EM2
Relahisn oset and Latice
Relaigm: et A and B be 2 pon-eMpty Le&
Ae AXB u Ris a slkatian Arom A n B
ORe AXB and ab)e R L ay a Rb_lais
elattd to8)-
04aiau a a Rb.
RCA XA i alaha nA
inbtd m AoA
AXB A: (,2) B (3,)
AXB (,8),(1,4), (2,3), (2, 4)
Ex A-ab,cB to B: 1,22{diteumia ekhau
S(aCa,2) ,Co,D,Cb,2)
A XB aRb
G R (a),(a,2),Co 2)
R Axb ab
aksb
Lin) Ra= &Cc, 1, c,2) (C,)r a Rub
)Ry2 9Cb,2) Ry EAXB
a Rgb
> ) Re = RaEAx
An is alo0o3 a9Ol atiou_
| Domain_andkange f Kelatian8
omain ut A,Bbe henpty 8eh aRb
2then domain ofR is a 9ubset _of A
t Damain of R 2alab) 6 R_
Rarge Kage af Ri a ubset of B_
ieRame o_R lblab) ER?_
n vehee_oRela~io
let_R_be a HelationAem A tn B Hun thenuRe d A_
is collecbidn of_all (b,a)_Suth Hhat (a,b)LR. dundkd byk
R b,a) Cab)ER
R (23), (1s)3 SC3,2), (5,4)
a A 12,5,13,R doliaad by xRy i 2sy-
R2L2).lL3),A,),e,2.(2,42,S,)}
R C2,0.(3,),4,)(3,2) Cu,2) (4,3) }
in wods R 2MRy f yk
uSbe a claktan_sn N_ inego1s, &uch that
t u 13
S (ab)| a,b EN, a+3h =13 F
MAx valuo of = 13_min 4
max alua o yE 4 2-3b
Laluss ,2,3, 4
paudhag ?10,7, 4
(D,D,(7,2), CY,32,(1,)
8 ),(2) (3. (M,D
y: 13-
3
u RAud be sllouu Radan o A 23
2) (2,3j (3,1) (3132
SC2,3),Cu3),(2,1D,3,3)3
Ro9 as),(3,3)
R1 D ,U2),23)3.D(S,3)03) 21)}
3 2,2) (2062)X
AA D1,2) (13) (2) (2,2)2,3,e0,N:3E
AKepePAENdatiou KelationS
Dioaphs A is finiE Lek and Re A
Ra _Can paaek
Druo a Anall sclo o eada elundutg A
Dro an asbaO Col edge M
uoKA a; Ao ai i aiRa
A Su23,4?
R ),L,2,(2,1,(2,3) GD,8,,(,1)
find uhkia
R a,),(s2,G.),4a_
.
* The madx _of a4Alakia
u can kpaeLent_a yelattau COnkinlaa Mauda elo4,
A a a2,ag-
B:hbabz b
nin a_Matx Ma LMi leeaA
Mi ai,b)CR Le Rbj
0 4 Cab)_ R iic aiR b
R C ,(230,(3,2) ,C33)
2 3 B
A$ 2 3 Me
B 2 3
3
A
M, 0 1 JA3
I4
TOTO
2
B-b ba bgb
Rab2, (aba)b) bs)(aab),@bsl_
XDepnatians
In-deore Let Rj8 a ulahion on A %aEA_Men
in-degau o a £AB tha nunmbe o b
Suth Haadt aRb l n0-b ðeR
Out-dagnee: lat Ris a elakoM @hA_ Hha outdog4R
aA_ tha no ab 2u6h tha
a
aRb ie no qblab kR
2 8 uls
TD| 2| 2 22
CD412||
e Find 0atax laton Qd 2elaho diq
A-L2S4St=6. aRb l b uutHplo f=
R (,2,1,3)C1u,)(S),4,2),(2,4),C3,2)G)SSs
2 O LO
3 0
1D
2 2 |3 2
OD 5
Find A And its diqqph. ist Dand 0D
Ar la,b,.cdek
a lo d e
Coo 0 L Ra,a, a,b), (be) b,a),
c,ale,e) Ldb),lal,c
Lea}
A_b de
L 2 2 2 2 I
OD|2
Eind clationde degined hy digaaph ad ukate atak
A 3 ,4,5
R:§C2) ,2,2), (2,3),(8,) 4,)(5,DSY
#Path in ulation aud Digsaph3
Suppsee Hoat R n_A
A ath of Longth a inR_pemn a wb s a faite Segusnct
aR R pR
Nok i Apatta that bginh and eudblita SaML ebmut
A o_lngth nuotus elenu q A
Det neLSR aily_distinct
ConngckiuitXalaton
A latian danokd bu K and dfins.d os R ta
a patla oany ugth rom n y ihR- i3
CallLd Connaceuy sLlation. es R
Aab,cde R- Saa),a,b),b)od)eck) d
_(smpudcR and R_
Digaph R:
R abCac) b Ccb E R
Cab)(a. b,A), (b,e) ,ace)f
RS[ab)w ig a pafh q au Lungih Artm
Laab],ad,a (ae),(bc ()he,Lea (
Path d ngth 1L
2),(.6).C23) (3,3) ,4)
Kayh d lonat2 uitex 2
(23)
332222 3,3
P332324
12z 2 3 22 3,2
6 Lu 3,6
33 3 3,5
8 3 3 3,3
3 4 3 34
3 >5 3,S
3 12-328 ,3)
z]P2 y2/
222229(23
6 3 3 3222 (
b S _ 6,5
22U25.
| 3 3 1 23
Adle at
2 Y 22
Eind aisole at G
DLaL0digsaph R
R C LD 2.3) (2,40 (a,3) (3,1D 3,4)Ca,5
fiAL MA
M 2 O
M ol o o6 o 1oeo11laol OO
7otot6
vil
SD G, (1,3) 1,) (1, 5) L62
(2,0(2,2) (33) (2) (215) (26)
(3,10 (82 (33) (3D (3,)(3,6
(uD (Y2) (13 (4M) (4S) ( 6)
EnpasitionRalation
A,.c be the 8etg ARBand BSc Loe ca
deinesu laton u CoMpagitou d Rand& ltten
8.R
SoR 2 (a.c |aRb, Sck
aeACEC han. a (SoR Ja _o, ona bEB
BSa,b,et C ,z
S0R (a,c)aRb, b8 c R-CL)Laa,c)2
CL,2pa,zl} Saghn)lel
4
Mg M
b o
MoR
3
A L23,9
RaC, (3,1)13,4) Y)L2,
S?L8) (2,13 (3,1) 3,2) (4, 2
Ro3 G 3) (3,3) (3,) CLD 2 )
SAR X() C, u) (21) (3.1) (Y 2) (3) S
RROR- ac) aRb, LRe
S C1)_(3,1) (3,2) 3,3) (L)CY4)
M
Ma 2leet+
3
R KOR_?I,D(30(3,) (1), 9,2) (u3)
3
M 2ll
OE -e-A2
R F )E R
) 1,a) (3) (2,4) C3,4)
RoR )( 3)3 D38)
(u)
e'oR C01M)(2,2(2,3)(3,2)(3,3)
(Xuu)_
22 3
R o
2
Aopeatiea Rlatiana
KepHexiue:i (nn)e R_V2 E A ie nRa vnéA-
diageal xelatio 2Ra t n£A buk 2Ky t A
donoted by A A=Sma) d néA
also
eauality ulatio
3 lepLexiu
MoteiARA (x¢lExiue )AGR
ARA imeeliunAAR = ¢.
5 Non-Summatnic 2R by Ra oMe (1) 6K
AsummanMe 21Ry 4RL t_(miy) ER
Nae This is a paiicula cse c nat
Summekznc
Anh-8ummgtn 1p nky yRn
R issmet L R: R
Ris ayM eROR
R R isaui -qm i RRCA
8Taansitiuu i gER aud qz CR ta
ln2)ER
Nok: MR:MR tas R s aunitu
Let A SL2,34 Ruteiut,Sum, 9Sum,anti Bym
aRSG) _2)(2,) (2) (3,3) ENA34,y)
AA LD,22) (3,3) (,O{ 2diagoual
AC R
Ris «lLxiua.
) 8 my) ER han ER_#au n 3um
R CL,1) (2,1) ti 2) (2,2) (42 U3)3,){,
R:R'_
R
_RoR
RaRR
RO 4A R Is not AnHaum
3
M
M
Ris ansitius
Eauivaloco Rolatton
ARA
a
RIs aRx
Repoxiue
eALuLvalaaasi itis
Sumatai.c 2Ry Ra
cToaniua 2Ry&Rz nRz-
iaculaxRalatjen:
ARA e úhulah ip aRb and bRc e
cRa
Odeaune
i clLLuL
Nok:
R. AWR
) AS,2,33 RCLDC12) (2,3
A:SL)L2,2)(3,3\L
Ropuex iuA dsLuA k a AUR
mnetie deguh
Symthe-3C
gCis R ieelf
_/
@ Tansitiu cleAo
T Smallat tansitiue xlatan hat ntaius R i
tranaitiuaCLesuh
iR R
i T.congh-tmnstiueLlatiaia Can_behund
bing axihalls alaaxthm.
LWanahall's alqasithn
A S113,ut R ,2) (2,D(2,2) (4,2) (3,)
(2) C ) e R but0) LR_hono
Risnot teanStiue.
Mas G s_psittouS:
3
R pos.itto
Insett4 at cembinatious
lc R):2) G2-
2
G 2,3
R 22
-e e- (2
GR) Li2(1)(2)
34
Cy O no 4 sut
6
R (1)u2) L2,) C2,1) (3,d3,2)(4)(YS CY3)3
Poset
ARA panhal o da i it is etaxiueanh&unmaAtic.
And tansiti (A,R is phialb ardaed get au peI
R Replexiue
(AR)2_pesek_
Taansit ue
R Real no.s Rilatisn _n R:K:R<R
Rppexiu asa aa)
Anti gqmsmstac a,b eR a s b
and b Sa
asb
Iransuhu q,b,c ER Asb and bs
aSC
(R,S) s pc9et Sinco is patia g,do
Ran 2 as aRbip alb
hLn pesoue tu Ris pahal ad
K xiu: a¬ z ala , ala
PniauM 4E ,béz i alb ond bla A=b
ansih4g ,bcEz if Qlb and blC = ac
XCompoable Lima_Osda
CeposablkIn a Pe8et (A, <, elasmat a gaid to be
CeMpasable
Nok n a oser eus Llomnt_10Ld not be
mpanable
ineox 0de) 90 14_eu eanmaut oA mpaJabl
i t is lta adod s Chain @ totally
e41:s) GdoA
:(z,D_X_Sind 23 e B15_ist onpau
1 (A.R) is posst than (AR is a o3t.
R:_aRa -
a EA_
aR a Y EA
e: aRb bRa a = b
bRa aRb 2ab Antisum
(ii R: Rb b Re
aRe
bAa cR'b_2 cRa
oanstus
(A,R a_poset.
Hasse diagsam oa ser
oagnam-paxhal dea 0an be implikedand s called
halBe diagaaM
udhen barttal_asdes i atal sdeh its Hase
a9taiglet linsaud (Aespaudiu4pSe ig Calld1
Chain
A 2 RCII) (,29(12) (22) (2,G +
3 3
2
Hasse daigaam
1 A L2,34,2 R ckuide
R:YD1a)).1HDL12) (2,2)(24) (2,12) (33)
(312)(D2) L2,12)}
12
&3. a,bcs A= PS) Potial SaE)
DrauD ee AG)
A:20,a,b,c, ahbc, ca abe {
abe
abe
alo
betes
tooRiug
(2,2)(2,4)(3,3)6,9(
R010.2)03) (,)
3AS12,3,4
2 3
A ab.det
(bib)lhc)b,d) Ch,e) G) lc.d)
R(a,) Cac) (ad) a.e)
Cce) Cd,d) (e,e)
Quasi Ooudes
ARA,R uas1 Csdes s tuausitius and LksepLIAIL
A n 2 ic xal,-55n 201 i s a quasi slation
xelexi u -
is eftoviu
Tansiu t yyz-
Posek ia0 moplugm-
A<) awd (A,S ba too_posalsaud
13aid D iSsnGApuip4 YA,btA,aSb
fa)s£Cb) in A'
A A sisomohi3M AS qus
CA,S)a i3oMaLpic Poseks
| AI2,3.5,6,10,1S,80bB
Maantaat alb
erA PCs)uhha S fela he tu pog£t_
C
Shao A,S) aud CAC arigamaLpuiS
LA Da3)(LLe),o) (LS)(1.30 (2) (2.0a,10)30
C3, (3.) GIS) (3,30) (5,)10)(5)s30)l6)63 =
(o16)CIo 90 (IS.1S) (iS,30) (3s, 30)
Te
PaseotA- |f(3)>E
CS) 9
FLC)-
TF(10)> er
C ALpondan ASO M O p L
-
Noko Tu00 pases au aomafale4 Aasua Hase dkagas
aR D Lectiu idoutoalL Lstaptlabels
A 213, 6 uday
(A):,D1,2)4,3) LO (22) 2,0 G3)3,) ¢
(A) 4,a,ba) }
in A
f CC) =a,b
#Extexnal element ot Yoaetg
u (As) br pose aEA
Nayi aalL eluomen a k Maximal ellMoud laM Hue
0Db9uCh aat asb_
Heighus Hess Hacs e diagram
Minima elomaut louak d Hasse
Maximal 3,5
4 Minima 6
uutm
AD:D(24)ROEZ8) Q,2u)(2,us) (3,3)
(3,6)C3,24) (3,u8)(y,u)8)CY2u](, 8
(66) CM) GuEO(S,8) (8,29) (8,48)
C24,24)OY,uS)_(43, 43)}
Max
Mia 2A3
