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The syntactic graph of a sofic shift is invariant under shift equivalence
Marie-Pierre Béal, Francesca Fiorenzi, Dominique Perrin
To cite this version:
Marie-Pierre Béal, Francesca Fiorenzi, Dominique Perrin. The syntactic graph of a sofic shift is
invariant under shift equivalence. International Journal of Algebra and Computation, World Scientific
Publishing, 2006, 16 (3), pp.443-460. �hal-00619735�
is invariant under shift equivalene
Marie-Pierre Beal
Franesa Fiorenzi
DominiquePerrin
Abstrat
We dene a new invariant for shift equivalene of so shifts. This
invariant,thatweallthesyntatigraphofasoshift,isthedireted
ayligraphofharateristigroupsofthenonnullregularD-lassesof
thesyntatisemigroupoftheshift.
Keywords: Automataandformallanguages,symbolidynamis.
1 Introdution
So shifts [22℄ are sets of bi-innite labels in a labeled graph. If the graph
an be hosen strongly onneted, the so shift is said to be irreduible. A
partiularsublassofsoshiftsisthelassofshiftsofnitetype,denedbya
nitesetofforbiddenbloks. TwososhiftsXandY areonjugateifthereisa
bijetiveblokmapfromXontoY. Itisanopenquestiontodeidewhethertwo
soshiftsareonjugate,eveninthepartiularaseofirreduibleshiftsofnite
type. Thereisanotionweakerthanonjugay,alledshiftequivalene(see[18,
Setion7.3℄). Therefore,invariantsfor shiftequivaleneare alsoinvariantsfor
onjugay.
Therearemanyinvariantsforonjugayofshifts,algebraiorombinatorial,
see [18, Chapter 7℄, [7℄, [17℄, [3℄. Forinstane the entropy is a ombinatorial
invariant whih gives the omplexity of allowed bloks in a shift. The zeta
funtion is another invariant whih ounts the numberof periodi orbitsin a
shift.
Inthis paper, wedene anewinvariant forshiftequivalene ofirreduible
soshifts. Thisinvariantisbasedonthestrutureofthesyntatisemigroup
of the language of nite bloks of the shift. An irreduible so shift has a
unique(up to isomorphismsof automata) minimaldeterministi presentation,
alled itsrightFisherover. The syntatisemigroup S(X)of anirreduible
soshiftX isthetransitionsemigroupofitsrightFisherover.
InstitutGaspard-Monge,UniversitedeMarne-la-Vallee,77454Marne-la-ValleeCedex2,
Frane. fbeal,fiorenzi,perringun iv- mlv. fr
relations (denoted R, L, H ;D;J) [20℄. Our invariant is the ayli direted
graphwhosenodesthenonnullregularD-lassesofS(X)labeledbytheirrank
andtheir harateristi group. The edgesorrespondto the partial order
J
betweenthese D-lasses. Weallitthesyntatigraphofthe soshift. The
resultanbeextendedtotheaseofreduiblesoshifts.
We rst prove the onjugay invariane of the syntati graph, and using
this, we prove the shift equivalene invariane. The proof of the onjugay
invariane is based on Nasu's Classiation Theorem for so shifts [19℄ that
extends William's one for shifts of nite type. This theorem says that two
irreduibleso shifts X;Y are onjugateif andonly ifthere is a sequeneof
symboliadjaenymatriesofrightFisheroversA = A
0
; A
1
;:::;A
l 1
;A
l
=
B,suhthat A
i 1 andA
i
are elementarystrong shiftequivalentfor1il,
whereAandB aretheadjaenymatriesoftherightFisheroversofX and
Y, respetively. This meansthat, for eah i, there aretwo symboli matries
U
i and V
i
suh that, after reoding the alphabets of A
i 1 and A
i
, we have
A
i 1
=U
i V
i and A
i
=V
i U
i
. Abipartiteshiftisassoiatedin anaturalwayto
apairofelementarystrongshiftequivalentandirreduiblesoshifts[19℄.
The key point in our invariant is the fat that an elementary strong shift
equivalene relationbetweenadjaeny matries implies someonjugay rela-
tionsbetweentheidempotentsinthesyntatisemigroupofthebipartiteshift.
Weshowthatpartiularlassesofirreduiblesoshiftsanbeharaterized
withthissyntatiinvariant: thelassofirreduibleshiftsofnitetypeandthe
lassofirreduibleaperiodisoshifts.
Arelatedinvariantharaterizingreduibleso shiftsand whihusessyn-
tatiproperties hasbeenpresentedin [11℄. It is alattiewhose vertiesrep-
resentthesub-synhronizingsubshiftsoftheshift. Somevertiesofthis lattie
orrespond to the vertiesof rank 1in our syntati graph. Other invariants
ofaso shift,asthederivedshiftspaesand thedepthoftheshift,aregiven
in[21℄.
Basidenitionsrelatedtosymbolidynamis aregiveninSetion2.1. We
referto [18℄ or[14℄ for moredetails. Seealso [15℄, [16℄, [5℄aboutso shifts.
Basidenitionsandpropertiesrelatedtonitesemigroupsandtheirstruture
are given Setion 2.2. We refer to [20, Chapter 3℄ for a moreomprehensive
expository. Nasu'sClassiationTheorem isrealledin Setion 2.4. Weprove
the onjugay invariane of the syntati graph in Setion 3. A omparison
betweenthissyntatiinvariantandsomeotherones whihare wellknown,is
giveninSetion4. InSetion 3.1,weextendtheresulttotheaseofreduible
so shifts. In Setion 5, wereall the denition of shift equivalenebetween
so shiftsand weprovethat the syntati graphisalso invariantunder shift
equivalene. PartofthispaperwaspresentedattheonfereneSTACS'04 [4℄.
2.1 So shifts and their presentations
LetAbeanitealphabet,i.e.aniteset ofsymbols. Theshiftmap:A Z
!
A Z
is dened by ((a
i )
i2Z ) = (a
i+1 )
i2Z , for (a
i )
i2Z 2 A
Z
. If A Z
is endowed
with the produt topology of the disrete topology on A, a shift is a losed
-invariantsubsetofA Z
.
IfX isashiftofA Z
andnapositiveinteger,thenthhigher power ofX is
theshiftof(A n
) Z
dened byX n
=f(a
in
;:::;a
in+n 1 )
i2Z j(a
i )
i2Z 2Xg.
A nite automaton is a nite multigraph labeled by A. It is denoted
A = (Q;E), where Q is anite set of states, and E anite set of edges la-
beledbyA. It isequivalentto asymboli adjaeny (QQ)-matrix A, where
A
pq
is thenite formalsum ofthe labelsof alltheedges from pto q. Aso
shift isthesetofthelabelsofallthebi-innitepathsonaniteautomaton. If
Aisaniteautomaton,wedenotebyX
A
theso shiftdenedbytheautoma-
tonA. Several automata andene thesame so shift. Theyare also alled
presentations orovers ofthesoshift. Wewillassumethatallpresentations
areessential: allstateshaveatleastoneoutgoingedgeandoneinomingedge.
An automaton is deterministi if for any given state and any given symbol,
thereisatmostoneoutgoingedgelabeledbythisgivensymbol. Asoshiftis
irreduible ifithasapresentationwithastronglyonnetedgraph. Irreduible
soshiftshaveaunique(uptoisomorphismsofautomata)minimaldetermin-
istipresentation,that isadeterministipresentationhavingthefewest states
among alldeterministi presentationsof the shift. This presentation is alled
theright Fisher over oftheshift.
Let A = (Q;E) be a nite deterministi (essential) automaton on the al-
phabet A. Eah nite word w of A
denes a partial funtion from Q to Q.
Thisfuntion sends thestatepto thestate q, ifwis thelabelof apathfrom
ptoq. The semigroupgeneratedbyallthese funtions isalled thetransition
semigroup oftheautomaton. WhenX
A
is notthefull shift,thesemigrouphas
anullelement,denoted 0,whihorrespondstowordswhiharenotfatorsof
anybi-innitewordofX
A
. Thesyntatisemigroup ofanirreduiblesoshift
isdenedasthetransitionsemigroupofitsrightFisherover.
Example1 The so shift presentedby the automaton of Figure 1 is alled
theevenshift. Its syntatisemigroupisdened bythetable in therightpart
ofthegure.
2.2 Struture of nite semigroups
Wereferto[20℄formoredetailsaboutthenotionsdened inthissetion.
Given a semigroup S, we denote by S 1
the following monoid: if S is a
monoid, S 1
=S. If S isnot amonoid, S 1
=S[f1gtogether with thelaw
denedbyxy=xy ifx;y2S and1x=x1=xforeahx2S 1
.
1 2 b
b a
a 1
b 2 1
ab 2
ba 1
bb 1 2
bab 2
aba
:
Figure1: TherightFisheroveroftheevenshiftanditssyntatisemigroup. Sine
aaandadenethesamepartialfuntionfromQtoQ,wehaveaa=ainthesyntati
semigroup. Wealso haveaba=0and, ingeneral,ab 2k +1
a=0for any nonnegative
integerk. Thewordbbistheidentityinthissemigroup.
WerealltheGreen'srelations whiharefundamental equivalenerelations
denedinasemigroupS. ThefourequivalenerelationsR,L,H ,J aredened
asfollows. Letx;y2S,
xRy , xS 1
=yS 1
;
xLy , S
1
x=S 1
y;
xJy , S 1
xS 1
=S 1
yS 1
;
xH y , xRy andxLy:
AnotherrelationDisdenedby:
xDy , 9z2S xRz andzLy:
InanitesemigroupJ =D. Wereallthedenition ofthequasi-order
J :
x
J
y , S
1
xS 1
S 1
yS 1
:
AnR-lassisanequivalenelassforarelationR(similarnotationsholdforthe
otherGreen'srelations). An idempotent is anelemente2S suh thatee=e.
A regular lass is alass ontainingan idempotent. Inaregular D-lass, any
H -lass ontaining an idempotent is a maximal subgroup of the semigroup.
Moreover,tworegularH -lassesontainedinasameD-lassareisomorphi(as
groups), see forinstane [20, Chapter 3Proposition 1.8℄. This groupis alled
theharateristigroup oftheregular D-lass. Thequasi-order
J
indues a
partial order between the D-lasses(still denoted
J
). The struture of the
transitionsemigroupSisoftendesribedbythesoalled\egg-box"pituresof
theD-lasses.
Wesaythattwoelementsx;y2S areonjugateifthereareelementsu;v2
S 1
suhthatx=uvandy=vu. TwoidempotentsbelongtoasameregularD-
lassifandonlyiftheyareonjugate,seeforinstane[20,Chapter3Proposition
1.12℄.
Therank ofxistheardinaloftheimageofx asapartialfuntion fromQto
Q. Thekernel ofx isthepartitioninduedbytheequivalenerelationover
thedomain of x where pq if and onlyp;q havethe sameimage byx. The
kernelofx isthusapartitionofthedomainof x. InFigure2,wedesribethe
egg-boxpituresfortheevenshiftofExample1.
12
1=2 b
b 2
1 2
1
a ab
2 ba
bab
0
Figure 2: Thesyntatisemigroupof theeven shift isomposedof threeD-lasses
D
1 , D
2 , D
3
, of rank 2, 1 and0, respetively, represented by the above tables from
left to right. Eahsquare inatable representsanH-lass. Eahrow represents an
R-lassandeaholumnanL-lass. Theommonkerneloftheelementsineahrow
iswrittenontheleftoftherow. Theommonimageoftheelementsineaholumnis
writtenabovetheolumn. Idempotentsaremarkedwiththesymbol. EahD-lass
ofthissemigroupis regular. Theharateristigroups ofD
1 ,D
2 , D
3
are Z=2Z,the
trivialgroupZ=ZandZ=Z,respetively.
2.3 The syntati graph of a so shift
LetX beanirreduiblesoshiftandS(X)itssyntatisemigroup. Itisknown
thatS(X)hasauniqueD-lassofrank1whihisregular(see[5℄or[6℄,seealso
[11℄).
Wedeneanitedireted ayligraphassoiatedwithX asfollows. The
set of verties of this graph is the set of non null regular D-lasses of S(X),
but theregularD-lass ofnullrank, ifthere is one. Eahvertexis labeledby
therankoftheD-lassanditsharateristigroup. Thereisanedgefrom the
vertexassoiatedwithaD-lassDtothevertexassoiatedwithaD-lassD 0
if
andonlyifD 0
J
D. Weallthis ayligraphthesyntatigraph ofX (see
Figure3foranexample). NotethattheregularD-lassofnullrank,ifthereis
one,is nottaken into aount in asyntatigraph. This islinkedto the fat
thatafullshift(i.e.theset ofallbi-innitewordsonanite alphabet)anbe
onjugateto anonfullshift.
2.4 Nasu's Classiation Theorem for so shifts
Inthis setion,wereallNasu'sClassiationTheoremforsoshifts[19℄(see
also[18,Theorem7.2.12℄),whihextends William'sClassiationTheoremfor
shiftsofnitetype(see[18,Theorem7.2.7℄).
Let X A Z
;Y B Z
be two shifts and m;a be nonnegative integers. A
map : X ! Y is a (m;a)-blok map (or (m;a)-fator map) if there is a
map Æ : A m+a+1
! B suh that ((a
i )
i2Z ) =(b
i )
i2Z
where Æ(a
i m :::a
i 1 a
i
a
i+1 :::a
i+a )=b
i
. Ablokmap isa(m;a)-blokmap forsomenonnegativein-
tegersm;a(respetivelyalleditsmemory and antiipation). Thewellknown
rank1,Z=Z
Figure3: ThesyntatigraphoftheevenshiftXofExample1.WehaveD
2
J D
1
sine,for instane,S(X) 1
abS(X) 1
S(X) 1
bS(X) 1
.
theoremofCurtis,HedlundandLyndon[10℄assertsthatontinuousmapsom-
mutingwiththeshiftmap, areexatlyblokmaps. Aonjugay isaone-to-
oneandontoblokmap(then,beingashiftompat,alsoitsinverseisablok
map).
Wenowdene thenotionofstrongshiftequivalenebetweentwosymboli
adjaenymatries. A symboli monomial isaformal produt ofseveralnon-
ommutingvariables. Inpartiular,theentriesofasymboliadjaenymatrix
areintegralombinationsof symbolimonomials. Inthis ategoryofmatries,
we write A $ B if A = B modulo a bijetion of their underlying symboli
monomials. Forexampleweanwrite
0 b
b+ 2a
$
0 a
a+d 2e
$
0 bb
bb+ 2b
:
Two symboli matries A and B with entries in A and B respetively, are
elementary strong shift equivalent if there is a pair symboli matries (U;V)
withentriesindisjointalphabetsU andV respetively,suhthatA$UV and
B$VU.
Anotherequivalentformulationofthisdenitionisthefollowing. LetAand
B be twonite alphabets. We denote by AB theset ofwordsab with a 2A
andb2B. Letf beamapfromAtoB. Themapf isextendedtoamorphism
fromniteformalsumsofelementsofAtoniteformalsumsofelementsofB.
Wesaythatf transforms asymboli(QQ)-matrixAintoasymboli(QQ)-
matrixB if B
pq
=f(A
pq
)for eah p;q 2Q. Twosymbolimatries Aand B
withentries in A and B respetively, areelementary strongshift equivalent if
thereis apairof symboli matries(U;V)withentriesin disjointalphabetsU
and V respetively, suh that there is aone-to-one map from A to UV whih
transforms A into UV, and there is a one-to-one map from B to VU whih
transformsB into VU.
TwosymboliadjaenymatriesAandB arestrongshiftequivalentwithin
right Fisher overs if there is a sequene of symboli adjaeny matries of
rightFisherovers
A=A
0
;A
1
;:::;A
l 1
;A
l
=B
i 1 i
equivalent.
Theorem 2(Nasu) Let X and Y be irreduible so shifts and letA and B
be the symboli adjaeny matries of the right Fisher overs of X and Y,
respetively. Then X and Y are onjugate if and only if A and B are strong
shiftequivalentwithin rightFisher overs.
Example3 Letusonsider thetwo(onjugate) irreduibleso shiftsX and
Y denedbytherightFisheroversinFigure 4.
1 2
b
b a
2 0
3 0 1
0 a
0
b 0
b 0
d 0
0
Figure4: TwoonjugateshiftsX andY.
Thesymboliadjaenymatriesoftheseautomata arerespetively
A=
a b
b 0
; B = 2
4 a
0
0 d 0
0
0 b 0
0 b 0
0 3
5
:
ThenAandB areelementarystrongshiftequivalentwith
U =
u
1
0 u
2
0 u
2 0
; V = 2
4 v
1 0
v
2 0
0 v
2 3
5
:
Indeed,
UV =
u
1 v
1 u
2 v
2
u
2 v
2 0
; VU = 2
4 v
1 u
1
0 v
1 u
2
v
2 u
1
0 v
2 u
2
0 v
2 u
2 0
3
5
:
Theone-to-one mapsfromA=fa;bgtoUV andfromB=fa 0
;b 0
; 0
;d 0
gtoVU
a u
1 v
1
b u
2 v
2
; a
0
v
1 u
1
b 0
v
2 u
2
0
v
2 u
1
d 0
v
1 u
2 :
An elementary strongshiftequivalene betweenA=(Q;E)and B=(Q 0
;E 0
),
enablestheonstrutionofanirreduiblesoshiftZ onthealphabetU[V as
follows. Thesoshift Z is dened by theautomatonC=(Q[Q 0
;F),where
thesymboliadjaenymatrixC ofCis
Q Q
0
Q
Q 0
0 U
V 0
:
TheshiftZ isalledthebipartiteshift dened byU;V (seeFigure5). Anedge
ofClabeledbyU goesfromastateinQtoastateinQ 0
. AnedgeofClabeled
byV goesfrom astate in Q 0
to astatein Q. Hene,apath ofCgoesfrom a
statein Q[Q 0
to a statein Q[Q 0
, itsdomain is inluded either in Q orin
Q 0
, and itsimage is inludedeither in Qorin Q 0
. If apath of Chasdomain
inludedin P and theimageinludedin P 0
,wesaythatithastype(P;P 0
).
RemarkthattheseondhigherpowerofZ isthedisjointunionofX andY
sine
C 2
=
UV 0
0 VU
:
NotealsothatCisarightFisherover(i.e.isminimal).
1
1 0
2 3
0
2 0 u
2
v
2
u
2 v
2 u
1 v
1
Figure5: ThebipartiteshiftZ oftheshiftsXandY inFigure4.Thewordu1v1has
type(Q;Q)andorrespondstothewordainX.
3 A syntati invariant for onjugay
Inthis setion,weprovethatthesyntatigraphisaninvariantfortheonju-
gayofirreduiblesoshifts.
gate,thentheirsyntatigraphsareisomorphiandtheisomorphism preserves
the labels.
Wegiveafewlemmas beforeprovingTheorem 4.
LetX(respetivelyY)beanirreduiblesoshiftwhosesymboliadjaeny
matrixof its right Fisher overis a (QQ)-matrix (respetively (Q 0
Q 0
)-
matrix)denotedbyA(respetivelybyB). WeassumethatAandBareelemen-
tarystrongshiftequivalentthroughapairofmatries(U;V). Theorresponding
alphabets aredenotedA, B,U,and V asbefore. Wedenotebyf aone-to-one
map from A to UV whih transforms A into UV and byg a one-to-one map
fromBto VU whihtransformsB intoVU. LetZ bethebipartiteirreduible
so shift assoiated to U;V. We denote by S(X) (respetively S(Y), S(Z))
thesyntatisemigroupofX (respetivelyY,Z).
Remarkthatw2S(Z)hastype(Q;Q)ifandonlyifw6=0andw2(f(A))
,
andwhastype(Q 0
;Q 0
)ifandonlyifw6=0andw2(g(B))
.
Lemma5 Elements ofS(Z)inasamenon nullH -lasshave thesame type.
Proof We showthe property forthe (Q;Q)-type. Let w2 H and w of type
(Q;Q). Ifw=w 0
v withw 0
;v2S(Z),thenw 0
hastype(Q;). Ifw=zw 0
with
z;w 0
2 S(Z), then w 0
hastype (;Q). Thus, wH w 0
implies that w 0
has type
(Q;Q).
TheH -lassesofS(Z)ontainingelementsoftype(Q;Q)(respetively(Q 0
;Q 0
))
arealled(Q;Q)-H -lasses (respetively(Q 0
;Q 0
)-H -lasses).
Letw=a
1 :::a
n
beanelementofS(X),wedenetheelementf(w)asf(a
1 )
:::f(a
n
). Notethatthisdenition isonsistentsineifa
1 :::a
n
=a 0
1 :::a
0
m in
S(X), then f(a
1
):::f(a
n
) =f(a 0
1
):::f(a 0
m
) in S(Z). Similarly we dene an
elementg(w)foranyelementwofS(Y).
Conversely,letwbeanelementofS(Z)belongingtof(A)
((UV)
). Then
w=f(a
1
):::f(a
n
), with a
i
2A. Wedene f 1
(w)asa
1 :::a
n
. Similarlywe
deneg 1
(w). Againthese denitionsand notationsare onsistent. Thus f is
asemigroupisomorphismfromS(X)tothesubsemigroupofS(Z)oftransition
funtions dened bythe wordsin (f(A))
. Notiethat f(0)=0if02S(X).
Analogously,g isasemigroupisomorphismfrom S(Y)to thesubsemigroupof
S(Z)oftransitionfuntionsdened bythewordsin(g(B))
.
Lemma6 Letw;w 0
2S(Z)of type(Q;Q). Then wH w 0
in S(Z)ifandonly if
f 1
(w)H f 1
(w 0
)in S(X).
Proof Letw=f(a
1
):::f(a
n
)and w 0
=f(a 0
1
):::f(a 0
m
),with a
i
;a 0
j
2A. We
havew=w 0
v withv2S(Z)ifandonlyifv =f( a
1
):::f( a
r
)witha
i
2Aand
f(a
1
):::f(a
n )=f(a
0
1
):::f(a 0
m )f( a
1
):::f( a
r
). Thisisequivalenttoa
1 :::a
n
=
a 0
1 :::a
0
m a
1 :::a
r
,thatis f 1
(w)S(Z) 1
f 1
(w 0
)S(Z) 1
. Analogously, wehave
w 0
=wv 0
with v 0
2 S(Z), if and onlyif f 1
(w 0
)S(Z) 1
f 1
(w)S(Z) 1
. This
provesthatwRw 0
inS(Z)ifandonlyiff 1
(w)Rf 1
(w 0
)inS(X). Inthesame
relationH .
Asimilarstatementholdsfor(Q 0
;Q 0
)-H -lasses.
Lemma7 Letw;w 0
2S(Z)oftype(Q;Q). Thenw
J w
0
inS(Z)ifandonly
iff 1
(w)
J f
1
(w 0
)in S(X). Thisimplies thatwJw 0
inS(Z)ifandonlyif
f 1
(w) J f 1
(w 0
)in S(X).
ProofTherststatementanbeproovedasinthepreviouslemma.
Similar results hold between S(Y) and S(Z). As a onsequene we get the
followinglemma.
Lemma8 The bijetionf betweenS(X)andtheelements ofS(Z)in (f(A))
,
indues a bijetion between the non nullH -lasses of S(X) andthe (Q;Q)-H -
lassesofS(Z). Moreoverthisbijetion keepstherelationsJ,
J
andthe rank
ofthe H -lasses.
Asimilarstatementholdsforthebijetiong.
We now ome to the main lemma, whih shows the link between the ele-
mentary strong shift equivalene of the symboli adjaeny matries and the
onjugay of some idempotents in the semigroup of the bipartite shift. This
linkisthekeypointoftheinvariant.
Lemma9 LetH bea regular (Q;Q)-H -lassof S(Z). Then there isa regular
(Q 0
;Q 0
)-H -lassin the sameD-lassasH.
ProofLete2S(Z)beanidempotentelementoftype(Q;Q). Letu
1 v
1 :::u
n v
n
in (UV)
suh that e = u
1 v
1 :::u
n v
n
. We dene e = v
1 :::u
n v
n u
1
. Thus
eu
1
=u
1
e inS(Z). Remark that edependsonthehoieof thewordu
1 v
1 :::
u
n v
n
representingein S(Z).
Ifw denotes v
1 :::u
n v
n
and v denotes u
1
, wehavee=vw ande=wv. It
followsthateandeareonjugate,thuse 2
=eande 2
areonjugate. Moreover
e 3
=wvwvwv =weev=wev=wvwv=e 2
:
Thus e 2
is an idempotentonjugate to theidempotente. As aonsequenee
ande 2
belongtoasameD-lassofS(Z)(seeSetion2),ande 2
6=0. Theresult
followssinee 2
isoftype(Q 0
;Q 0
).
Note that the number of regular (Q;Q)-H -lasses and the number of regular
(Q 0
;Q 0
)-H -lassesin asameD-lassof S(Z),maybedierentin general.
WenowproveTheorem4.
Proof[of Theorem4℄ByNasu'sTheorem [19℄weanassume, withoutlossof
generality,that the symboliadjaeny matries ofthe rightFisheroversof
Z asabove.
Let D be anon nullregular D-lass of S(X). Let H be aregularH -lass
of S(X) ontainedin D. Let H 00
= f(H). By Lemma 8, the groupsH and
H 00
areisomorphi. LetD 00
theD-lassofS(Z)ontainingH 00
. ByLemma 9,
thereisat leastoneregular(Q 0
;Q 0
)-H -lassK 00
in D 00
, whih isisomorphito
H 00
. Let H 0
=g 1
(K 00
) andletD 0
be theD-lassof S(Y)ontainingH 0
. By
Lemma8,thegroupsH 0
and K 00
areisomorphi. Hene thegroupsH andH 0
areisomorphi.
By Lemmas 8 and 9, we have that the above onstrution of D 0
from D
isa bijetivefuntion 'from thenon nullregularD-lasses ofS(X)onto the
nonnull regularD-lassesof S(Y). Moreovertheharateristigroupof D is
isomorphito theharateristigroupof '(D) and, byLemma 8, therankof
Disequaltotherankof'(D).
We nowonsider two non null regular D-lassesD
1 and D
2
of S(X). By
Lemma8andLemma 9,D
1
J D
2
ifandonlyif'(D
1 )
J '(D
2
). Itfollows
thatthesyntatigraphsofS(X)andS(Y)areisomorphithroughthebijetion
'.
3.1 The reduible ase
Nasu'sClassiationTheoremholdsforreduiblesoshiftsbytheuseofright
Kriegeroversinsteadof right Fisherovers[19℄. This enablestheextension
ofourresulttotheaseofreduiblesoshifts.
LetXA Z
beashift. Wedene
X =fx jx2Xg;
where for x 2 A Z
, wedenote by x the left innite word :::x
2 x
1 x
0 . The
equivalenerelationonX isdened asfollows. Letx;y2X ,
xy,fu2A +
jxu2X g=fu2A +
jyu2X g:
IfX isasoshift,theequivalenelassesofarenitelymany[15℄. Theright
KriegeroverofX isdenedastheautomatonlabeledbyAinwhihthestates
are the -lasses[x℄ with x 2 X , and there is a an edge labeled a from [x℄
to[xa℄ ifxa2X . The analogousof Theorem2for (possibly)reduible so
shiftsisthefollowing.
Theorem 10 [19, Theorem 3.3℄ Let X and Y be so shifts and let A and
B bethe symboli adjaeny matries of the right Krieger overs of X and Y,
respetively. Then X and Y are onjugate if and only if A and B are strong
shiftequivalentwithin rightKriegerovers.
Hene weandene the syntati graphof areduibleshift X asthegraphof
theregularD-lassesofthetransitionsemigroupofitsrightKriegerover.The
resultofTheorem4isextendedasfollowsforreduiblesoshifts.
theirsyntatigraphs areisomorphiandthe isomorphismpreserves thelabels.
An eetive proedure to onstrut the right Kriegerover of a so shift is
desribedin [19℄. First,oneonstrutsthe(unique) minimaldeterministi au-
tomatonwith oneinitial state reognizing thelanguage ofnite bloks of the
shift. Next,one erasesall thestateswhih are nottheend of anyleft-innite
path. ThisautomatonturnsouttobetherightKriegeroveroftheshift. For
instane, therightKriegeroverof theeven shiftin Figure 1,is illustratedin
Figure6.
2 3
1 b
b
b a
a
Figure6: TherightKrieger overoftheevenshift X desribedinFigure1. Notie
that,althoughtheshiftX isirreduible,therightFisheroverofX doesnotoinide
withitsrightKriegerover.
4 How dynami is this invariant?
Inthis setion, we briey omparethe syntati invariantwith other lassial
onjugayinvariants. Wereferto[18℄ fortheirdenitionsand properties.
First,onean remark that thesyntatiinvariantdoesnot aptureall the
dynamis. Twoso shifts anhavethe samesyntatigraphand adierent
entropy,asshownintheexampleofFigure 7.
Theomparisonwith thezetafuntionismoreinteresting. Reallthat the
zetafuntionofashiftX is(X)=exp P
n1 p
n z
n
n
,wherep
n
isthenumberof
bi-innitewordsx 2X suh that n
(x)=x. Wegivein Figure 8anexample
of two irreduibleso shifts whih havethesame zetafuntion and dierent
syntatigraphs.
Thefollowingharaterizationofirreduibleshiftsofnitetypetsnaturally
in ourframework. It isof oursewell-knownandanbeobtainedforinstane
fromtheharaterizationofsyntatisemigroupsofloallanguages(see[9℄),and
theharaterizationof syntatisemigroupsofirreduible soshifts(see[5℄),
oralsofrom[11℄.
1 2 b
b
1 2
b
b
Figure7: ThetwoabovesoshiftsX;Y havethesamesyntatigraphanddierent
entropies. Indeed,wehaveb=inthesyntatisemigroupofY. HenetheshiftsX
andY havethesamesyntatisemigroup.
1 2
a
a
b b
x
y
1 2
a
d
b
x
y
Figure 8: Two so shifts X;Y whih have the same zeta funtion 1
1 4z+z 2
(see
for instane [18, Theorem 6.4.8℄, or [2 ℄for the omputation of the zeta funtionof
a so shift), and dierent syntati graphs. Indeed the syntati graph of X is
(rank2;Z=2Z) ! (rank1;Z=Z)while the syntati graph of Y has only one node
(rank1;Z=Z).Thustheyarenotonjugate. NotiethatY isashift ofnitetype.
Proposition 12 Anirreduiblesoshiftsisof nitetypeifandonlyitssyn-
tatigraph isreduedtoonenode ofrank1representing the trivialgroup.
Proof LetX beanirreduible shiftofnitetype. Itis wellknownthatX is
onjugatetoanedgeshift,that isasoshiftwithapresentationinwhih the
labelsoftheedgesarealldierent(or,inotherwords,anitemultigraphwhih
isnotlabeled). Hene,byTheorem 4,weansupposethatX isanedgeshift.
LetS(X)bethesyntatisemigroupofX. EahnonnullelementofS(X)has
rank1beauseit determinesaninitial stateandaterminalstate. Moreoverit
anbeeasilyseenthat forx;y 2S(X)nf0gwehavexRyifandonlyifxandy
havethesamedomain, andxLy ifandonlyiftheyhavethesameimage. This
meansthat S(X)ontainsonly one D-lass of rank 1and that the H -lasses
ontainexatlyoneelement.
For the onverse, suppose that the syntati graph of an irreduible so
shift X is redued to one node of rank 1 representing the trivial group. By
[18,Theorem 3.4.17℄,itsuÆesto provethat allsuÆientlylongandnon null
wordsin thesyntati semigroupS(X) of X have rank1. By[20, Chapter 3
Proposition 1.12℄, we havethat all suÆiently longwordsin S(X) are of the
1,wehavethateahnonnullwordoftheformuvwhasrank1.
Anotherinterestinglassofirreduiblesoshiftsanbeharaterizedwith
thesyntatiinvariant. Itisthelassofaperiodisoshifts[1℄.
Let x 2 X, we denote by period(x) theleast positive integer n suh that
n
(x)=xifsuhanintegerexists. Itisequalto1otherwise.
LetX;Y betwoshiftsandlet:X !Y beablokmap. Themapissaid
aperiodiifperiod(x)=period((x))foranyx2X withniteperiod. Roughly
speaking,suhafatormapdoesnotmakeperiodsderease.
A soshift isaperiodi ifitis theimage ofashiftof nite type underan
aperiodiblokmap. An aperiodi presentation isapresentationin whihfor
everyu2A +
,wheneverthereisayling pathlabeledu n
p
1 u
!p
2 u
!::: u
!p
n u
!p
1
;
onehasp
i
=p
1
foreahi=2;:::;n.
Proposition 13 A so shift is aperiodi if and only if it has an aperiodi
presentation.
Proof Let X be an aperiodi so shift. Hene X = (Y), where is an
aperiodi blok map and Y a shift of nite type. Notie that we analways
suppose that Y is an edge shiftand that has no memory norantiipation.
HeneapresentationAofX isgivenbythenotlabeledpresentationGofY in
whihthelabelofanedgeeistheletter(e)(weidentifytheblokmapwith
theloal ruleÆ dening it). Moreover,weanalwayssuppose that thereis at
mostoneedgefrom agivenstatepto agiven stateq in G. Wehavethat the
presentationAisaperiodi. Indeed,let
p
1 u
!p
2 u
!::: u
!p
n u
!p
1
beapathinA. Notiethatweanalwayssupposethattheongurationu 1
2
X, obtainedbyrepeatinginnitelymany timestheworduin bothdiretions,
hasperiodh=juj. Moreover,inGthere isapathv
p
1 e
(1)
1 :::e
(1)
h
!p
2 e
(2)
1 :::e
(2)
h
!::: e
(n 1)
1 :::e
(n 1)
h
!p
n e
(n)
1 :::e
(n)
h
!p
1
;
suh that (e (i)
1 :::e
(i)
h
) = u, for eah i = 1;:::;n. Being (v 1
) = u 1
, the
ongurationv 1
2Y musthaveperiodh. Thisimpliese (i)
1 :::e
(i)
h
=e (1)
1 :::e
(1)
h
foreahi=2;:::;n. Inpartiularwehavep
i
=p
1
foreahi=2;:::;n.
Fortheonverse,supposethatX isasoshiftwithanaperiodipresenta-
tionA. LetY betheedgeshiftwhosepresentationistheunderlyinggraphGof
A. Let:Y !X bethelabelingmap. Wehavethat isaperiodi. Indeed
letx beaongurationofY withperiod n. Hene inGthereis thepath
p
1 e
1
!p
2 e
2
!::: e
n 1
!p
n e
n
!p
1
;
wherex=(e
1 e
2 :::e
n
) :Hene(x)=(a
1 a
2 :::a
n
) ,wherea
i
isthelabelof
theedgee
i
. Supposethata
1 :::a
n
=(a
1 :::a
h )
k
,withhk =n. Hene inAwe
havethepath
p
i a
i :::a
h a
1 :::a
i 1
!p
h+i a
i :::a
h a
1 :::a
i 1
!:::p
(k 1)h+i a
i :::a
h a
1 :::a
i 1
!p
i
;
for eah i = 1;:::;h. Being the presentation of X aperiodi, one has p
i
=
p
h+i
==p
(k 1)h+i
foreahi=1;:::;h. Thismeansthattheedges
p
i ei
!p
i+1
p
h+i eh+i
!p
h+i+1
.
.
.
p
(k 1)h+i e
(k 1)h+i
!p
(k 1)h+i+1
;
havesameinitial state,samenal stateand samelabel(where thestatep
n+1
isdenedas p
1
). Thus theyoinide andthisimpliesh=nandk=1. Hene
period((x))=n.
Proposition 14 Theright Fisheroverof anirreduible aperiodi soshift
isanaperiodi presentation.
ProofLet A be an aperiodi presentation of a so shift X. Let us assume
thatAisnotdeterministi. WeomputefromAadeterministipresentationB
bythewellknownsubsetonstrution (seefor instane[18, Setion 3.3℄). We
showthatBisanaperiodipresentation.
Supposethatin Bthereisaylingpathlabeledbyu n
P
1 u
!P
2 u
!::: u
!P
n u
!P
1
;
whereuisawordandeahP
i
isastateofBidentiedwithasubsetofthestates
ofA. LetP andQbetwosubsetsofthestatesofA. ReallthatinBthereisa
uniquepathfromPtoQlabeledu,ifandonlyifQisthesetofallstatesqinA
forwhihthere isatleastonestatepin P andapathinAfromptoqlabeled
u. Ifsuhapathexists,thestateQisdenotedbyPu. Itfollowsthat,foreah
statep
j 2P
1
,thereisaleftinnitepath(q
j;1 (i+1) u
!q
j;1 i )
i0
labeledby
!
u
(that is theleft innite word obtainedby repeating innitelymany times the
worduontheleft),whereq
j;1
=p
j andq
j;i 2P
imodn
foreahi0. Sinethe
numberofstatesisnite,therearetwopositiveintegermandlandanitepath
in A suh that q
j;1 (m+l) u
l
! q
j;1 m u
m
! q
j;1
= p
j with q
j;1 (m+l)
=q
j;1 m .
SineAisaperiodi,oneanset l=1. Moreover,oneanalwayssupposethat
mdoesnotdepend onj. Letk=(1 m)modn. WedenotebyQ
k
thesetof
allstatesq
j;1 m
. ThusQ
k
u =P
1
. Moreover,wehave
Q
k
Q
k
uQ
k u
2
Q
k u
m
=P
1
Q
k u
m+1
=P
2
Q
k u
m+2
=P
3
:::
Q
k u
m+n+1
=P
1 :
ItfollowsthatP
1 P
2 P
3
P
n P
1
,andnallyP
1
=P
2
==P
n .
The right Fisher over of theshift is obtained by statemerging of states
ofB havingthe samefuture. Thus, ifBis anaperiodipresentation, itsright
Fisher over also. It is known that the right Fisher over of an irreduible
shifthasastronglyonnetedgraph.
Aharaterizationofirreduibleaperiodisoshiftsisthefollowing.
Proposition 15 Anirreduible soshiftisaperiodiifandonlyifitssynta-
tigraphontainsonly trivialgroups.
Proof LetX beanirreduible aperiodiso shift andletS(X)bethesyn-
tatisemigroup of X. If e is anidempotentof S(X) and u 2 S(X)is suh
thatuH e,thereexistsn1suhthatu n
=e. BeingtherightFisheroverof
X aperiodi,thefuntionuoinideswitheateahstatepsuhthate(p)=p.
If e(p) 6= p and u(p) = q, we have that q is in the image of e beause this
latteroinides withtheimage of u. Hene e(q)=q and thenu(q)=q. This
impliese(p)=u n
(p)=u n 1
(q)=q=u(p). Thusu=e. Hene alltheregular
H -lassesofS(X)aretrivial.
For the onverse, suppose that the syntati graph of an irreduible so
shiftX hasonlytrivialgroups. Letu n
bethelabelof ayle
p
1 u
!p
2 u
!::: u
!p
n u
!p
1 :
Withoutlossofgenerality,weanassumethatu n
isidempotent(indeedthereis
alwaysapowerofu n
whihisidempotent). Beingu n+1
;:::;u 2n 1
inthesame
H -lassof u n
, they must oinide. From u n+i
=u n
we deduep
i+1
=p
1 for
eahi=1;:::;n 1.
Shutzenberger's haraterization of aperiodi languages (see for instane
[20,Chapter4Theorem2.1℄)assertsthatthesetofbloksofanaperiodiso
shiftisaregularstar freelanguage.
5 An invariant for shift equivalene
Wenow provethat ourinvariantfor strongshift equivaleneis alsoan invari-
antof shiftequivalene. Althoughshiftequivaleneis deidable,even forso
lene of so shifts, whih is equivalent to eventualonjugay, maybeuseful.
Mostknownonjugayinvariantsarealsoinvariantsforshiftequivalene.
Twosymboli adjaenymatriesAandB withentriesinAand Brespe-
tively, areshift equivalent with lag l,where l isapositiveinteger, ifthere is a
pairof symboliadjaenymatries(U;V)withentriesindisjointalphabetsU
andV respetively,suhthat (see[8℄)
A l
$UV; B l
$VU;
AU $UB; VA$BV:
Twomatriesareshift equivalent ifthereisapositiveintegerlsuh that they
areshiftequivalentwithlagl. Strongshiftequivaleneimpliesshiftequivalene
buttheonverseisfalse[13℄.
Inthefollowingtheoremweprovethatourinvariantisalsoinvariantunder
shiftequivalene.
Theorem 16 LetX andY betwososhifts. IfX andY areshiftequivalent,
then their syntati graphs are isomorphi and the isomorphism preserves the
labels.
Proof LetA (respetivelyB), be thesymboli adjaenymatrixof theright
FisheroverofX (respetivelyofY)ifX andY areirreduible,oroftheright
KriegeroverofX(respetivelyofY)ifXandY arereduible. SupposethatA
andB areshiftequivalentwithlag
l . NotiethatAand B areshiftequivalent
withlaglforeahl
l. Moreover,beingA l
elementarystrongshiftequivalent
toB l
,theyhavethesamesyntatigraphbyTheorem4.
Hene itsuÆesto provethatfor eah symboliadjaenymatries Aand
B,thereisabigenoughintegerl,suh thatAandA l
havethesamesyntati
graph,andB andB l
havethesamesyntatigraph.
LetS(X)bethesyntatisemigroupofX. Foreahidempotente2S(X),
letw
e 2A
beawordrepresentinge. Foreahx2S(X)suhthat xH e,there
isapositiveintegerh
x;e
suhthat u hx;e
=e(reallthat aregularH -lassis a
nitegroup).Wedothesameforeahidempotente 0
2S(Y)andeahy 2S(Y)
suhthat yH e 0
in S(Y). Let
h
X
= Y
e2S(X)
e 2
=e jw
e j
Y
x;e2S(X)
e 2
=e; xHe h
x;e :
Leth=h
X h
Y
landl=h+1. Notethatl
l.
We provethat A and A l
havethe same syntati graph. Thesame proof
holdsforB andB l
. LetS(A l
)bethe syntatisemigroupofA l
. First,notie
that thewordsrepresenting elementsof S(A l
)are wordslabelled in A l
. Thus
S(A l
)isaisomorphitoasubsemigroupofS(X)andaGreen'srelationinS(A l
)
isstillaGreen'srelationin S(X).
Lete be anidempotentof S(X)andletD be itsregularD-lass in S(X).
Sinee l
=e, theidempotente isalsoanidempotentofS(A l
),and theregular
D-lassofS(A)ontainingeisontainedinD. Moreover,iftwoidempotentse
andeareontainedin thesameD-lassof S(X),thentheyarealso ontained
in thesameD-lass of S(A l
). Indeed letx;y 2S(X)suhthat e=xey. Let
u(resp. v) awordin A
representingx (resp.y). Wehave,sine eand eare
idempotents,
e=uw
e (juj+jw
e j)
h
jw
e j
+1
evw
e
(jvj+jw
e j)
h
jw
e j
+1
;
and
juw
e (juj+jw
e j)
h
jwej +1
j=juj+(juj+jw
e
j)h+jw
e
j=(juj+jw
e j)l:
Inthesamewayonehasthatlalsodividesjvw
e
(jvj+jwej) h
jwej +1
j. Hene eande
areinthesameD-lassofS(A l
).
Hene to eahregular D-lass in S(X) orresponds exatlyoneregularD-
lassinS(A l
)andthepartial orderrelation
J
iskept.
It remainsto provethat foreahidempotent e, theregularH -lassesH
S(X)and
H S(A l
)ontaininge,oinide. Clearly
H H. Fortheonverse,
ifx 2H, we havethat x h
=e (reallthat h
x;e
divides h),and henex h+1
=
x l
=x. Sinex l
2S(A l
),wehavethat x2
H.
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