research article on the menelaus and ceva 6-figures in the fibered projective...

Research ArticleOn the Menelaus and Ceva 6-Figures in the FiberedProjective Planes

AyGe Bayar and Suumlheyla Ekmekccedili

Department of Mathematics and Computer Science Eskisehir Osmangazi University 26480 Eskisehir Turkey

Correspondence should be addressed to Ayse Bayar akorkmazoguedutr

Received 18 December 2013 Accepted 28 February 2014 Published 31 March 2014

Academic Editor Chun-Gang Zhu

Copyright copy 2014 A Bayar and S Ekmekci This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The fibered versions of Menelaus and Ceva 6-figures in the fibered projective plane are given and the conditions to the fiberedversions of Menelaus and Ceva 6-figures in the fibered projective plane with base plane that is projective plane are determined

1 Introduction

In 1965 Zadeh [1] introduced the concept of fuzzy set Sincethen research on the theory and applications of fuzzy setshas been growing steadily Several basic geometric conceptshave been considered by various authors in a fuzzy context[2ndash4] An attempt to develop a kind of fuzzy projectivegeometry deduced from fuzzy vector space was made byKuijken et al [5] This provided a link between the fuzzyversions of classical theories that are very closely relatedIn these papers for the most suitable definition of fuzzygeometry they encountered the following problem Thereare two possibilities to fuzzify the points of geometry Eitherone assigns a unique degree of membership to each ofthem or one assigns several degrees of membership to allof them They explored the second possibility and it turnsout that the geometric structures involved are much richerand defined fibered projective planes and looked at fiberedversions of theorems of Desargues and Pappus [6] The roleof the triangular norm in the theory of fibered projectiveplanes and fibered harmonic conjugates and a fibered versionof Reidemeisterrsquos condition were given in [7] The authorsconsidered different triangular norms in the theory of fiberedprojective planes When the minimum operator is used thisimplies that if the underlying projective plane is finite then afinite number of 119891-points generate a finite fibered projectiveplane However for other triangular norms the fibered planemay be infinite For instance if we take 119886 and 119887 = 119886119887

the ordinary product then we see that as soon as at leastone membership degree 120572 is different from 1 we obtaininfinitely many membership degrees and therefore infinitelymany119891-points Another observation to bemade is that not alltriangular norms are suitable here For instance if we take theLukasiewicz norm given by 119886and119887 = max0 119886+119887minus1 then nopoint can have amembership degree different from 1 Hence afibered projective planewith the Lukasiewicz triangular normis equivalent to a crisp projective plane

Kaya and Ciftci introduced the 6-figures of Menelaus andCeva in Moufang projective planes in [8] In 1992 in orderto state the theorems of Menelaus and Ceva Klamkin andLiu needed the notion of signed distance in projective planeP2(R) In 2008 Kelly [9] generalized the Klamkin and Liu

[10] result to projective planes P2(F) where F is the field

of characteristic not equal to two But in an arbitrary fielddistance is not defined Thus he used the scalars that areassociated with the cross-ratio to prove these theorems inprojective planes P

2(F)

In this paper since the definition of cross-ratio has notextended to fibered projective plane yet we propose tocontribute to fuzzy geometry by looking at the fibered versionofMenelaus andCeva 6-figures in the fibered projective planewhich are given and the conditions to the fibered version ofMenelaus and Ceva 6-figures in the fibered projective planewith base plane that is projective plane are determined Webelieve that such a concept will play an important role toestablish a genuine fuzzy geometry

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 803173 5 pageshttpdxdoiorg1011552014803173

2 Abstract and Applied Analysis

2 Preliminaries

We first recall some basic notions from fuzzy set theory andfibered geometry We denote by and a triangular norm on the(real) unit interval [0 1] that is a symmetric and associativebinary operator satisfying (119886 and 119887) le (119888 and 119889) whenever 119886 le 119888and 119887 le 119889 and 119886 and 1 = 119886 for all 119886 119887 119888 119889 isin [0 1]

Definition 1 (see [6]) Let P = (119875 119861 ∘) be any projectiveplane with point set 119875 and line set 119861 that is 119875 and 119861 are twodisjoint sets endowed with a symmetric relation ∘ (called theincidence relation) such that the graph (119875 cup 119861 ∘) is a bipartitegraph with classes 119875 and 119861 and such that two distinct points119901 119902 inP are incident with exactly one line every two distinctlines 119871119872 are incident with exactly one point and every lineis incident with at least three points A set 119878 of collinear pointsis a subset of 119875 each member of which is incident with acommon line 119871 Dually one defines a set of concurrent linesWe now define fibered points and fibered lines which arebriefly called 119891-points and 119891-lines

Definition 2 (see [6]) Suppose that 119886 isin 119875 and 120572 isin]0 1] Thenan 119891-point (119886 120572) is the following fuzzy set on the point set 119875ofP

(119886 120572) 119875 997888rarr [0 1] 119886 997888rarr 120572

119909 997888rarr 0 if 119909 isin 119875 119886 (1)

Dually one defines in the same way the 119891-line (119871 120573) for119871 isin 119861 and 120573 isin]0 1] The real number 120572 above is called themembership degree of the 119891-point (119886 120572) while the point 119886 iscalled the base point of it the same for 119891-lines

Definition 3 (see [6]) The two119891-lines (119871 120572) and (119872 120573) with120572 and 120573 gt 0 intersect in the unique 119891-point (119871 cap 119872 120572 and 120573)Dually the 119891-points (119886 120582) and (119887 120583) with 120582and120583 gt 0 span theunique 119891-line (⟨119886 119887⟩ 120582 and 120583)

Definition 4 (see [6]) A (nontrivial) fibered projective planeFP consists of a set 119865119875 of 119891-points ofP and a set 119865119861 of 119891-lines of P such that every point and every line of P are thebase point and base line of at least one 119891-point and 119891-linerespectively (with at least one membership degree differentfrom 1) and such thatFP = (119865119875 119865119861) is closed undertakingintersections of 119891-lines and spans of 119891-points Finally a setof 119891-points are called collinear if each pair of them span thesame 119891-line Dually a set of 119891-lines are called concurrent ifeach pair of them intersects in the same 119891-point

Theorem 5 (see [6]) Suppose that one has a fibered projectiveplane FP with base plane P that is Desarguesian Choosethe three 119891-points (119886

1 1205721) (1198862 1205722) and (119886

3 1205723) in FP with

noncollinear base points and the three other 119891-points (1198871 1205731)

(1198872 1205732) and (119887

3 1205733) with noncollinear base points such that

the lines ⟨119886119894 119887119894⟩ for 119894 isin 1 2 3 are concurrent in a point 119901 of

P with 119886119894

= 119887119894

= 119901 = 119886119894 Then the three 119891-lines (⟨119886

119894 119886119895⟩ 120572119894and120572119895)

and (⟨119887119894 119887119895⟩ 120573119894and 120573119895) (for 119894 = 119895 and 119894 119895 isin 1 2 3) intersect in

three collinear 119891-points

3 Fibered Version of Menelaus andCeva 6-Figures

The Alexandrian Greek mathematician Menelaus and theseventeenth-century Italian mathematician Ceva are invari-ably mentioned together Menelausrsquo theorem which involvesa test for the collinearity of three points and Cevarsquos theoremwhich involves a test for the concurrency of three lines arefrequently called the twin theorem These theorems shouldhave been discovered together and it is not insignificantthat such a long period separates Menelaus and CevaDuring the 1500 years that separate the two there waslittle development in mathematics About the year AD 100Menelaus of Alexandria extended a then well-known lemmato spherical triangles in his Sphaerica the high point of Greektrigonometry It is this lemma for the plane that today bearsthe name ofMenelaus Of theCeva brothers the lesser knownTommaso wrote on the cycloid while Giovanni resurrectedthe forgotten Menelausrsquo theorem and published it in 1678along with the twin theorem now known as Cevarsquos theorem

Wenow recall some observations regarding the triangularnorm and the Menelaus and Ceva 6-figures

In [7] the following theorems give an answer to thequestion when does a finite fibered projective plane existgiven a particular triangular norm

Theorem 6 (see [7]) Let P be a finite crisp projectiveplane and and a triangular norm Then there exists some finitenontrivial fibered projective plane FP with base plane P ifand only if there exists an idempotent element 120572 isin]0 1] thatis 120572 and 120572 = 120572

Theorem 7 (see [7]) Let P be an arbitrary crisp projectiveplane and and a triangular norm Then there exists somenontrivial fibered projective plane FP with base plane P ifand only if one of the following holds

(i) there exists an idempotent element 120572 isin]0 1[(ii) there exists an element 120572 isin [0 1[ with the property

that 120573 and 1205731015840 gt 120572 for all 120573 1205731015840 isin]120572 1]

In [7] the definition and theorems related to119891-harmonicconjugate in a fibered projective plane whose base planesatisfies Fanorsquos axiom are given

Definition 8 (see [7]) Suppose that one has a fibered projec-tive planeFPwith base projective planeP Choose the four119891-points (119886

1 1205721) (1198862 1205722) (1198863 1205723) and (119886

4 1205724) in FP none

of the three base points of which are collinearThese119891-pointsare called 119891-vertices The configuration that consists of thesefour119891-points the six119891-lines (119860

119894119895 120573119894119895

) = (⟨119886119894 119886119895⟩ 120572119894and120572119895)

for 119894 = 119895 119894 119895 isin 1 2 3 4 (which we call 119891-sides) and thethree119891-points (119860

119894119895and119860119896119897

1205721and1205722and1205723and1205724) with 119894 119895 119896 119897 =

1 2 3 4 (the 119891-diagonal points) is called an 119891-completequadrangle

Theorem 9 (see [7]) Suppose that one has a fibered projectiveplaneFP with base planeP Let 119886 119887 and 119888 be three collinearpoints in P and let (119886 120572) (119887 120573) (119888 120574) be three 119891-points ofFP Suppose that the point119889 on ⟨119886 119887⟩ obtained by intersecting

Abstract and Applied Analysis 3

⟨119886 119887⟩ with the join of the two diagonal points different from 119888in any complete quadrangle where 119886 119887 are vertices and 119888 is adiagonal point is independent of the chosen quadrangle Thenthe 119891-point (119889 120575) obtained by intersecting (⟨119886 119887⟩ 120572 and 120573) withthe 119891-join of the two 119891-diagonal points different from (119888 120574) inany 119891-complete quadrangle where (119886 120572) (119887 120573) are 119891-verticesand (119888 120574) is an 119891-diagonal point is independent of chosen 119891-complete quadrangle

The 119891-point (119889 120575) of previous theorem if it exists iscalled the fourth 119891-conjugate to ((119886 120572) (119887 120573) (119888 120574))

Theorem 10 (see [7]) Given a triangular norm and thenand is the minimum operator if and only if in any fiberedprojective plane FP for every quadruple ((119886 120572) (119887 120573)(119888 120574) (119889 120575)) of 119891-points with collinear base points the 119891-point (119889 120575) is the fourth 119891-conjugate to ((119886 120572) (119887 120573) (119888 120574))whenever (119888 120574) is the fourth 119891-conjugate to ((119886 120572) (119887 120573)(119889 120575))

The original definitions of Menelaus and Ceva 6-figuresare given in [11]

Definition 11 (see [8]) LetP be a projective plane A 6-figureinP is a sequence of six distinct points (119886

111988621198863 119887111988721198873) such

that 119886111988621198863constitutes a nondegenerate triangle with 119887

1isin

⟨1198862 1198863⟩ 1198872isin ⟨1198861 1198863⟩ 1198873isin ⟨1198861 1198862⟩ The points 119886

1 1198862 1198863 1198871

1198872 1198873are called vertices of this 6-figure Such a configuration

is said to be aMenelaus 6-figure or aCeva 6-figure if 1198871 1198872 and

1198873are collinear or if ⟨119886

1 1198871⟩ ⟨1198862 1198872⟩ ⟨1198863 1198873⟩ are concurrent

respectively

We now consider some classical configurations and the-orems and extend them to fibered projective planes forsatisfying one of the conditions (i) and (ii) in Theorem 7suitably

Definition 12 LetFP be a fibered projective plane with baseplane P Choose three 119891-points (119886

119894 120572119894) 119894 isin 1 2 3 in FP

with noncollinear base points and the other three 119891-points(119887119896 120573119896) 119896 isin 1 2 3 with (119887

119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895) for

119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If the 119891-points (119887119896 120573119896) are 119891-

collinear the configuration that consists of these six 119891-pointsis called an 119891-Menelaus 6-figure It is called 119891-Menelaus linespanned with 119891-points (119887

119896 120573119896) for 119896 = 1 2 3

Theorem 13 Suppose that one has a fibered projective planeFP with base plane P Let 119886

1 1198862 1198863be three noncollinear

points in P and let (1198861 1205721) (1198862 1205722) (1198863 1205723) be three f-points

of FP Suppose that the point 1198873on ⟨1198861 1198862⟩ is obtained by

intersecting ⟨1198861 1198862⟩ with the join of two chosen points 119887

1and

1198872where 119887

1on ⟨1198862 1198863⟩ and 119887

2on ⟨1198861 1198863⟩ Then the f-point

(1198873 1205733) obtained by intersecting (⟨119886

1 1198862⟩ 1205721and 1205722) with the 119891-

join of the two 119891-points (1198871 1205731) and (119887

2 1205732) where (119887

1 1205731)

(1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723) are 119891-collinear

is independent of the chosen 119891-points (1198871 1205731) and (119887

2 1205732)

Proof Since the three 119891-points (1198861 1205721) (1198863 1205723) (1198872 1205732) and

the three 119891-points (1198862 1205722) (1198863 1205723) (1198871 1205731) are 119891-collinear

1205721and 1205723= 1205721and 1205732= 1205723and 1205732and 120572

2and 1205723= 1205722and 1205731= 1205723and 1205731

One calculates that 1205733= 1205721and 1205722and 12057223 and hence the 119891-point

(1198873 1205733) is independent of the choice of the 119891-points (119887

1 1205731)

and (1198872 1205732)

Theorem 14 Suppose that one has a fibered projective planeFP with base plane P Choose the three 119891-points (119886

1 1205721)

(1198862 1205722) and (119886

3 1205723) in FP neither of the three base points

of which is collinear Let the 119891-point (1198873 1205733) be obtained

by intersecting (⟨1198861 1198862⟩ 1205721and 1205722) with the 119891-join of the

two 119891-points (1198871 1205731) and (119887

2 1205732) where (119887

1 1205731) (1198862 1205722)

(1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723) are 119891-collinear Then the

configuration that consists of the six 119891-points (119886119894 120572119894) (119887119894 120573119894)

119894 isin 1 2 3 is an 119891-Menelaus 6-figure

Proof Since the three 119891-points (1198871 1205731) (1198872 1205732) and (119887

3 1205733)

are 119891-collinear from Definition 12 the configuration thatconsists of the six 119891-points (119886

119894 120572119894) (119887119894 120573119894) 119894 isin 1 2 3 is an

119891-Menelaus 6-figure

Theorem 15 Let FP be a fibered projective plane with baseplane P Choose three 119891-points (119886

119894 120572119894) 119894 isin 1 2 3 in FP

with noncollinear base points and with (119887119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and

120572119896) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If 119887

1 1198872 and 119887

3in P

are collinear then the three 119891-points (119887119895 120573119895) 119895 isin 1 2 3

are 119891-collinear if and only if 12057221and 1205722and 1205723= 1205721and 12057222and 1205723=

1205721and 1205722and 12057223

Proof A configuration is picked such that three 119891-points(1198861 1205721) (1198862 1205722) and (119886

3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)

for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 Suppose the three 119891-points(119887119895 120573119895) 119895 isin 1 2 3 are 119891-collinear Since three 119891-points

(119887119895 120573119895) are119891-collinear and the three119891-points (119886

119894 120572119894) (119886119895 120572119895)

and (119887119896 120573119896) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 are 119891-collinear

120573119894and 120573119895= 120573119894and 120573119896and 120572

119894and 120572119895= 120572119894and 120573119896= 120572119895and 120573119896 Then clearly

12057221and 1205722and 1205723= 1205721and 12057222and 1205723= 1205721and 1205722and 12057223

Conversely if 12057221and 1205722and 1205723= 1205721and 12057222and 1205723= 1205721and 1205722and 12057233

are satisfied 1205731and1205732= 1205731and1205733= 1205732and1205733 Then three 119891-points

(1198871 1205731) (1198872 1205732) and (119887

3 1205733) are 119891-collinear

The Fibered Menelaus Condition (FMC) LetFP be a fiberedprojective plane with base plane P Choose three 119891-points(119886119894 120572119894) 119894 isin 1 2 3 in FP with noncollinear base points

and the other three 119891-points (119887119896 120573119896) 119896 isin 1 2 3 with

(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3

The configuration that consists of these six 119891-points isMenelaus 6-figure if and only if 1205722

1and 1205722and 1205723= 1205721and 12057222and 1205723=

1205721and 1205722and 12057223

Theorem 16 Let and be a triangular norm Let FP be anynontrivial fibered projective plane with base plane P andtriangular norm and Then there is at least triangular norm andsuch thatFP satisfies FMC

Proof When the minimum operator for and is used it impliesthat 120572

1 1205722 and 120572

3are idempotent elements inTheorem 15 So

FMC is satisfied Clearly using the operator defined by 119886and119887 =12 for all 119886 119887 ge 12 and 119886 and 119887 = min119886 119887 otherwise is atriangular norm then also FMC is satisfied


Pappusrsquo theorem is an important theorem in geometryFirstly Kuijken and VanMaldeghem gave the fibered versionof Pappusrsquo theorem using the minimum operator for and [6]

It is known that Pappusrsquo theorem can be proved by usingthe Menelaus theorem Now we give the fibered version ofPappusrsquo theorem by using the 119891-Menelaus 6-figure

Theorem 17 Let FP be a fibered projective plane with baseplane P The fibered version of the Pappus configuration isobtained by using five 119891-Menelaus configurations

Proof We choose two triples of 119891-points (119886119894 120572119894) (119887119894 120573119894)

and (119888119894 120574119894) with collinear base points for 119894 = 1 2 and

such that neither of the three of the base points 1198861

1198871 1198862 1198872is collinear Then three intersection 119891-points

are (119886 120572) = (⟨1198862 1198871⟩ 1205722and 1205731) cap (⟨119886

1 1198882⟩ 1205721and 1205742)

(119887 120573) = (⟨1198862 1198871⟩ 1205722

and 1205731) cap (⟨119887

2 1198881⟩ 1205732

and 1205741)

and (119888 120574) = (⟨1198872 1198881⟩ 1205732

and 1205741) cap (⟨119886

1 1198882⟩ 1205721

and 1205742)

Let (1198981 1205831) = (⟨119887 119888⟩ 120573 and 120574) cap (⟨119887

1 1198882⟩ 1205731

and 1205742)

(1198982 1205832) = (⟨119886 119888⟩ 120572 and 120574) cap (⟨119886

2 1198881⟩ 1205722

and 1205741) and

(1198983 1205833) = (⟨119886 119887⟩ 120572 and 120573) cap (⟨119886

1 1198872⟩ 1205721and 1205732) If the

triples of 119891-collinear points (1198981 1205831) (1198882 1205742) (1198871 1205731)

(1198982 1205832) (1198862 1205722) (1198881 1205741) (119898

3 1205833) (1198872 1205732) (1198861 1205721)

(1198861 1205721) (1198881 1205741) (1198871 1205731) and (119887

2 1205732) (1198862 1205722) (1198882 1205742)

are 119891-Menelaus 6-figure with the three 119891-points (119886 120572)(119887 120573) (119888 120574) then 120583

1and 1205742

= 1205831and 1205731

= 1205742and 1205731

1205832and 1205722= 1205832and 1205741= 1205722and 1205741 1205833and 1205732= 1205833and 1205721= 1205721and 1205732

1205721and 1205741= 1205721and 1205731= 1205731and 1205741 and 120572

2and 1205742= 1205722and 1205732= 1205732and 1205742

It is easily seen that 1205831and 1205832= 1205831and 1205833= 1205832and 1205833 The triple of

119891-points (1198981 1205831) (1198982 1205832) (1198983 1205833) is 119891-Menelaus 6-figure

with the three 119891-points (119886 120572) (119887 120573) (119888 120574)


119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)

for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If the 119891-lines (⟨119886119894 119887119894⟩ 120572119894and

120573119894) 119894 = 1 2 3 are 119891-concurrent the configuration that

consists of these six119891-points is called an119891-Ceva 6-figureTheintersection point of the 119891-lines (⟨119886

119894 119887119894⟩ 120572119894and 120573119894) 119894 = 1 2 3

is called 119891-Ceva point



points in P and let (1198861 1205721) (1198862 1205722) (1198863 1205723) be three 119891-

points of FP Let points 1198871and 119887

2be chosen such that

1198871on ⟨119886

2 1198863⟩ and 119887

2on ⟨119886

1 1198863⟩ Suppose that the point 119887

3

on ⟨1198861 1198862⟩ is obtained by intersecting ⟨119886

1 1198862⟩ with the join

(⟨1198861 1198871⟩ cap ⟨119886

2 1198872⟩) and 119886

3 Then the 119891-point (119887

3 1205733) obtained

by intersecting (⟨1198861 1198862⟩ 1205721and 1205722) with the 119891-join of the two

119891-points (⟨1198861 1198871⟩ 1205721and 1205731) cap (⟨119886

2 1198872⟩ 1205722and 1205732) and (119886

3 1205723)

where (1198871 1205731) (1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723)

are 119891-collinear is independent of the chosen 119891-points (1198871 1205731)

and (1198872 1205732)

Proof Since three 119891-points (1198861 1205721) (1198863 1205723) (1198872 1205732) and

three 119891-points (1198862 1205722) (1198863 1205723) (1198871 1205731) are 119891-collinear 120572

1and

1205723

= 1205721and 1205732

= 1205723and 1205732and 120572

2and 1205723

= 1205722and 1205731

=

1205723and 1205731 One calculates that 120573

3= 1205721and 1205722and 12057223 and hence

the 119891-point (1198873 1205733) is independent of the chosen 119891-points

(1198871 1205731) and (119887

2 1205732)

Theorem 20 Let FP be a fibered projective plane withbase plane P Choose three 119891-points (119886

119894 120572119894) 119894 isin 1 2 3

in FP with noncollinear base points and with (119887119895 120573119895) isin

(⟨119886119894 119886119896⟩ 120572119894and 120572119896) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If the lines

⟨1198861 1198871⟩ ⟨1198862 1198872⟩ and ⟨119886

3 1198873⟩ in P are concurrent then three

119891-lines (⟨119886119894 119887119894⟩ 120572119894and 120573119894) for 119894 isin 1 2 3 are 119891-concurrent if

and only if 12057221and 1205722and 1205723= 1205721and 12057222and 1205723= 1205721and 1205722and 12057223


3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)

for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 Suppose that three 119891-lines(⟨119886119894 119887119894⟩ 120572119894and120573119894) for 119894 isin 1 2 3 are 119891-concurrentThen three

membership degrees in 119891-concurrent point 120572119894and 120572119895and 120573119894and 120573119895

119894 = 119895 119894 119895 = 1 2 3 are equal Since three 119891-points (119887119895 120573119895) isin

(⟨119886119894 119886119896⟩ 120572119894and 120572119896) 120572119894and 120572119895= 120572119894and 120573119896= 120572119895and 120573119896for 119894 = 119895 = 119896

119894 119895 119896 = 1 2 3 are valid one can easily get 12057221and 1205722and 1205723=

1205721and 12057222and 1205723= 1205721and 1205722and 12057223

Conversely by using119891-points (119887119895 120573119895) (119886119894 120572119894) and (119886

119896 120572119896)

that are collinear for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 in 12057221and 1205722and

1205723= 1205721and 12057222and 1205723= 1205721and 1205722and 12057223 it is shown that the three

values 120572119894and 120572119895and 120573119894and 120573119895 119894 = 119895 119894 119895 = 1 2 3 are equal

Corollary 21 (the fibered Ceva condition (FCC)) Let FPbe a fibered projective plane with base plane P Choose three119891-points (119886

119894 120572119894) 119894 isin 1 2 3 in FP with noncollinear base

points and the other three 119891-points (119887119896 120573119896) 119896 isin 1 2 3 with

(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 The

configuration that consists of these six119891-points is Ceva 6-figureif and only if 1205722


The following theorem shows that 119891-Ceva 6-figures canbe obtained as a corollary of 119891-Menelaus 6-figures


119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)

for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 the three lines ⟨119886119894 119887119894⟩ are

concurrent in P If the configuration that consists of these six119891-points is 119891-Menelaus 6-figure it is 119891-Ceva 6-figure

Proof Let the configuration picked such that the three119891-points (119886

1 1205721) (1198862 1205722) and (119886

3 1205723) and (119887

119895 120573119895) isin

(⟨119886119894 119886119896⟩ 120572119894and 120572119896) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 be Menelaus

6-figure Three membership degrees in 119891-intersection pointof three 119891-lines (⟨119886

119894 119887119894⟩ 120572119894and 120573119894) are 120572

119894and 120572119895and 120573119894and 120573119895

119894 = 119895 119894 119895 = 1 2 3 It is easily seen that these are equalSo three 119891-lines (⟨119886

119894 119887119894⟩ 120572119894and 120573119894) for 119894 isin 1 2 3 are

119891-concurrent

The reverse of this theorem is not true inFPFano projective plane denoted by 119875119866(2 2) consists of

seven points and seven lines Fano projective plane is onlyexample that is both Menelaus 6-figure and Ceva 6-figureEven if the base plane P of FP is Fano plane the reverseof the process is not always valid inFP


Theorem 23 Let and be a triangular norm Let FP be anynontrivial fibered projective plane with base plane P thatis Fano plane Let three 119891-points be (119886

119894 120572119894) 119894 isin 1 2 3

in FP with noncollinear base points and the other three 119891-points (119887

119896 120573119896) 119896 isin 1 2 3 with (119887

119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)


concurrent in P If the configuration that consists of these six119891-points is 119891-Ceva 6-figure it cannot be 119891-Menelaus 6-figure

Proof The configuration is picked such that 119891-points(1198861 05) (119886

2 05) (119886

3 05) and (119887

1 06) (119887

2 07) (119887

3 08) is

119891-Ceva 6-figure in FP But using the minimum operatorfor and it is easily seen that the 119891-points (119887

1 06) (119887

2 07) and

(1198873 08) are not 119891-collinear

4 Conclusion

In this paper the fibered versions of Menelaus and Ceva 6-figures in the fibered projective plane are defined Althoughthe conditions of the fibered versions of Menelaus andCeva 6-figures are similar the fibered versions of some oftheir properties in base projective plane cannot hold infibered projective plane 119891-Ceva 6-figure is obtained from119891-Menelaus 6-figure automatically Fano projective plane isboth Menelaus 6-figure and Ceva 6-figure But 119891-Menelaus6-figure cannot be obtained from 119891-Ceva 6-figure in fiberedprojective plane with Fano plane We have seen that thetriangular norms have important role in the fiber versions oftheorems related to theory

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] Z Akca A Bayar S Ekmekci and H Van Maldeghem ldquoFuzzyprojective spreads of fuzzy projective spacesrdquo Fuzzy Sets andSystems vol 157 no 24 pp 3237ndash3247 2006

[3] Z Akca A Bayar and S Ekmekci ldquoOn the classification offuzzy projective lines of fuzzy 3-dimensional projective spacerdquoCommunications Mathematics and Statistics vol 55 no 2 pp17ndash23 2006

[4] S Ekmekci Z Akca and A Bayar ldquoOn the classification offuzzy projective planes of fuzzy 3-dimensional projective spacerdquoChaos Solitons amp Fractals vol 40 no 5 pp 2146ndash2151 2009

[5] L Kuijken H VanMaldeghem and E Kerre ldquoFuzzy projectivegeometries from fuzzy vector spacesrdquo in Information Processingand Management of Uncertainty in Knowledge-Based SystemsA Billot et al Ed pp 1331ndash1338 Editions Medicales etScientifiques Paris France 1998

[6] L Kuijken and H Van Maldeghem ldquoFibered geometriesrdquoDiscrete Mathematics vol 255 no 1ndash3 pp 259ndash274 2002

[7] A Bayar S Ekmekci and Z Akca ldquoA note on fibered projectiveplane geometryrdquo Information Sciences vol 178 no 4 pp 1257ndash1262 2008

[8] R Kaya and S Ciftci ldquoOn Menelaus and Ceva 6-figures inMoufang projective planesrdquo Geometriae Dedicata vol 19 no 3pp 295ndash296 1985

[9] F B Kelly Ceva and Menelaus in Projective Geometry Univer-sity of Louisuille 2008

[10] M S Klamkin and A Liu ldquoSimultaneous generalizations of thetheorems of Ceva and Menelausrdquo Mathematics Magazine vol65 no 1 pp 48ndash52 1992

[11] F S Cater ldquoOn Desarguesian projective planesrdquo GeometriaeDedicata vol 7 no 4 pp 433ndash441 1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Preliminaries

We first recall some basic notions from fuzzy set theory andfibered geometry We denote by and a triangular norm on the(real) unit interval [0 1] that is a symmetric and associativebinary operator satisfying (119886 and 119887) le (119888 and 119889) whenever 119886 le 119888and 119887 le 119889 and 119886 and 1 = 119886 for all 119886 119887 119888 119889 isin [0 1]

Definition 1 (see [6]) Let P = (119875 119861 ∘) be any projectiveplane with point set 119875 and line set 119861 that is 119875 and 119861 are twodisjoint sets endowed with a symmetric relation ∘ (called theincidence relation) such that the graph (119875 cup 119861 ∘) is a bipartitegraph with classes 119875 and 119861 and such that two distinct points119901 119902 inP are incident with exactly one line every two distinctlines 119871119872 are incident with exactly one point and every lineis incident with at least three points A set 119878 of collinear pointsis a subset of 119875 each member of which is incident with acommon line 119871 Dually one defines a set of concurrent linesWe now define fibered points and fibered lines which arebriefly called 119891-points and 119891-lines

Definition 2 (see [6]) Suppose that 119886 isin 119875 and 120572 isin]0 1] Thenan 119891-point (119886 120572) is the following fuzzy set on the point set 119875ofP

(119886 120572) 119875 997888rarr [0 1] 119886 997888rarr 120572

119909 997888rarr 0 if 119909 isin 119875 119886 (1)

Dually one defines in the same way the 119891-line (119871 120573) for119871 isin 119861 and 120573 isin]0 1] The real number 120572 above is called themembership degree of the 119891-point (119886 120572) while the point 119886 iscalled the base point of it the same for 119891-lines

Definition 3 (see [6]) The two119891-lines (119871 120572) and (119872 120573) with120572 and 120573 gt 0 intersect in the unique 119891-point (119871 cap 119872 120572 and 120573)Dually the 119891-points (119886 120582) and (119887 120583) with 120582and120583 gt 0 span theunique 119891-line (⟨119886 119887⟩ 120582 and 120583)

Definition 4 (see [6]) A (nontrivial) fibered projective planeFP consists of a set 119865119875 of 119891-points ofP and a set 119865119861 of 119891-lines of P such that every point and every line of P are thebase point and base line of at least one 119891-point and 119891-linerespectively (with at least one membership degree differentfrom 1) and such thatFP = (119865119875 119865119861) is closed undertakingintersections of 119891-lines and spans of 119891-points Finally a setof 119891-points are called collinear if each pair of them span thesame 119891-line Dually a set of 119891-lines are called concurrent ifeach pair of them intersects in the same 119891-point

Theorem 5 (see [6]) Suppose that one has a fibered projectiveplane FP with base plane P that is Desarguesian Choosethe three 119891-points (119886

1 1205721) (1198862 1205722) and (119886

3 1205723) in FP with

noncollinear base points and the three other 119891-points (1198871 1205731)

(1198872 1205732) and (119887

3 1205733) with noncollinear base points such that

the lines ⟨119886119894 119887119894⟩ for 119894 isin 1 2 3 are concurrent in a point 119901 of

P with 119886119894

= 119887119894

= 119901 = 119886119894 Then the three 119891-lines (⟨119886

119894 119886119895⟩ 120572119894and120572119895)

and (⟨119887119894 119887119895⟩ 120573119894and 120573119895) (for 119894 = 119895 and 119894 119895 isin 1 2 3) intersect in

three collinear 119891-points

3 Fibered Version of Menelaus andCeva 6-Figures

The Alexandrian Greek mathematician Menelaus and theseventeenth-century Italian mathematician Ceva are invari-ably mentioned together Menelausrsquo theorem which involvesa test for the collinearity of three points and Cevarsquos theoremwhich involves a test for the concurrency of three lines arefrequently called the twin theorem These theorems shouldhave been discovered together and it is not insignificantthat such a long period separates Menelaus and CevaDuring the 1500 years that separate the two there waslittle development in mathematics About the year AD 100Menelaus of Alexandria extended a then well-known lemmato spherical triangles in his Sphaerica the high point of Greektrigonometry It is this lemma for the plane that today bearsthe name ofMenelaus Of theCeva brothers the lesser knownTommaso wrote on the cycloid while Giovanni resurrectedthe forgotten Menelausrsquo theorem and published it in 1678along with the twin theorem now known as Cevarsquos theorem

Wenow recall some observations regarding the triangularnorm and the Menelaus and Ceva 6-figures

In [7] the following theorems give an answer to thequestion when does a finite fibered projective plane existgiven a particular triangular norm

Theorem 6 (see [7]) Let P be a finite crisp projectiveplane and and a triangular norm Then there exists some finitenontrivial fibered projective plane FP with base plane P ifand only if there exists an idempotent element 120572 isin]0 1] thatis 120572 and 120572 = 120572

Theorem 7 (see [7]) Let P be an arbitrary crisp projectiveplane and and a triangular norm Then there exists somenontrivial fibered projective plane FP with base plane P ifand only if one of the following holds

(i) there exists an idempotent element 120572 isin]0 1[(ii) there exists an element 120572 isin [0 1[ with the property

that 120573 and 1205731015840 gt 120572 for all 120573 1205731015840 isin]120572 1]

In [7] the definition and theorems related to119891-harmonicconjugate in a fibered projective plane whose base planesatisfies Fanorsquos axiom are given

Definition 8 (see [7]) Suppose that one has a fibered projec-tive planeFPwith base projective planeP Choose the four119891-points (119886

1 1205721) (1198862 1205722) (1198863 1205723) and (119886

4 1205724) in FP none

of the three base points of which are collinearThese119891-pointsare called 119891-vertices The configuration that consists of thesefour119891-points the six119891-lines (119860

119894119895 120573119894119895

) = (⟨119886119894 119886119895⟩ 120572119894and120572119895)

for 119894 = 119895 119894 119895 isin 1 2 3 4 (which we call 119891-sides) and thethree119891-points (119860

119894119895and119860119896119897

1205721and1205722and1205723and1205724) with 119894 119895 119896 119897 =

1 2 3 4 (the 119891-diagonal points) is called an 119891-completequadrangle

Theorem 9 (see [7]) Suppose that one has a fibered projectiveplaneFP with base planeP Let 119886 119887 and 119888 be three collinearpoints in P and let (119886 120572) (119887 120573) (119888 120574) be three 119891-points ofFP Suppose that the point119889 on ⟨119886 119887⟩ obtained by intersecting







111988621198863 119887111988721198873) such


1isin


1 1198862 1198863 1198871




1 1198871⟩ ⟨1198862 1198872⟩ ⟨1198863 1198873⟩ are concurrent

respectively



119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895) for

119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If the 119891-points (119887119896 120573119896) are 119891-


119896 120573119896) for 119896 = 1 2 3






1and

1198872where 119887

1on ⟨1198862 1198863⟩ and 119887



1 1198862⟩ 1205721and 1205722) with the 119891-


2 1205732) where (119887

1 1205731)

(1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723) are 119891-collinear


2 1205732)



1205721and 1205723= 1205721and 1205732= 1205723and 1205732and 120572

2and 1205723= 1205722and 1205731= 1205723and 1205731



1 1205731)

and (1198872 1205732)


1 1205721)

(1198862 1205722) and (119886




two 119891-points (1198871 1205731) and (119887

2 1205732) where (119887

1 1205731) (1198862 1205722)

(1198863 1205723) and (119887





3 1205733)


119894 120572119894) (119887119894 120573119894) 119894 isin 1 2 3 is an



119894 120572119894) 119894 isin 1 2 3 in FP


119894 119886119896⟩ 120572119894and

120572119896) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If 119887

1 1198872 and 119887

3in P



1205721and 1205722and 12057223


3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)



119894 120572119894) (119886119895 120572119895)


120573119894and 120573119895= 120573119894and 120573119896and 120572





(1198871 1205731) (1198872 1205732) and (119887

3 1205733) are 119891-collinear



(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3


1and 1205722and 1205723= 1205721and 12057222and 1205723=

1205721and 1205722and 12057223



1 1205722 and 120572











are (119886 120572) = (⟨1198862 1198871⟩ 1205722and 1205731) cap (⟨119886

1 1198882⟩ 1205721and 1205742)

(119887 120573) = (⟨1198862 1198871⟩ 1205722

and 1205731) cap (⟨119887

2 1198881⟩ 1205732

and 1205741)

and (119888 120574) = (⟨1198872 1198881⟩ 1205732

and 1205741) cap (⟨119886

1 1198882⟩ 1205721

and 1205742)

Let (1198981 1205831) = (⟨119887 119888⟩ 120573 and 120574) cap (⟨119887

1 1198882⟩ 1205731

and 1205742)

(1198982 1205832) = (⟨119886 119888⟩ 120572 and 120574) cap (⟨119886

2 1198881⟩ 1205722

and 1205741) and

(1198983 1205833) = (⟨119886 119887⟩ 120572 and 120573) cap (⟨119886

1 1198872⟩ 1205721and 1205732) If the


(1198982 1205832) (1198862 1205722) (1198881 1205741) (119898

3 1205833) (1198872 1205732) (1198861 1205721)

(1198861 1205721) (1198881 1205741) (1198871 1205731) and (119887

2 1205732) (1198862 1205722) (1198882 1205742)


1and 1205742

= 1205831and 1205731

= 1205742and 1205731


1205721and 1205741= 1205721and 1205731= 1205731and 1205741 and 120572

2and 1205742= 1205722and 1205732= 1205732and 1205742





119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




119894 119887119894⟩ 120572119894and 120573119894) 119894 = 1 2 3







1198871on ⟨119886

2 1198863⟩ and 119887

2on ⟨119886


3



(⟨1198861 1198871⟩ cap ⟨119886

2 1198872⟩) and 119886

3 Then the 119891-point (119887

3 1205733) obtained


119891-points (⟨1198861 1198871⟩ 1205721and 1205731) cap (⟨119886

2 1198872⟩ 1205722and 1205732) and (119886

3 1205723)

where (1198871 1205731) (1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723)


and (1198872 1205732)



1and

1205723

= 1205721and 1205732

= 1205723and 1205732and 120572

2and 1205723

= 1205722and 1205731

=


3= 1205721and 1205722and 12057223 and hence


(1198871 1205731) and (119887

2 1205732)


119894 120572119894) 119894 isin 1 2 3



⟨1198861 1198871⟩ ⟨1198862 1198872⟩ and ⟨119886





3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)






1205721and 12057222and 1205723= 1205721and 1205722and 12057223


119896 120572119896)







(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 The





119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




1 1205721) (1198862 1205722) and (119886

3 1205723) and (119887

119895 120573119895) isin



119894 119887119894⟩ 120572119894and 120573119894) are 120572

119894and 120572119895and 120573119894and 120573119895


119894 119887119894⟩ 120572119894and 120573119894) for 119894 isin 1 2 3 are

119891-concurrent





119894 120572119894) 119894 isin 1 2 3


119896 120573119896) 119896 isin 1 2 3 with (119887

119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




2 05) (119886

3 05) and (119887

1 06) (119887

2 07) (119887

3 08) is


1 06) (119887

2 07) and


4 Conclusion




References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra















111988621198863 119887111988721198873) such


1isin


1 1198862 1198863 1198871




1 1198871⟩ ⟨1198862 1198872⟩ ⟨1198863 1198873⟩ are concurrent

respectively



119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895) for

119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If the 119891-points (119887119896 120573119896) are 119891-


119896 120573119896) for 119896 = 1 2 3






1and

1198872where 119887

1on ⟨1198862 1198863⟩ and 119887



1 1198862⟩ 1205721and 1205722) with the 119891-


2 1205732) where (119887

1 1205731)

(1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723) are 119891-collinear


2 1205732)



1205721and 1205723= 1205721and 1205732= 1205723and 1205732and 120572

2and 1205723= 1205722and 1205731= 1205723and 1205731



1 1205731)

and (1198872 1205732)


1 1205721)

(1198862 1205722) and (119886




two 119891-points (1198871 1205731) and (119887

2 1205732) where (119887

1 1205731) (1198862 1205722)

(1198863 1205723) and (119887





3 1205733)


119894 120572119894) (119887119894 120573119894) 119894 isin 1 2 3 is an



119894 120572119894) 119894 isin 1 2 3 in FP


119894 119886119896⟩ 120572119894and

120572119896) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 If 119887

1 1198872 and 119887

3in P



1205721and 1205722and 12057223


3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)



119894 120572119894) (119886119895 120572119895)


120573119894and 120573119895= 120573119894and 120573119896and 120572





(1198871 1205731) (1198872 1205732) and (119887

3 1205733) are 119891-collinear



(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3


1and 1205722and 1205723= 1205721and 12057222and 1205723=

1205721and 1205722and 12057223



1 1205722 and 120572











are (119886 120572) = (⟨1198862 1198871⟩ 1205722and 1205731) cap (⟨119886

1 1198882⟩ 1205721and 1205742)

(119887 120573) = (⟨1198862 1198871⟩ 1205722

and 1205731) cap (⟨119887

2 1198881⟩ 1205732

and 1205741)

and (119888 120574) = (⟨1198872 1198881⟩ 1205732

and 1205741) cap (⟨119886

1 1198882⟩ 1205721

and 1205742)

Let (1198981 1205831) = (⟨119887 119888⟩ 120573 and 120574) cap (⟨119887

1 1198882⟩ 1205731

and 1205742)

(1198982 1205832) = (⟨119886 119888⟩ 120572 and 120574) cap (⟨119886

2 1198881⟩ 1205722

and 1205741) and

(1198983 1205833) = (⟨119886 119887⟩ 120572 and 120573) cap (⟨119886

1 1198872⟩ 1205721and 1205732) If the


(1198982 1205832) (1198862 1205722) (1198881 1205741) (119898

3 1205833) (1198872 1205732) (1198861 1205721)

(1198861 1205721) (1198881 1205741) (1198871 1205731) and (119887

2 1205732) (1198862 1205722) (1198882 1205742)


1and 1205742

= 1205831and 1205731

= 1205742and 1205731


1205721and 1205741= 1205721and 1205731= 1205731and 1205741 and 120572

2and 1205742= 1205722and 1205732= 1205732and 1205742





119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




119894 119887119894⟩ 120572119894and 120573119894) 119894 = 1 2 3







1198871on ⟨119886

2 1198863⟩ and 119887

2on ⟨119886


3



(⟨1198861 1198871⟩ cap ⟨119886

2 1198872⟩) and 119886

3 Then the 119891-point (119887

3 1205733) obtained


119891-points (⟨1198861 1198871⟩ 1205721and 1205731) cap (⟨119886

2 1198872⟩ 1205722and 1205732) and (119886

3 1205723)

where (1198871 1205731) (1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723)


and (1198872 1205732)



1and

1205723

= 1205721and 1205732

= 1205723and 1205732and 120572

2and 1205723

= 1205722and 1205731

=


3= 1205721and 1205722and 12057223 and hence


(1198871 1205731) and (119887

2 1205732)


119894 120572119894) 119894 isin 1 2 3



⟨1198861 1198871⟩ ⟨1198862 1198872⟩ and ⟨119886





3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)






1205721and 12057222and 1205723= 1205721and 1205722and 12057223


119896 120572119896)







(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 The





119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




1 1205721) (1198862 1205722) and (119886

3 1205723) and (119887

119895 120573119895) isin



119894 119887119894⟩ 120572119894and 120573119894) are 120572

119894and 120572119895and 120573119894and 120573119895


119894 119887119894⟩ 120572119894and 120573119894) for 119894 isin 1 2 3 are

119891-concurrent





119894 120572119894) 119894 isin 1 2 3


119896 120573119896) 119896 isin 1 2 3 with (119887

119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




2 05) (119886

3 05) and (119887

1 06) (119887

2 07) (119887

3 08) is


1 06) (119887

2 07) and


4 Conclusion




References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















are (119886 120572) = (⟨1198862 1198871⟩ 1205722and 1205731) cap (⟨119886

1 1198882⟩ 1205721and 1205742)

(119887 120573) = (⟨1198862 1198871⟩ 1205722

and 1205731) cap (⟨119887

2 1198881⟩ 1205732

and 1205741)

and (119888 120574) = (⟨1198872 1198881⟩ 1205732

and 1205741) cap (⟨119886

1 1198882⟩ 1205721

and 1205742)

Let (1198981 1205831) = (⟨119887 119888⟩ 120573 and 120574) cap (⟨119887

1 1198882⟩ 1205731

and 1205742)

(1198982 1205832) = (⟨119886 119888⟩ 120572 and 120574) cap (⟨119886

2 1198881⟩ 1205722

and 1205741) and

(1198983 1205833) = (⟨119886 119887⟩ 120572 and 120573) cap (⟨119886

1 1198872⟩ 1205721and 1205732) If the


(1198982 1205832) (1198862 1205722) (1198881 1205741) (119898

3 1205833) (1198872 1205732) (1198861 1205721)

(1198861 1205721) (1198881 1205741) (1198871 1205731) and (119887

2 1205732) (1198862 1205722) (1198882 1205742)


1and 1205742

= 1205831and 1205731

= 1205742and 1205731


1205721and 1205741= 1205721and 1205731= 1205731and 1205741 and 120572

2and 1205742= 1205722and 1205732= 1205732and 1205742





119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




119894 119887119894⟩ 120572119894and 120573119894) 119894 = 1 2 3







1198871on ⟨119886

2 1198863⟩ and 119887

2on ⟨119886


3



(⟨1198861 1198871⟩ cap ⟨119886

2 1198872⟩) and 119886

3 Then the 119891-point (119887

3 1205733) obtained


119891-points (⟨1198861 1198871⟩ 1205721and 1205731) cap (⟨119886

2 1198872⟩ 1205722and 1205732) and (119886

3 1205723)

where (1198871 1205731) (1198862 1205722) (1198863 1205723) and (119887

2 1205732) (1198861 1205721) (1198863 1205723)


and (1198872 1205732)



1and

1205723

= 1205721and 1205732

= 1205723and 1205732and 120572

2and 1205723

= 1205722and 1205731

=


3= 1205721and 1205722and 12057223 and hence


(1198871 1205731) and (119887

2 1205732)


119894 120572119894) 119894 isin 1 2 3



⟨1198861 1198871⟩ ⟨1198862 1198872⟩ and ⟨119886





3 1205723) and (119887

119895 120573119895) isin (⟨119886

119894 119886119896⟩ 120572119894and 120572119896)






1205721and 12057222and 1205723= 1205721and 1205722and 12057223


119896 120572119896)







(119887119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and120572119895) for 119894 = 119895 = 119896 119894 119895 119896 = 1 2 3 The





119894 120572119894) 119894 isin 1 2 3 in FP


119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




1 1205721) (1198862 1205722) and (119886

3 1205723) and (119887

119895 120573119895) isin



119894 119887119894⟩ 120572119894and 120573119894) are 120572

119894and 120572119895and 120573119894and 120573119895


119894 119887119894⟩ 120572119894and 120573119894) for 119894 isin 1 2 3 are

119891-concurrent





119894 120572119894) 119894 isin 1 2 3


119896 120573119896) 119896 isin 1 2 3 with (119887

119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




2 05) (119886

3 05) and (119887

1 06) (119887

2 07) (119887

3 08) is


1 06) (119887

2 07) and


4 Conclusion




References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 120572119894) 119894 isin 1 2 3


119896 120573119896) 119896 isin 1 2 3 with (119887

119896 120573119896) isin (⟨119886

119894 119886119895⟩ 120572119894and 120572119895)




2 05) (119886

3 05) and (119887

1 06) (119887

2 07) (119887

3 08) is


1 06) (119887

2 07) and


4 Conclusion




References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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