10
ivqaR⊆qapRandRa⊆Rqaimpliespqa−.
vIfcaxandbyathenba−cis invariant under all choice of 1-inverse of a.
vi
pwqa− ⇐⇒ ⎧ ⎨ ⎩
qapwRqaRand w qap−,
Proof. Whenever two symmetric results are involved we shall prove the first leaving the
second.
iLetbya, since a is regular.
baba−ayaa−ayab. 3.12
Thusbya⇒bab.
Letbabthen forz∈Rb
zxb for somex∈R
xba∈Ra. 3.13
Thusbba⇒Rb⊆Ra.
Sincebis regular, byLemma 3.6, b xb ∈ Rb,since Rb ⊆ Ra,b xb ya.Thus
Rb⊆Ra⇒bya.
Henceiholds.
iiLetx∈a{1}then byLemma 3.6andProposition 3.7we have
ax2 ax xa a. 3.14
Hence,axRaRandRxaRa.
Conversely, letaxRaRandax2 ax.
Then,aR⊆axR⇒axaabyi
⇒x∈a{1}. 3.15
iiiInterchangexand a iniitheniiiholds. ivLetqaR⊆qapRandRa⊆Rqa
qaR⊆qapR⇒qapqaqabyi 3.16
Ra⊆Rqa⇒aqaa,that is, a is regular withq∈a{1}.
Now,
qapqaqa,
aqapqaaqa,
apqaa.
3.17
Thusivholds.
vLetcaxandbyafor somex, y∈R
ba−cyaa−axyax. 3.18
Which is independent ofa−andba−cis invariant for all choice ofa−of a. viLetpwqa−⇒aapwqaByDefinition 3.1.
From the statementii, we have
RaRpwqa⊆Rqa⊆Ra
⇒RaRqa. 3.19
Thereforeq∈a{1}byii. Now,
apwqaa⇒qapwaqaqa,
qaR⊆qapwR⊆qaR,
qaRqapwR,
qapwqaqa,
aqapwqaaqa,
apwqaa,
⇒pwq∈a{1}.
3.20
Thusviholds.
Corollary 3.17. For a, b in a regular incline one has the following:
RaRb⇐⇒aRbR⇐⇒ab. 3.21
Proof. SinceRaRb, Ra⊆Rb, and Rb⊆Ra.
ByTheorem 3.16iwe have
Ra⊆Rb⇒aab⇒a≤b by Property 2.5
Rb⊆Ra⇒bba⇒b≤a by Property 2.5. 3.22
Thereforeab. In a similar manner we can showaRbR⇒ab.
Remark 3.18. We note thatCorollary 3.17fails for regular matrices over an incline. Let us considerB1 1
1 0
andA1 0 1 1
P B,whereP 0 1 1 0
.
SinceP2I
2, P AB. HereAandBare regular.
By Proposition 2.4 in8, page 297,RA⊆RBandRB⊆RA.
HenceRA RBbutA /B.
It is well known that9, page 26ifa−is a particularg-inverse of a in a ring with unit, then the general solution of the equationaxa ais given bya−h−a−ahaa−,wherehis arbitrary. Here we shall generalize this for incline.
Theorem 3.19. Let a ∈ R and a− be any particular 1-inverse of a then ag−{1} {a− h/his arbitrary element inR}is the set of allg-inverses of a dominatinga−. Furthermore,a{1} ∪ag−{1}, union over allg-inverses of a.
Proof. LetAdenote the set{a−h/his arbitrary element inR}. Suppose thatxis arbitrary element ofag−{1}thenx≥a−which impliesxk≥a−kfork∈Rand byProperty 2.4we havea−k≥a−.
Thereforexk≥a−k≥a−
Pre- and postmultiplication by a we get
axaaka≥aa−ka≥aa−a,
aaka≥aa−ka≥a 3.23
ByDefinition 3.1.
ByProperty 2.5aka≤a, henceaakaa.
a≥aa−ka≥a,
aa−kaa. 3.24
Thereforea−k∈a{1}.
Thus for eachx∈ag−{1}there exists an element inA. Henceag−{1} ⊆ A.
On the other hand for anyy∈ A, ya−h≥a−byProperty 2.4. FromDefinition 3.1andProperty 2.5, we get
ayaaa−haaahaa. 3.25
Hencey∈ag−{1},which impliesA ⊆ag−{1}. ThereforeA 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-
T
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 Hartwig4, Kim and Roush8and Meenakshi6,
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 in6,8.
An involution-T of an incline R is an involutary anti-automorphism, that is,aTT a,abTaTbT,abT bTaT, aT 0 if and only ifa0 for alla, b∈R.
Definition 4.1. An elementa∈Ris said to be a projection ifaTaa2, that is a is symmetric
and idempotent.
Definition 4.2. Forain an incline R with involution-T, we say thatx∈Ris a 3-inverse of a if axT ax,and we say thaty∈Ris a 4-inverse of a ifyaTya.
Definition 4.3. An elementx ∈ Ris said to be Moore-Penrose inverse of a, ifxsatisfies the following:iaxaa,iixaxx,iii axT ax,andiv xaT xa, denoted asa†.
In2it is stated that for an element a in an incline with involution-T,a†exists if and only if aHaaTa. Here we derive equivalent condition for the existence ofa† in terms of the
weaker relation aLaaTa.
First we shall show that Green’s equivalence relation on an incline R reduces to equality of elements in R.
Lemma 4.4. Fora, b∈Rthe following hold:
iaLb⇒ab,iiaRb⇒ab.
Converse holds for elements in a regular incline or incline with unit.
Proof. iIfaLbthen byDefinition 2.6there existx, y ∈ Rsuch thatxa bandyb a. By
Property 2.5we havexab⇒b≤aandyba⇒a≤b. Therefore,aLb⇒ab.
iiThis can be proved in a similar manner and hence omitted.
The converse holds foraregular incline. For, ifa, bare regular, then byLemma 3.6a bybbyandbaxaaxfor somex, y∈a{1}. Henceab⇒aLbandaRb.aLband
aRbtrivially hold for incline with unit.
Theorem 4.5. Let R be an incline with involution-T. For a∈Rthe following are equivalent:
ia is a projection,
iia has 1-3 inverse,
iiia has 1-4 inverse,
iva†exists and equals a,
vaTax= aThas a solution in R,
vixaaT = aThas a solution in R,
viia is regular and aT∈a{1},
viiiaLaaTa,
Proof. i⇔iiLet a be a projection, byDefinition 4.1a is symmetric idempotent. a is regular
follows fromProposition 3.7. Thus a has 1-inversexsayand byLemma 3.6a axxa. Since a is symmetric,aaT. Thereforexis a 1-3 inverse of a.
Thus a has 1–3 inverses. Coverersly if a has 1–3 inverses, then again byLemma 3.6
there existsx∈R, such thataax xaandax axT. Hence a is symmetric idempotent. Thusiholds.
i⇔iiiThis can be proved along the same lines as that ofi⇔iii, hence omitted.
i⇔ivThis equivalence can be proved directly by verifying that a satisfy the four equations inDefinition 4.3.
ii⇔vLet a has 1–3 inverses,xsaythen
aTaxaTax aTaxT aTxTaT axaTaT,
aTaxaT,
4.1
byDefinition 4.2.
Conversely, ifaTaxaT, thenaT ≤aTa≤aT ⇒aT aTaand thereforeaaTaand a
is symmetric. Hence the given conditionaTaxaTreduces to axaT
Now,axa axa aTa aandax aT a axTx ∈ a{1,3}. Thus a has 1–3
inverses.
iii⇔viThis can be proved in the same manner and hence omitted. vii⇔ia is regular andaT ∈a{1}
⇔ a is regular andaaaTa
⇔ a is idempotent andaaTaaaTbyProposition 3.7andLemma 3.6 ⇔a is symmetric and idempotent
⇔a is a projection.
vii⇔viii⇔ixfollow fromLemma 4.4.
Remark 4.6. It is well known that 4 for an element a in a ∗-regular ring if a† exists then
a† a1,4a a1,3. We observe that for an elementain an incline with involution-T if ais regular, then byLemma 3.6it follows thata{1,3}a{1,4}. Ifa†exists it is unique and given bya†a1,3a a1,3.
Remark 4.7. Let us consider the incline R inRemark 3.15under the identity involution-T on
R. Here each element in R is symmetric and the 3-inverse of the elementdis R and 4-inverse also the same.
Hence 0, dare the only projections inR
d†dd{1,2}/d{1,3}d{1,4}d{1}{b, c, d,1},
b·d·cdd† ∀b, c∈d{1,4}.
Theorem 4.8. Let R be an incline with involution-T. For any element a ∈ R and x ∈ a{1,3}
given, then ag{1,3} {xh/his arbitrary element inR} is the set of all {1,3} inverses of a dominatingx.
Proof. This can be proved along the same lines as that ofTheorem 3.19and 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 ring4and for elements in a Fuzzy algebra8. We
have proved the results by usingProperty 2.5without using star cancellation law.
In 2 it is stated that an element is regular if and only if D class contains an idempotent. ByLemma 4.4theDclass{b/bDa} {a} and by Proposition 3.1 a is regular if and only if a is idempotent.
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