J. Semigroup Theory Appl. 2020, 2020:3 https://doi.org/10.28919/jsta/4369 ISSN: 2051-2937
INVERSE SEMIGROUP AMALGAMS WITH A LOWER BOUNDED CORE
PAUL BENNETT∗
Singapore
Copyright c2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract.We consider inverse semigroup amalgams[S1,S2;U]such that for anyu∈Uande∈E(Si)withu≥einSi,
wherei∈ {1,2}, there exists f ∈E(U)withu≥f ≥einSi; we say thatUis lower bounded inS1andS2. We construct
and describe the Sch¨utzenberger automata ofS1∗US2and give conditions for decidable word problem. The homomorphisms
of the Sch¨utzenberger graphs ofS1∗US2are studied and conditions are given forS1∗US2to be completely semisimple. In
the case whenS1andS2have decidable word problems andUis finite, we show thatS1∗US2has decidable word problem.
Keywords:inverse semigroups; amalgams; Sch¨utzenberger automata; word problem; completely semisimple; subgroups.
2010 AMS Subject Classification:20M18.
1.
INTRODUCTIONIt was proved by Hall [11] (1975) that any amalgam of inverse semigroups is strongly embedded
into an inverse semigroup. It follows that any amalgam[S1,S2;U]is strongly embedded into the amalgamated free productS1∗US2, in the variety of inverse semigroups. Since Hall’s result, the structure ofS1∗US2and conditions for decidability of the word problem, in the general case, have been open problems, although some special cases have been studied.
∗Corresponding author
E-mail address: paul [email protected]
Received November 13, 2019
It was shown by Birget, Margolis and Meakin [3] (1991) thatS1∗US2can have undecidable word problem. Haataja, Margolis and Meakin in [10] (1996) considered full amalgams. Lower bounded
amalgams were studied by the author in [1] and [2] (1997). Stephen [18] (1998) reproved Hall’s result.
Cherubini, Meakin and Piochi [6] (1997) showed that the amalgamated free product of free inverse
semigroups can have decidable word problem. In [7] (2005), they showed that amalgams of finite
inverse semigroups have decidable word problem. Further work was done by Cherubini, Jajcayov´a,
Mazzuchelli, Nuccio and Rodaro in [5], [15], [8], [9] and [4] (2008–2015).
In Algorithm4.20, a method is given for constructing the Sch¨utzenberger automata ofS1∗US2. Theorem4.26describes the Sch¨utzenberger automata ofS1∗US2. Results are given concerning homomorphisms of Sch¨utzenberger graphs, which lead to conditions forS1∗US2to be completely semisimple. In Theorem4.40, a list of conditions is given forS1∗US2to have decidable word problem. As an example, Corollary4.41shows thatS1∗US2has decidable word problem whenS1andS2have decidable word problems andU is finite.
2.
PRELIMINARIESA semigroupSis called aninverse semigroupif, for every elements∈S, there is a unique elements−1, called theinverse of s, such thatss−1s=sands−1ss−1=s. The setE(S) ={e∈S:e2=e}is called thesemilattice of idempotentsofS. Thenatural partial orderofSis defined bya≤bif and only
ifa=eb, for somee∈E(S), fora,b∈S. A subsemigroupU ofSis called aninverse subsemigroup
ofSif the inverse of each element ofU is also contained inU. For results on inverse semigroups, see
Howie [12] and Petrich [14].
Apresentationfor an inverse semigroupSis a pairhX |Ri, whereX is a non-empty set andRis
a binary relation on(X∪X−1)+ such thatS= (∼ X∪X−1)+/τ, lettingτ denote the congruence
generated byRand the Vagner congruenceρ. We say that the inverse semigroupSispresentedby the
generators X andrelations Rand writeS=InvhX|Ri.
The presentationhX|Riis then studied by considering theSch¨utzenberger automatonA(X,R,w)
ofw, forw∈(X∪X−1)+. The automatonA(X,R,w)has underlying graphSΓ(X,R,w), consisting of verticesRwτ, theR-class ofScontainingwτ, and an edge fromstotlabeled byy, ifs,t∈Rwτ and
respectively. If a presentation has been specified then we also denotehX |Ri,SΓ(X,R,w),A(X,R,w)
byhSi,SΓ(S,w),A(S,w), respectively. For results on presentations, see Stephen [16], [17] and [18].
For a non-empty setX, aninverse word graphΓover X is a connected graph with edges labeled
overX∪X−1, such that for every edge fromv1tov2labeled byy, there is aninverse edgefromv2to
v1labeled byy−1. The inverse word graphΓisdeterministicif no two distinct edges have the same
initial vertex and label. The vertex and edge sets are denotedV(Γ)andE(Γ), respectively.
AV-equivalenceηis an equivalence relation onV(Γ). ThequotientofΓunderη is defined to be
the graphΓ/ηwith verticesV(Γ/η) =V(Γ)/η and an edge fromv1η tov2η, labeled byy, if there is an edge fromv1tov2inΓlabeled byy. The quotientΓ/η is also an inverse word graph overX and η induces a homomorphism fromΓontoΓ/η.
Ifv1,v2∈V(Γ)andwlabels a path, or edge, fromv1tov2then we write this asv1→wv2. We have a pathv1η→wv2ηinΓ/η if and only if there are pathsx1→w1y1,x2→w2y2,. . .,xn→wn yn inΓ, wheren≥1,v1ηx1,y1ηx2,. . .,yn−1ηxn,ynηv2andw1w2· · ·wn=w.
TheV-equivalenceη is calleddeterminisingifΓ/η is deterministic. Theleast determinising
V-equivalence containingη is defined as the intersectionη∗of all determinisingV-equivalences that
containη. Thedeterminised formofΓis the quotientΓ/id∗, whereidis the identity relation.
Result 2.1. [17, Lemma 4.3, Theorem 4.4]The determinised formΓ/id∗of an inverse word graphΓ over X is a well-defined deterministic inverse word graph over X . For v1,v2∈V(Γ), we have v1id∗=v2id∗if and only if there is a path v1→wv2, for some word w freely reducible to1.
The composition ofV-equivalencesη◦id∗is equal toη∗. Thus, for verticesv1,v2∈V(Γ), we havev1η∗v2if and only if there is a pathv1η→wv2η inΓ/η, wherewis freely reducible to 1. Thenv1η∗v2if and only if there are pathsx1→w1 y1,x2→w2y2,. . .,xn→wn yninΓ, wheren≥1,
v1ηx1,y1ηx2,. . .,yn−1ηxn,ynηv2andw1w2· · ·wnis freely reducible to 1.
Further, we have a pathv1η∗→wv2η∗inΓ/η∗if and only if there exist pathsx1η →w1y1η,
x2η →w2y
2η,. . .,xnη→wn ynη inΓ/η, wheren≥1,v1η∗x1,y1η∗x2, . . ., yn−1η∗xn, ynη∗v2 and w1w2· · ·wn=w. Thus, using the above, we a pathv1η∗→wv2η∗ in Γ/η∗if and only if there exist pathsx1→w1 y
1,x2→w2 y2,. . .,xn→wn yninΓ, wheren≥1,v1ηx1,y1ηx2,y2ηx3,. . .,
A(birooted) inverse automaton over X is a tripleA = (α,Γ,β), whereΓis an inverse word
graph overX andα,β ∈V(Γ). The verticesα andβ are called theinitialandterminal rootsofA,
respectively. We also denote the vertices ofA byV(A). Thelanguage L[A]ofA is the set of all
words that label paths fromα toβ. Ifη is aV-equivalence on the vertices ofΓthen the quotient
automatonA/η is given by(α η,Γ/η,β η). Thedeterminised formofA is the quotientA/id∗.
Result 2.2. [17, Theorem 2.5]If A and A0 are deterministic inverse automata over X with L[A]⊆L[A0]then there is a unique homomorphism fromA intoA0. Thus if L[A] =L[A0]then
we haveA ∼=A0.
An inverse automatonA overX hasdecidable languageif there exists an algorithm that decides, on
inputw∈(X∪X−1)+, whether or notw∈L[A].
Result 2.3. [17, Theorems 3.1 and 3.9]For S=InvhX |Ri, the language L[A(X,R,w)]consists of all words w1∈(X∪X−1)+ such that w1≥w in S. We have w=w1 in S if and only if A(X,R,w)∼=A(X,R,w1).
Result 2.4. The word problem for S=InvhX|Riis decidable if and only if the Sch¨utzenberger automata ofhX|Rihave decidable languages.
Proof. The proof follow from Result2.3.
An inverse automatonA over X is called anapproximate automaton ofA(X,R,w)if we
haveL[A]⊆L[A(X,R,w)]and there existsw1∈L[A]withw1=winS=InvhX|Ri, written
A A(X,R,w). An inverse word graphΓoverX is called anapproximate graphif we have
(α,Γ,β) A(X,R,w), for someα,β ∈V(Γ)and somew. Multiplication of disjoint automata
A1= (α1,Γ1,β1)andA2= (α2,Γ2,β2)is defined byA1×A2= (α1η,(Γ1∪Γ2)/η,β2η), where
η is theV-equivalence generated by{(β1,α2)}.
Result 2.5. [17, Theorem 5.2] If we have A1 A(X,R,w1) and A2 A(X,R,w2) then A1×A1 A(X,R,w1w2).
Thelinear automatonofw=y1y2· · ·yn∈(X∪X−1)+, foryk∈X∪X−1, is the inverse automaton
(αw,Γw,βw), with verticesv0=αw,v1,. . .,vn−1,vn=βwand edgesvk−1→yk vk,vk→y
−1
k vk−1,
If(α,Γ,β)is an inverse automaton overX and(r,s)is a relation inRsuch thatΓcontains a path
v1→r v
2, and no pathv1→sv2, then we perform anelementary expansion, relative tohX |Ri, by taking the disjoint unionΓ∪Γs and then forming the automaton(α η,(Γ∪Γs)/η,β η), whereη is
theV-equivalence generated by{(v1,αs),(v2,βs)}. We refer to this construction assewing onthe linear automaton ofsfromv1tov2.
If we take the disjoint union ofΓwith any automaton(α1,Γ1,β1)and then take the quotient by the
V-equivalence generated by{(v1,α1),(v2,β1)}, then we also refer to this operation assewing on
(α1,Γ1,β1)fromv1tov2. Ifv1=v2then we refer to this operation assewing on(α1,Γ1,β1)atv1. IfΓ has two edgesv1→yv2 and v1→yv3, for somey∈X∪X−1, then we can perform an
elementary determination, relative tohX |Ri, by taking the quotient ofΓby theV-equivalence
generated by{(v2,v3)}. An elementary expansion or determination of an inverse automaton is just an elementary expansion or determination of the underlying graph.
Result 2.6. [17, Lemmas 4.1, 5.5, 5.6]IfBis obtained from the automatonA by an elementary expansion or elementary determination, relative to hX|Ri, then L[A]⊆L[B]. Further, if
A A(X,R,w)thenB A(X,R,w).
Result 2.7. If B is the determinised form of the automaton A, relative to hX |Ri, then L[B]⊆ {z∈(X∪X−1)+:zτ≥yτ, for some y∈L[A]}.
Proof. We haveB=A/id∗. Thus ifz∈L[B]then there existsy∈L[A]that is freely reducible
toz. Hencezτ≥yτ for somey∈L[A].
Result 2.8. IfA A(X,R,w)andB is the determinised form ofA thenB A(X,R,w).
Proof. The proof follows from Results2.3and2.7.
A deterministic inverse automaton overX isclosed, relative tohX|Ri, if no elementary expansions
can be performed.
Result 2.9. [17, Theorem 5.10]IfA A(X,R,w)andA is deterministic and closed, relative tohX |Ri, thenA =∼A(X,R,w).
IfΓis an inverse graph overX then we saythere is a path from vertex v1to vertex v2labeled
by s∈S, and writev1→sv
IfΓis closed, relative tohX |Ri, and we have a pathv1→wv2, for somew∈(X∪X−1)+ with
wτ=s, then we have a pathv1→yv2, for everyy∈(X∪X−1)+withyτ≥s.
As defined in Stephen [17], anexpansion, relative tohX|Ri, consists of performing an elementary
expansion and then taking the determinised form.
Result 2.10. [16]The category of all inverse automata over X is cocomplete.
That is, every directed system of inverse automata overX has a direct limit. For an inverse automaton
A overX, theclosed form of A,relative tohX |Ri, is the direct limit of the directed system of all
automata obtained fromA by finite elementary expansions and determinations, relative tohX|Ri.
Result 2.11. [18, Theorem 3.3]The automatonA(X,R,w)is the closed form, relative tohX|Ri, of the linear automaton of w.
Result 2.12. [18, Lemma 3.4]IfBis the closed form of the automatonA, relative tohX |Ri, then we have L[B] ={z∈(X∪X−1)+ :zτ≥yτ, for some y∈L[A]}.
LetS1=InvhX1|R1iandS2=InvhX2|R2i, whereX1∩X2= /0. Then the free productS1∗S2, in the variety of inverse semigroups, has presentationhX|Ri, whereX=X1∪X2andR=R1∪R2. If
Γis an inverse word graph overX then each edge ofΓis labeled fromXi∪Xi−1, for somei∈ {1,2}, and is said to becoloredbyi. A subgraph ofΓismonochromaticif all its edges have the same color. A
lobeofΓis a maximal monochromatic connected subgraph ofΓ. The coloring of edges extends to
coloring of lobes. Two lobes are said to beadjacentif they share common vertices, calledintersections.
A path inΓis calledsimpleif it contains no repeated vertex, other than perhaps its first and last, in
which case it is asimple cycle. The graphΓis calledcactoidif it has finitely many lobes and every
simple cycle is monochromatic. An inverse automaton is calledcactoidif is underlying graph is cactoid.
Result 2.13. [13, Theorem 4.1]The Sch¨utzenberger automata of the free product S1∗S2=InvhX|Ri
are precisely, up to isomorphism, the cactoid inverse automata over X , where the lobes are isomorphic
to Sch¨utzenberger graphs of eitherhX1|R1iorhX2|R2i.
Construction 2.14. [13, Section 3] LetA = (α,Γ,β)be a cactoid inverse automaton overX
(v,∆,v) A(Xi,Ri,y), for somev∈V(∆)andy∈(Xi∪Xi−1)+, and let(v1,∆1,v1)∼=A(Xi,Ri,y) be disjoint from∆. Construct the quotient A0= (α η,(Γ∪∆1)/η,β η), whereη is the least V-equivalence identifyingvwithv1and determinising∆and∆1.
Result 2.15. [13, Propositions 3.1, 3.2, 3.3] If A0 is obtained from A by an application of Construction2.14thenA0is a cactoid inverse automaton over X with lobes that approximate graphs
relative to eitherhX1|R1iorhX2|R2i. Further, ifA A(X,R,w)thenA0 A(X,R,w).
Result 2.16. [13, Theorem 3.4]Starting with the linear automaton of w, any sequence obtained by repeated applications of Construction2.14terminates finitely inA(X,R,w).
Result 2.17. [13, Corollary 3.5]The free product S1∗S2=InvhX |Rihas decidable word problem if S1=InvhX1|R1iand S2=InvhX2|R2iboth have decidable word problems.
3.
DIRECTED SYSTEMS OF INVERSE AUTOMATANotation 3.1. Suppose we have an operation,Construction Q, say, that we can apply to inverse automata overX. LetIdenote the set of all automata obtained from a deterministic automatonA by
finitely many applications of ConstructionQ. Define a relation byB≤C if and only ifB=C orC
is obtained fromBby finite applications of ConstructionQ, forB,C ∈I.
Lemma 3.2. Suppose Construction Q satisfies the following, forB,C,D∈I:
(A) B≤C impliesC is a deterministic and L[B]⊆L[C].
(B) B≤C andB≤D implyC ≤E andD ≤E, for someE ∈I.
Then the automata of I form a directed system. The direct limit is the quotient of the disjoint union
tB∈IB, under the V -equivalenceη defined by v1ηv2if and only ifB≤D andC ≤D, for some
D ∈I, where the images of v1and v2are identified inD, for v1∈V(B), v2∈V(C)andB,C ∈I.
For any w in the language of the direct limit, there existsB in the directed system with w∈L[B].
Proof. It is immediate that the relation≤is reflexive and transitive. IfB≤C andC ≤B then
condition (A) impliesB∼=C, by Result2.2. Hence≤defines a partial order onI. IfC,D ∈I
thenA ≤C andA ≤D implyC ≤E andD ≤E, for someE ∈I, by condition (B). Thus(I,≤)
IfB,C ∈I andB≤C then we have a unique homomorphismαB,C :B→C, whereαB,B is
the identity map onB, by Result2.2. IfB,C,D∈I andB≤C ≤D thenαB,C◦αC,D =αB,D,
by uniqueness. Thus the automata inI, with the homomorphismsαB,C, determine a directed system.
It is immediate that η is reflexive and symmetric. Suppose v1ηv2 and v2ηv3, for some
v1∈V(B),v2∈V(C)andv3∈V(D), whereB,C,D ∈I. There existE,F ∈I withB,C ≤E andC,D≤F, wherev1andv2are identified inE, andv2andv3are identified inF. There exists
G ∈I withE,F ≤G. Thusv1andv3are identified inG and sov1ηv3. Henceη is transitive.
Thereforeηdefines aV-equivalence. PutH = (tB∈IB)/η.
Supposev1→xv3is an edge inBandv2→xv4is an edge inC, wherev1ηv2, for someB,C ∈I. There existsD ∈Isuch thatB,C ≤D, where the images ofv1andv2are identified inD. Thenv3
andv4are identified inD, sinceD is deterministic. Thusv3andv4are identified inH . HenceH is deterministic. For eachB∈I, we have a unique homomorphismβB:B→H , induced byη. For
allB,C ∈I withB≤C, we haveαB,C◦βC =βB, by uniqueness.
Suppose we have an inverse automaton H0 over X and a homomorphism γB:B→H 0,
for each B∈I, such that αB,C ◦γC =γB, for all B,C ∈I with B≤C. If v1ηv2, where
v1∈V(B)and v2∈V(C), for some B,C ∈I, then there exists D∈I withB,C ≤D, such that(v1)αB,D = (v2)αC,D. Thus we have(v1)γB= (v1)αB,D◦γD = (v2)αC,D◦γD = (v2)γC. Hence we have a mapδ :V(H)→V(H 0), defined by(vη)δ = (v)γB, forv∈V(B)andB∈I.
ForB∈Iand any edgev1→yv
2inB, we define(v1η→yv2η)δ = (v1→yv2)γB. Then we have a homomorphismδ :H →H 0withβB◦δ =γB, forB∈I. Now supposeδ0:H →H0is a
homomorphism such thatβB◦δ0=γB, forB∈I. Then(v)βB◦δ0= (vη)δ0= (v)γB, for all
v∈V(B), and it follows thatδ0=δ. HenceH is the direct limit of the directed system.
Supposew=x1x2· · ·xn∈L[H ]. PutA = (α,Γ,β). Then there existsBk∈Icontaining an edgevk→xk y
k, for eachk, withα ηv1,ykηvk+1, fork≤n−1, andynη β. Puty0=α,vn+1=β
and B0=Bn+1=A. Then, for 0≤k≤n, there exists Ck∈I with Bk,Bk+1≤Ck and the images of the verticesyk andvk+1 are identified inCk. By induction onn, there existsB∈I withC0,C1, . . . ,Cn≤B. The images ofykandvk+1are identified inB, for 0≤k≤n. Thus we
In Stephen [16], it was shown that the direct limit of the directed system of all automata obtained
from the linear automaton ofwby finitely many expansions, relative tohX|Ri, is the automaton
A(X,R,w). This result is included in the following.
Corollary 3.3. LetA be a deterministic inverse automaton over X and let R1, R2denote sets of relations. Then we have a directed system of all automata obtained fromA by finite applications of
performing an elementary expansion, relative tohX |R1i, and taking the closed form, relative tohX |R2i. The direct limitE is the closed form ofA, relative tohX|R1∪R2i. Thus if we have
A A(X,R1∪R2,w)thenE ∼=A(X,R1∪R2,w).
Proof. IfC is obtained fromB by performing an elementary expansion, relative tohX |R1i, and then taking the closed form, relative tohX |R2i, thenC is deterministic andL[B]⊆L[C]. Thus condition (A) of Lemma3.2is satisfied.
Letv1,v2,v3,v4denote vertices ofB. LetC1 be obtained fromB by sewing on the linear automaton ofsfromv1tov2. Then letC2be the closed form ofC1, relative tohX |R2i. Next letC3 be obtained fromC2by sewing on the linear automaton oft from the image ofv3to the image ofv4. Then letC4be the closed form ofC3, relative tohX |R2i. Similarly, letD1be obtained fromB by sewing on the linear automaton oft fromv3tov4. Then letD2be the closed form ofD1, relative tohX |R2i. Next letD3be obtained fromD2by sewing on the linear automaton ofsfrom the image ofv1to the image ofv2. Then letD4be the closed form ofD3, relative tohX |R2i.
It is immediate thatL[B]⊆L[D1]. NowL[D1]⊆L[D2], by Result2.12. ThusL[B]⊆L[D2]. It follows thatL[C1]⊆L[D3]. Then, from Result2.12, we haveL[C2]⊆L[D4]. It follows that
L[C3]⊆L[D4], since there is a path labeled byt from the image ofv3to the image ofv4inD4. Then, again from Result2.12, we haveL[C4]⊆L[D4]. A similar proof shows thatL[D4]⊆L[C4]. HenceC4∼=D4, by Result2.2. It now follows that condition (B) of Lemma3.2is satisfied.
Hence, from Lemma3.2, we have a directed system of all automata obtained fromA by finite
applications of performing an elementary expansion, relative tohX|R1i, and then taking the closed form, relative tohX |R2i. LetE denote the direct limit of this directed system and letF denote the closed form ofA, relative tohX |R1∪R2i.
Letw∈L[E]. Then, by Lemma3.2, there exists an automatonB in the directed system forE such
tohX |R1i, and taking the closed form, relative tohX |R2i, a finite number of times. IfA0is obtained fromA by an elementary expansion, relative tohX|R1i, thenz∈L[A0]impliesz≥yinInvhX|R1i, for somey∈L[A], by Result2.12. IfA00 is the closed form ofA0, relative tohX |R2i, then
z∈L[A00]impliesz≥yinInvhX|R2i, for somey∈L[A0], by Result2.12. It now follows that
z∈L[B]impliesz≥yinInvhX |R1∪R2i, for somey∈L[A]. Thusw≥yinInvhX|R1∪R2i, for somey∈L[A]. HenceL[E]⊆L[F], by Result2.12.
Conversely, letw∈L[F]. There existsC in the directed system forF withw∈L[C]. ThenC is
obtained fromA by a finite sequence of elementary expansions and elementary determinations, relative
tohX |R1∪R2i. Now ifA0is obtained fromA by either an elementary expansion or an elementary determination, relative tohX|R1∪R2i, then there exists an automatonB obtained fromA by an elementary expansion, relative tohX |R1i, and taking the closed form, relative tohX |R2i, withL[A0]⊆L[B]. IfA00 is obtained fromA0by an elementary expansion or an elementary
determination, relative tohX |R1∪R2i, then there exists an automatonB0where eitherB0=B or
B0is obtained fromB by an elementary expansion, relative tohX |R
1i, and taking the closed form, relative tohX |R2i, withL[A00]⊆L[B0]. It follows that there existsB00 in the directed system forE withL[C]⊆L[B00]and sow∈L[E]. ThusL[F]⊆L[E]. HenceE ∼=F, by Result2.2.
SupposeA A(X,R1∪R2,w). By Results2.3and 2.12, we haveE A(X,R1∪R2,w). SinceE is closed, relative tohX |R1∪R2i, we haveE ∼=A(X,R1∪R2,w), by Result2.9.
Corollary 3.4. LetA be a deterministic inverse automaton over X . Let V1be a subset of the vertices
ofA and let R1, R2be sets of relations. Then we have a directed system of all automata obtained from A by finite applications of an elementary expansion relative tohX|R1i, between vertices of V1or
their images, and taking the closed form relative tohX|R2i.
Proof. The proof is similar to that in Corollary3.3.
4.
A GENERALISATION OF LOWER BOUNDED AMALGAMSDefinition 4.1. We extend the terminology of [1] and [2] by calling an inverse subsemigroupU lower bounded in Sif for anyu∈U ande∈E(S)withu≥ethere exists f ∈E(U)withu≥ f ≥e. The
FIGURE 1. The lower bounded subsemigroup condition.
In this section, let[S1,S2;U;φ1,φ2]be an amalgam of inverse semigroups whereU is lower bounded inS1andS2.
Notation 4.2. Fori∈ {1,2}, we assume thatSihas a given inverse semigroup presentation, which we refer to byhSii. We then denote the Sch¨utzenberger automaton ofs∈Si, relative tohSii, byA(Si,s).
If∆is isomorphic to a Sch¨utzenberger graph ofhSiiandv∈V(∆)then we letei(v)denote the unique
idempotent ofE(Si)such that(v,∆,v)∼=A(Si,ei(v)).
We have an inverse semigroup presentationhS1∗S2i=hSi∪S2ifor the free productS1∗S2, in the variety of inverse semigroups. We denote the Sch¨utzenberger automaton ofz, relative tohS1∗S2i, by
A(S1∗S2,z), for any wordzin the generators ofS1andS2.
Foru∈E(U), letwi(u)be a word in the generators ofSithat equalsuinSi, fori∈ {1,2}. We
have a presentationhS1∗US2i=hS1∪S2|Wifor the amalgamated free productS1∗US2, where
W ={(w1(u),w2(u)):u∈U}. We assume thatw1(u)andw2(u)are calculable, foru∈U. We let
A(S1∗US2,z)denote the Sch¨utzenberger automaton ofz, relative tohS1∗US2i.
The following generalises the definitions of [1, Sections 2 and 3].
Definition 4.3. LetΓbe an inverse word graph over the generators ofS1andS2. Ifvis a vertex ofΓ that belongs to a lobe colored byithen we denote this lobe by∆i(v), fori∈ {1,2}. Alobe pathis a
finite sequence of lobes∆1,∆2, . . . ,∆n, where∆k is adjacent to∆k+1, for 1≤k≤n−1. The path is
reducedif it is not of the form∆1,∆2,∆1and all the lobes in the sequence are distinct, except possibly the first and last. There is a unique reduced lobe path between any two lobes if and only if there are no
non-trivial reduced lobe loops. Thelobe graphofΓis the graph with vertices consisting of the lobes of
whenever∆1and∆2are adjacent inΓ. The lobe graph of an inverse automaton, over the generators of
S1andS2, is the lobe graph of the underlying graph. The lobe graph ofΓis a tree if and only if there
are no non-trivial reduced lobe loops. The graphΓis cactoid if and only if the lobe graph ofΓis a finite
tree and adjacent lobes share precisely one intersection. A lobe isextremalif it is adjacent to precisely
one other lobe.
The graphΓhas theidempotent property relative tohSii, fori∈ {1,2}, if for every loopv→sv
inΓ, wheres∈Si, there is a loopv→ev, for somee∈E(Si)withs≥einSi. An inverse automaton
over the generators ofS1andS2has the idempotent property relative tohSiiif its underlying graph does. The Sch¨utzenberger graphs ofhSiihave the idempotent property relative tohSii. A product of
automata with the idempotent property relative tohSiialso has the idempotent property relative tohSii.
We say thatΓhas theidempotent propertyif it has the idempotent property relative tohS1iandhS2i. An inverse automaton over the generators ofS1andS2has the idempotent property if its underlying graph does. The Sch¨utzenberger graphs ofhS1∗S2ihave the idempotent property, by Result2.13. SupposeΓis a cactoid graph with the idempotent property. LetΓ0be obtained fromΓby closing a
lobe∆ofΓcolored byi, relative tohSii. Letv∈V(∆)and letv0denote the image ofvinΓ0. Then
w∈L[(v0,∆i(v0),v0)]impliesw≥yinSi, for somey∈L[(v,∆,v)], by Result2.12. In which case,
w≥y≥einSi, for somee∈E(Si)withe∈L[(v,∆,v)], by the idempotent property. Thus∆i(v0)has
the idempotent property relative tohSii. Ifvis an intersection of∆then(v0,∆j(v0),v0)is isomorphic to
a productΠx(x,∆j(x),x), where the product is taken over all intersectionsxof∆that are identified
withv0inΓ0, where∆j(x)denotes the lobe ofΓcolored by jthat containsx. Thus∆j(v0)has the
idempotent property relative tohSji. It follows thatΓ0also has the idempotent property. More generally,
ifΓis a cactoid graph with the idempotent property then the closed form ofΓ, relative tohS1∗S2i, also has the idempotent property.
The graphΓhas theequality propertyif, for every intersectionv, there is loopv→uvin∆1(v)if and only if there is loopv→uvin∆
2(v), for allu∈U. An inverse automaton over the generators of
S1andS2has the equality property if its underlying graph does.
For any intersectionvofΓ, the set ofrelated pairsofvconsists of(v,v)and all pairs(v1,v2)of vertices for which we have a pathv→uv
1in∆1(v)and a pathv→uv2in∆2(v), for someu∈U. If
propertyif the related pairs of any two intersections, not belonging to the same pair of lobes, share no
common coordinates. An inverse automaton over the generators ofS1andS2has the separation property if its underlying graph does.
Lemma 4.4. LetΓbe an inverse word graph over the generators of S1and S2that is closed, relative
tohS1∗S2i, and has the equality property. Then, for any intersection v ofΓ, the set of related pairs of v defines a partial one-one map between V(∆1(v))and V(∆2(v)).
Proof. Suppose(v1,v2)and(v01,v2)are related pairs ofv. Then we have pathsv→uv
1,v→u
0 v01in
∆1(v)and pathsv→uv2,v→u
0
v2in∆2(v), for someu,u0∈U1. SinceΓhas the equality property, we then have a loopv→u0u−1 vin
∆1(v). SinceΓis closed, relative tohS1∗S2i, we must have
v1=v01. Similarly, if(v1,v2)and(v1,v02)are related pairs ofvthenv2=v02. We have a partial one-one mapV(∆1(v))→V(∆2(v)), defined byv1→v2, for each related pair(v1,v2)ofv.
The following generalises [1, Construction 2.1].
Construction 4.5. LetA be a cactoid inverse automaton over the generators ofS1andS2that is closed, relative tohS1∗S2i. Supposevis an intersection ofA and we have a loopv→f vin∆
i(v), for
some f ∈E(U), and no loopv→f vin
∆j(v), for somei∈ {1,2}and j=3−i. LetA0be the
closed form relative tohS1∗S2iof the automaton obtained fromA by sewing on the linear automaton ofwj(f)at the intersectionv. In Figure2, the circles and dots represent lobes and vertices ofA,
arrows represent paths and the dashed arrow represents the linear automaton ofwj(f).
Lemma 4.6. LetA be a cactoid inverse automaton over the generators of S1and S2that is closed, relative tohS1∗S2i.
(i) We have a directed system of all automata obtained fromA by finite applications of
Construction4.5.
(ii) The direct limitBis cactoid and closed, relative tohS1∗S2i.
(iii) We have a graph homomorphism from the lobe tree ofA onto the lobe tree ofB.
(iv) IfA A(S1∗US2,w)thenB A(S1∗US2,w).
(v) IfA has the idempotent property thenB has the idempotent property.
(vi) IfBhas the idempotent property thenBhas the equality property.
Proof. Only (vi) assumes thatU is lower bounded inS1andS2: (i) We have a directed system from Corollary3.4.
(ii) Put A = (α1,Γ1,β1) and B= (α2,Γ2,β2). Since A is cactoid, it follows that any automaton in the directed system is also cactoid. ThusBmust be cactoid. Letv1→rv
2be a path inB, where(r,s)is some relation ofS1orS2. Letα2→w1v1andv2→w2 β2be paths inB. Then, from Lemma3.2, there is some automatonA0in the directed system with
w1rw2∈L[A0]. SinceA0is closed, relative tohS1∗S2i, we havew1sw2∈L[A0]. Thus
w1sw2∈L[B]. HenceB is closed, relative tohS1∗S2i.
(iii) ForA0 in the directed system forB, the homomorphism fromA into A0 induces a
homomorphism from the lobe tree of A onto the lobe tree of A0. It follows that the
homomorphism fromA intoB induces a homomorphism from the lobe tree ofA onto the
lobe tree ofB.
(iv) SupposeA A(S1∗US2,w). Letw0∈L[B]. From Lemma3.2, there is someA0in the directed system forBwithw0∈L[A0]. It follows, from Results2.3,2.6and2.12, that
A0 A(S
1∗US2,w). Thusw0∈L[A(S1∗US2,w)]. HenceB A(S1∗US2,w). (v) SupposeA has the idempotent property. Ifzdefines an idempotent ofE(Si), wherei∈ {1,2},
then, fory∈L[(αz,Γz,βz)/η], we havey≥zinSi, where(αz,Γz,βz)is the linear automaton
ofzandη is theV-equivalence generated by(αz,βz). Thus any automaton obtained fromA
by sewing on the linear automaton ofz, at some vertex, has the idempotent property. From
tohS1∗S2i, of a cactoid graph. It follows that any automaton in the directed system forB has the idempotent property. If we have a pathα2→w1 vand a loopv→svinB, wheres∈Si andi∈ {1,2}, then, lettingα1→wβ1be a path inA, there is some automatonA0in the directed system withw1sw−11w∈L[A0], by Lemma3.2. Since the automatonA0has the idempotent property, we havew1ew−11w∈L[A0], for somee∈E(Si)withs≥einSi. We then have a loopv→evinB. HenceBhas the idempotent property.
(vi) SupposeBhas the idempotent property. Letvbe an intersection ofBand suppose we
have a loopv→wi(u)vin∆
i(v), for someu∈U and somei∈ {1,2}. Put j=3−i. From
the idempotent property, we have a loopv→evin
∆i(v), for somee∈E(Si)withu≥e.
SinceU is lower bounded inSi, there exists f ∈E(U)withu≥ f ≥einSi. SinceBis
closed, relative tohS1∗S2i, we have a loopv→wi(f)vin∆i(v). Letv1denote an intersection of A that is a preimage ofvand letα1→wβ1 andα1→zv1be paths inA. We have
z·wi(f)·z−1w∈L[B]. There is some automatonA0= (α10,Γ01,β10)in the directed system forB such thatz·wi(f)·z−1w∈L[A0], by Lemma3.2. ThusA0contains a pathα10 →zv0
and a loopv0→wi(f)v0, wherev0is an intersection. We can obtainA00 fromA0by an
application of Construction4.5such that we have a loopv00→wj(f)v00 in∆
j(v00), lettingv00
denote the image ofv0inA00. Thus we have a loopv→wj(f)vin∆
j(v). Sinceu≥ f inU,
we have a loopv→wj(u)vin∆
j(v). HenceBhas the equality property.
Lemma 4.7. Let(α2,Γ2,β2)be the direct limit of the directed system of all automata obtained from
(α1,Γ1,β1), by finite applications of Construction4.5. Letγ1,δ1∈V(Γ1)and letγ2,δ2denote their
respective images inΓ2. Then(γ2,Γ2,δ2)is the direct limitC of the directed system of all automata
obtained from(γ1,Γ1,δ1), by finite applications of Construction4.5.
Proof. Let γ1→w1 α1 and β1→w2 δ1 be paths in Γ1. Let y∈L[(γ2,Γ2,δ2)]. Then we have
w−11yw−21∈L[(α2,Γ2,β2)]and so there is some automaton(α10,Γ01,β10)in the directed system for
(α2,Γ2,β2)withw1−1yw−21∈L[(α10,Γ01,β10)], from Result3.2. Now(α10,Γ01,β10)is obtained from
deterministic, we havey∈L[(γ10,Γ10,δ10)]. Thusy∈L[C]and soL[(γ2,Γ2,δ2)]⊆L[C]. Similarly,
L[C]⊆L[(γ2,Γ2,δ2)]. Hence(γ2,Γ2,δ2)∼=C, by Result2.2. Lemma 4.8. LetBdenote the direct limit of the directed system of all automata obtained fromA, by finite applications of Construction4.5. Then there existsA0in the directed system forBsuch that
the homomorphism fromA0intoBinduces an isomorphism of the lobe trees.
Proof. PutA = (α1,Γ1,β1)and letw∈L[A]. Suppose the homomorphism fromA intoB maps distinct lobes∆1and∆2 ofA into a lobe ofB. Let the images ofv1∈V(∆1)andv2∈V(∆2) be identified inB. Letα1→w1v1,α1→w2 v2be paths inΓ1. Then we havew1w2−1w∈L[B]. There exists A0 in the directed system for B with w1w2−1w∈L[A0], from Result 3.2. Put
A0= (α0
1,Γ01,β10). There are pathsα10 →w1 v01andα10 →w2 v01inΓ01. Thus the images ofv1andv2 inA0are identified. Hence the homomorphism fromA intoA0maps∆1and∆2into a lobe ofA0. SinceA has finitely many lobes, it follows that there existsA0in the directed system forBsuch that
any lobes ofA that are identified inB are also identified inA0. Thus the lobe trees ofA0andB
must be isomorphic.
Construction 4.9. LetB= (α2,Γ2,β2)be a cactoid inverse automaton over the generators of
S1andS2that is closed, relative tohS1∗S2i. Supposev2is an intersection,∆i(v2)is extremal, wherei∈ {1,2},α2,β2∈/V(∆i(v2))\{v2}, and for any loopv2→yv2in∆i(v2)there exists a loop
v2→f v
2in∆j(v2), where j=3−iand f ∈E(U)withy≥ f inSi. Then putB0= (α∼ 2,Σ2,β2), whereΣ2is the subgraph ofΓ2consisting of all the lobes except∆i(v2).
Lemma 4.10. SupposeB0is obtained fromBby Construction4.9. If we haveB A(S1∗US2,w)
thenB0 A(S1∗US2,w).
Proof. We adopt the notation of Construction4.9. We haveL[B0]⊆L[B]⊆L[A(S1∗US2,w)]. Let
w0∈L[B]such thatw0=winS1∗US2. Ifw0∈L[B0]then it is immediate thatB0 A(S1∗US2,w). Otherwise we havew0=z0y1z1· · ·ynzn, for somen≥1, some pathsα2→z0 v2,v2→zkv2,v2→zn β2 inΣ2, wherek≤n−1, and some loopsv2→ykv2in∆i(v2), wherek≤n, allowingz0andznto be 1. Then, by assumption, there is a loopv2→fkv
2in∆j(v2), where fk∈E(U), j=3−iand
yk≥ fkinSi, for 1≤k≤n. Putz=z0·wj(f1)·z1· · · ·wj(fn)·zn. Thenw0≥zinS1∗US2, where
Corollary 4.11. For any word w, there exists a word w0such that we have w=w0in S1∗US2, the automatonA(S1∗S2,w0)has at most as many lobes asA(S1∗S2,w)and the direct limit of the directed system of all automata obtained fromA(S1∗S2,w0), by finite applications of Construction4.5,
has the property that Construction4.9cannot be applied.
Proof. Let B denote the direct limit of the directed system of all automata obtained from
A =A(S1∗S2,w), by finite applications of Construction4.5. By Lemma4.6, the automatonBhas at most as many lobes asA.
SupposeB0is obtained fromB by an application of Construction4.9. Adopt the notation of
Construction4.9. By Lemma4.6, we haveB A(S1∗US2,w). Hence, from Lemma4.10, we haveB0 A(S1∗US2,w). Thus there exists a wordz∈L[B0]such thatz=winS1∗US2. By Lemma4.8, there existsA0= (α10,Γ01,β10)in the directed system forBsuch that the homomorphism
fromA0intoB induces an isomorphism of the lobe trees. From Lemma3.2, there existsA00
in the directed system forB such that z∈L[A00]. Since we have a directed system, we can
assumeA0=A00.
Letv01denote the unique intersection ofA0that is a preimage ofv2. Then putC = (α10,Σ01,β10), where Σ01 is the subgraph of Γ01 consisting of all lobes, except for ∆i(v01). The automata
C = (α10,Σ01,β10), A0= (α10,Γ01,β10), B0= (α2,Σ2,β2) and B= (α2,Γ2,β2) are illustrated in Figure3. For some wordw0, we haveC ∼=A(S1∗S2,w0). NowL[C]⊆L[B0]. Also,z∈L[B0]
FIGURE 3. The automataC,A0,B0andB.
S1∗US2andC has fewer lobes thanB. LetD denote the direct limit of the directed system of all automata obtained fromC, by finite applications of Construction4.5. By Lemma4.6, the automatonD
has at most as many lobes asC. ThusD has fewer lobes thanB. SinceB has finite lobes, we can
continue in this manner until no application of Construction4.9can be applied.
The following generalises [1, Construction 3.3].
Construction 4.12. LetB= (α2,Γ2,β2)be a cactoid inverse automaton over the generators ofS1 andS2that is closed, relative tohS1∗S2i. Suppose there are pathsv1→wi(u)v
0andv1→wj(u)v2 inΓ2, wherev0andv1are two intersections, for someu∈U,i∈ {1,2}and j=3−i. Figure4 illustrates the situation. Since the lobe graph ofΓ2is a tree, the unique reduced lobe path from a lobe of
FIGURE 4. Construction4.12illustrated.
Γ2to∆i(v1)either contains∆j(v1)or does not. LetΣidenote the subgraph ofΓ2containing∆i(v1) and any lobe where the unique reduced lobe path to∆i(v1)does not contain∆j(v1). Similarly, letΣj denote the subgraph ofΓ2containing∆j(v1)and any lobe where the unique reduced lobe path to
∆j(v1)does not contain∆i(v1). ThusΣi∪Σj=Γ2andΣi∩Σjconsists ofv1.
LetΣ∗i andΣ∗j denote disjoint copies ofΣiandΣj, respectively. Ifα26=v1then letα∗denote the unique image ofα2inΣ∗i ∪Σ∗j. Itα2=v1then letα∗denote the image ofv1inΣ∗i. Defineβ∗ similarly. Then letη denote theV-equivalence onΣ∗i ∪Σ∗j generated by{(v0,v2)}, lettingv0andv2 denote their unique images inΣ∗i andΣ∗j, respectively. PutC = (α∗η,(Σ∗i ∪Σ∗j)/η,β∗η). LetB0
denote the closed form ofC relative tohS1∗S2i.
(i) The automatonB0is cactoid and has fewer lobes thanB.
(ii) IfB A(S1∗US2,w)thenB0 A(S1∗US2,w). (iii) IfBhas the idempotent property then so doesB0.
Proof. We adopt the notation of Construction4.12.
(i) Sincev2is identified withv0, the automatonC is cactoid and has fewer lobes thanB. Thus
B0is cactoid and has fewer lobes thanB.
(ii) Suppose B A(S1∗US2,w). Let α2→zv1 denote a path in either Σi or Σj. Since
L[B]⊆L[A(S1∗US2,w)], the languagesL[(v0,Σi,v0)]andL[(v2,Σi,v2)]are contained in L[A(S1∗US2,u−1z−1ww−1zu)]and we haveww−1zuRww−1in S1∗US2. Then the languageL[(v0η,(Σ∗1∪Σ∗j)/η,v0η)]is contained inL[A(S1∗US2,u−1z−1ww−1zu)]. Since we have a pathα∗η→zuv0η in(Σ∗1∪Σ∗j)/η, the languageL[(α∗η,(Σ∗1∪Σ∗j)/η,α∗η)]is contained inL[A(S1∗US2,ww−1)]. We show there existsy∈L[C]withy=winS1∗US2. It will follow thatC A(S1∗US2,w)and soB0 A(S1∗US2,w), by Result2.12.
Letz∈L[B]such thatz=winS1∗US2. Ifα2→zβ2is contained inΣithen puty=z. If
α2→zβ2is contained inΣjthen puty=z0zz1, wherez0=wi(u)·(wj(u))−1ifα2=v1, elsez0=1, andz1=wj(u)·(wi(u))−1ifβ2=v1, elsez1=1. Then we havey∈C and
y=winS1∗US2.
Now supposeα2→zβ2is not contained inΣiand not contained inΣj. Thenα2→zβ2is the concatenation of subpathsα2→w0v1,v1→w1v1,. . .,v1→wn β2, for somen≥2, where each subpath is either contained inΣior contained inΣj. If the subpath labeled bywk
is contained inΣithen putyk=wk. If the subpath labeled bywkis contained inΣjthen
putyk=z0wkz1, wherez0=wi(u)·(wj(u))−1if the subpath starts atv1, elsez0=1, and
z1=wj(u)·(wi(u))−1if the subpath ends atv1, elsez1=1. Puty=y1y2· · ·yn. Then we havey∈C andy=winS1∗US2.
(iii) SupposeB has the idempotent property. Then the automata(v0,Σi,v0)and(v2,Σi,v2)have the idempotent property. It follows thatC has the idempotent property. ThenB0has the
idempotent property, sinceC is cactoid and so the idempotent property is preserved under the
operation of taking the closed form, relative tohS1∗S2i, from the workings in Definition4.3.
Corollary 4.14. For any word w, there is a word w0with w=w0in S1∗US2,A(S1∗S2,w0)has at most as many lobes asA(S1∗S2,w)and the direct limit of the directed system of all automata obtained fromA(S1∗S2,w0), by finite applications of Construction4.5, has the separation property.
Proof. LetA =A(S1∗S2,w). LetB= (α2,Γ2,β2)denote the direct limit of the directed system of all automata obtained fromA, by finite applications of Construction4.5. By Lemma4.6, the
automatonB is cactoid and has the equality property, with at most as many lobes asA.
SupposeB does not have the separation property. Then there are distinct intersectionsv0andv1of
Band a pathv1→wi(u)v
0in∆i(v1), for someu∈U andi∈ {1,2}. Put j=3−i. SinceB has the equality property, there is a pathv1→wj(u)v
2in∆j(v1). Letα2→zv1be a path inB.
By Lemma 4.8, there exists A0= (α10,Γ01,β10) in the directed system for B such that the
homomorphism fromA0intoBinduces an isomorphism of the lobe trees. By Lemma3.2, there exists
A00 in the directed system forB withz·w
i(u)·wj(uu−1)·wi(u−1)·wj(uu−1)·z−1w∈L[A00]. Since we have a directed system, we can assume thatA0=A00. Thus we have pathsα10 →zv01, v01→wi(u)v0
0andv01→wj(u)v02inA0, wherev00,v01andv20 are preimages ofv0,v1andv2, respectively, andv00andv01are two intersections ofA0.
LetA000be obtained fromA0by an application of Construction4.12. ThenA000 has fewer lobes
thanA0and soA000has fewer lobes thanB. ThusA000 has fewer lobes thanA, by Lemma4.6. From
Construction2.14and Result2.15, we haveA000∼=A(S1∗S2,w0), for some wordw0. Also, from Lemma4.13, we haveA000 A(S1∗US2,w).
LetB0denote the direct limit of the directed system of all automata obtained fromA000, by finitely
many applications of Construction4.5. ThenB0has at most as many lobes asA000, by Lemma4.6.
HenceB0has fewer lobes thanB. Continuing in this manner, we reach such an automatonA000 where
the direct limitB0has the separation property.
The following generalises the definitions of [1, Sections 4 and 5].
Definition 4.15. LetΓbe an inverse word graph over the generators ofS1andS2that is closed, relative tohS1∗S2i. We say that an intersectionvofΓhasidentified related pairsif every related pair
assimilatedbyv. Figure5illustrates a graph where the related pairs are not identified and a graph where
the lobes are assimilated. If∆1(v)and∆2(v)are assimilated byvthen they are assimilated by every
FIGURE 5. Unidentified related pairs and assimilated lobes.
intersection of∆1(v)and∆2(v). The graphΓhas theassimilation propertyif any two adjacent lobes are assimilated by a common intersection. The assimilation property implies the separation property, An
inverse automaton over the generators ofS1andS2that is closed, relative tohS1∗S2i, has the assimilation property if its underlying graph does.
IfΓhas the equality property then the related pairs of any intersectionvdefine a partial one-one map
betweenV(∆1(v))andV(∆2(v)), by Lemma4.4. IfΓhas the equality and separation properties andv is the only intersection of∆1(v)and∆2(v), then we canassimilate∆1(v)and∆2(v)by taking the quotient ofΓby theV-equivalenceη generated by the related pairs ofv. SinceΓhas the separation
property, each lobe ofΓis mapped isomorphically onto a lobe ofΓ/η, underη. It follows thatΓ/ηis
closed, relative tohS1∗S2i, and has the equality and separation properties.
More generally, supposeΓhas the equality and separation properties and adjacent lobes ofΓhave
precisely one intersection. Then theassimilated formofΓis the quotient ofΓby theV-equivalenceη
generated by the related pairs of every intersection. Similarly, each lobe ofΓis mapped isomorphically
onto a lobe ofΓ/η, underη, and the quotientΓ/η is closed, relative tohS1∗S2i, with the equality, separation and assimilation properties. The assimilated form of an inverse automaton, over the
generators ofS1andS2, is defined by taking the assimilated form of the underlying graph. The graphΓisopuntoidif it has the idempotent, equality and assimilation properties and has no
non-trivial reduced lobe loops (equivalently, the lobe graph is a tree). Asubopuntoid subgraphof an
opuntoid graphΓis a connected subgraph that is also opuntoid and formed by a collection of the lobes
opuntoid if its underlying graph is opuntoid. A subautomaton is subopuntoid if its underlying graph is a
subopuntoid subgraph.
LetΓbe an opuntoid graph. A vertexv∈V(Γ)is abudif it is not an intersection and there is a loop
v→f vinΓ, for some f ∈E(U). We say that the opuntoid graphΓiscompleteif it has no buds. An
opuntoid automaton over the generators ofS1andS2is complete if its underlying graph is complete. Lemma 4.16. LetB denote a cactoid inverse automaton over the generators of S1and S2that
is closed, relative tohS1∗S2i, and has the idempotent, equality and separation properties. Let C denote the assimilated form ofB. Then C is opuntoid and B A(S1∗US2,w)implies
C A(S1∗US2,w).
Proof. Put B= (α,Γ,β). Let η denote theV-equivalence generated by the related pairs of
all the intersections ofΓ. It follows from the discussion in Definition 4.15that C =B/η is
opuntoid. Supposew0∈L[C]. Then there exist pathsx1→w1 y
1,x2→w2 y2,. . .,xn→wn yninB, where α ηx1, ykηxk+1, for 1≤k≤n−1, ynη β and the wordw1w2· · ·wn is equal to w0. Put
y0=α and xn+1=β. For 0≤k≤n, assuming yk6=xk+1, there is a path yk→ak xk+1 in B whereak=w1(u−1)·w2(u)orak=w2(u−1)·w1(u), for someu∈U. Then we havew0≥w00 in
S1∗US2, wherew00=a0w1a1w2a2· · ·wnan∈L[B]. Thus, assumingB A(S1∗US2,w), we haveL[C]⊆L[A(S1∗US2,w)]and it follows thatC A(S1∗US2,w).
The following generalises [1, Construction 5.1].
Construction 4.17. LetD be an opuntoid automaton that is closed, relative tohS1∗S2i. Ifvis a bud of a lobe∆i(v)ofD, wherei∈ {1,2}, then, putting j=3−i, we form the automatonE fromD by
sewing on atvthe linear automaton ofwj(f), for every f ∈E(U)that labels a loop atvin∆i(v). In
Figure6, the dashed arrows represent the linear automata that are sewed on atvand the dashed circle
represents the new lobe created. LetE0denote the closed form ofE, relative tohS1∗S2i. Letv0be the image ofvinE. ThenE0is obtained fromE by closing∆j(v0), relative tohSji. Letv00be the image
ofv0inE0. LetD0be the quotient ofE0by theV-equivalence generated by the related pairs ofv00.
Lemma 4.18. LetD be an opuntoid automaton and letD0be obtained fromD by an application of Construction4.17. ThenD0is an opuntoid automaton andD is a subopuntoid subautomaton ofD0.
FIGURE 6. Construction4.17illustrated.
Proof. We adopt the notation of Construction4.17. PutD = (α,Γ,β). LetA(wj(f))denote the
linear automaton ofwj(f), for f ∈E(U). Letα0andβ0denote the respective images ofα andβ
inE0. Letη denote theV-equivalence generated by the related pairs ofv00.
Ifw00∈L[(v00,∆j(v00),v00)]thenw00 ≥w0inSj, for somew0∈L[(v0,∆j(v0),v0)], by Result2.12.
Noww0∈L[Πnl=1A(wj(fl))], for some fl∈E(U), where fl labels a loop at vin ∆i(v). Thus
w00≥ f1f2· · ·fninSj, where f1f2· · ·fnlabels a loop atvin∆i(v)and so f1f2· · ·fnlabels a loop at
v00 in∆j(v00). It follows that∆j(v00)has the idempotent property. If we also havew00∈U then
w00≥ f1f2· · ·fnimplies thatw00 labels a loop atvin∆i(v). SinceD has the idempotent and equality properties, it now follows thatD0has the idempotent and equality properties.
SinceD has the assimilation property, if we have a pathv→uv
1in∆i(v), for someu∈U, thenv1 cannot be an intersection. ThusE0must have the separation property. ThenD0is obtained fromE0by
assimilating the lobes containingv00. SinceD is opuntoid, it now follows thatD0is opuntoid andD is a
subopuntoid subautomaton ofD0.
Supposew0∈L[D0]. Then there exist pathsx1→w1 y
1,x2→w2 y2,. . .,xn→wn yninE0, where
α0ηx1,ykηxk+1, for 1≤k≤n−1,ynη β and the wordw1w2· · ·wnis equal tow0. Puty0=α and
xn+1=β. Then, for 0≤k≤n, we have eitheryk=xk+1, in which case putbk=1, or there is a pathyk→bkx
k+1inE0wherebk=w1(u−1)·w2(u)orbk=w2(u−1)·w1(u), for someu∈U. Thusw0≥zinS1∗US2, wherez=b0w1b1w2b2· · ·wnbn∈L[E0].
By Result 2.12, ifz∈L[E0]then z≥z0 inS1∗S2, for some z0∈L[E]. Ifz0∈/L[D]then we have z0=y1a1w1a2· · ·wm−1amy2, for some y1∈L[(α,Γ,v)], ak∈L[(v0,∆j(v0),v0)], for all k,
inSj, wheregk labels a loop atvin∆i(v), for eachk. It follows thatz0≥z00 inS1∗US2, where
z00=y1g1w1g2· · ·wm−1gmy2∈L[D]. Hence we havez≥z00inS1∗US2, for somez00 ∈L[D]. We have shown that ifw0∈L[D0]then we havew0≥z00 inS1∗US2, for somez00∈L[D]. Hence if
D A(S1∗US2,w)thenL[D0]⊆L[A(S1∗US2,w)]and soD0 A(S1∗US2,w).
Lemma 4.19. LetD be an opuntoid automaton. Then we have a directed system of all automata obtained from the automatonD by finite applications of Construction4.17. The direct limitE is a
complete opuntoid automaton that is closed, relative tohS1∗US2i. Thus ifD A(S1∗US2,w)then
we haveE ∼=A(S1∗US2,w).
Proof. PutD= (α1,Γ1,β1). Letv1andv2denote buds ofΓ1. SupposeDkis obtained fromD applying Construction4.17atvk, fork=1,2. We show that there is an automatonD3that is obtained from bothD1andD2by an application of Construction4.17. By Lemma4.18, the automatonD is a subopuntoid subautomaton ofDk. ThusL[D]⊆L[Dk], fork=1,2. SinceD is embedded intoDk,
we letv1andv2denote their images inDk.
Suppose we do not have a pathv1→wi(u)v2inD, for anyu∈U andi∈ {1,2}. Thenv2is a bud ofD1andv1is a bud ofD2. It follows that the automatonD3obtained fromD1by applying Construction4.17at the budv2is isomorphic to the automaton obtained fromD2by applying Construction4.17at the budv1.
Suppose we have a pathv1→wi(u)v2 inD, for some u∈U andi∈ {1,2}. Thenv1 and v2 belong to a lobe ofD colored byi. Let∆denote the lobe ofD1containingv1andv2and colored by j=3−i. We havez∈L[(v1,∆,v1)]if and only ifz≥ f1f2· · ·fninSj, for some fk∈E(U)that label loops atv1 inD, using workings similar to those in the proof of Lemma4.18. We show thatz∈L[(v2,∆,v2)]if and only if we havez≥g1g2· · ·gninSj, for somegk∈E(U)that label loops atv2inD. It will then follow thatD1∼=D2. Supposez∈L[(v2,∆,v2)]. Thenuzu−1labels a loop atv1in∆and souzu−1≥ f1f2· · ·fn, for some fk∈E(U)that label loops atv1inD. Thus
z≥u−1f1uu−1f2u· · ·u−1fnu, wheregk=u−1fku∈E(U)labels a loop atv2inD. Conversely, supposez≥g1g2· · ·gninSj, for somegk∈E(U)that label loops atv2inD. Thenug1g2· · ·gnu−1
It now follows that conditions (A) and (B) of Result3.2are satisfied. We have a directed system of
all automata obtained fromD by finite applications of Construction4.17. LetE = (α2,Γ2,β2)be the direct limit.
Letv1→r v
2be a path inE, where(r,s)is a relation ofS1orS2. Letα2→z1v1andv2→z2β2be paths inE. There existsD0in the directed system forE such thatz1rz2∈L[D0], by Result3.2. Since
D0is closed, relative tohS
1∗S2i, we havez1sz2∈L[D0]. Thusz1sz2∈L[E]. HenceE is closed, relative tohS1∗S2i.
Suppose we have a loopv→svinE, wheres∈S
iandi∈ {1,2}. Letα1→wβ1be a path inD and letα2→zvbe a path inE. Then we havezsz−1w∈L[D0], for someD0in the directed system forE, by Result3.2. PutD0= (α10,Γ01,β10). Thus we have a pathα10 →zv0and a loopv0→sv0inD0.
SinceD0has the idempotent property, by Lemma4.18, we have a loopv0→ev0inD0, for some
e∈E(Si)withs≥e. Thus we have a loopv→evinE. HenceE has the idempotent property.
Suppose v is an intersection of E such that we have a loop v→wi(u)v, for some u∈U
and i∈ {1,2}. Put j=3−i. Let α1→wβ1 be a path in D and letα2→zv be a path in E. Then we havez·wi(u)·z−1w∈L[D0], for someD0in the directed system forE, by Result3.2. PutD0= (α10,Γ01,β10). Sincew∈L[D0], we must have a pathα10 →zv0and a loopv0→wi(u)v0inD0.
Ifv0is an intersection then we have a loopv0→wj(u)v0, sinceD0has the equality property, by
Lemma4.18. Otherwise, the vertexv0is a bud and we can obtain an automatonD00 fromD0with the
equality property, by Construction4.17, such that we have a loopv00→wj(u)v00, lettingv00 denote the
image ofv0inD00. Thus we have a loopv→wj(u)vinE. HenceE has the equality property.
Supposevis an intersection ofE and let(v1,v2)denote a related pair ofv, other than(v,v). Let α1→wβ1be a path inD and letα2→zv be a path inE. We have pathsv→w1(u)v1 and
v→w2(u)v
2inE, for someu∈U. Hence we havez·w1(uu−1)·w2(uu−1)·z−1w∈L[D0], for some automatonD0in the directed system forE, by Result3.2. PutD0= (α10,Γ01,β10). Then we have
pathsα10 →zv0,v0→w1(u)v0
1andv0→w2(u)v02inD10. SinceD0is opuntoid, with identified related pairs, by Lemma4.18, we must havev01=v20. Thusv1=v2and soE has identified related pairs.
Supposev3is also an intersection of∆1(v)and∆2(v). Letv→s1v3be a path in∆1(v)and let
v→s2 v
pathsα100 →zv00,v00→s1 v00
3 andv00→s2 v003 inD100. Thus(v003,v003)is a related pair ofv00, sinceD00has the assimilation property, by Lemma4.18. Hence(v3,v3)is a related pair ofvand soE has the assimilation property.
Suppose the lobe graph ofE has a reduced lobe loop∆1,∆2, . . . ,∆n=∆1, for somen≥4. Then the lobes∆1,∆2, . . . ,∆n−1 are distinct. Letvk denote an intersection common to∆k and ∆k+1, for 1≤k≤n−1. Putvn=v1. Letvk→wkv
k+1be a path in∆k+1, for 1≤k≤n−1. Letα1→wβ1 be a path inD and letα2→yv1be a path inE. We haveyw1w2· · ·wn−1y−1w∈L[D0], for some automatonD0in the directed system forE, by Result3.2. Then the lobes ofD0containing the loop
labeled byw1w2· · ·wn−1is a non-trivial reduced lobe loop and we have a contradiction, sinceD0is opuntoid. Hence the lobe graph ofE contains no non-trivial reduced lobe loops. We have now proved
thatE is opuntoid.
Supposevis a bud ofE of a lobe∆i(v), for somei∈ {1,2}. Letv→wi(f)vbe a loop in∆i(v), for
some f ∈E(U). Letα1→wβ1be a path inΓ1and letα2→zvbe a path inE. There is someD0in the directed system such thatz·wi(f)·z−1w∈L[D0], by Result3.2. PutD0= (α10,Γ01,β10). Then we
have a pathα10 →zv0and a loopv0→wi(f)v0 inD0, wherev0is a bud ofD0. We can obtain an
automatonD00 fromD0, by an application of Construction4.17, such thatv00 is an intersection ofD00,
lettingv00 denote the image ofv0inD00. We have a contradiction, since this impliesvis an intersection.
ThusE is a complete opuntoid automaton.
Letv1→w1(u)v
2be a path inE, for someu∈U. Then we have a loopv1→w1(uu
−1)
v1inE, since
E is closed, relative tohS1i. SinceE is complete, the vertexv1must be an intersection. SinceE has the equality property, we have a loopv1→w2(uu−1)v
1inE. We have a pathv1→w2(u)v02inE, since
E is closed, relative tohS2i. Then the assimilation property implies thatv2=v02. Hence we have a path
v1→w2(u)v
2inE. Similarly, if we have pathv1→w2(u)v2inE, for someu∈U, then we have a pathv1→w2(u)v2. We have shown thatE is closed, relative tohS1∗US2i.
Algorithm 4.20. Givenwin the generators ofS1 andS2, the Sch¨utzenberger automaton ofw, relative tohS1∗US2i, is constructed as follows:
(i) ConstructA =A(S1∗S2,w), by Construction2.14. ThenA is cactoid, has the idempotent property and we haveA A(S1∗US2,w).
(ii) The direct limit B of the directed system of all automata obtained fromA, by finite
applications of Construction4.5, is cactoid, has the idempotent and equality properties, has at
most as many lobes asA and we haveB A(S1∗US2,w), by Lemma4.6.
(iii) An application of Construction4.12toBresults in an automatonB0that is cactoid, has the
idempotent and equality properties, has fewer lobes thanBand we haveB0 A(S1∗US2,w), by Lemma4.13.
(iv) Steps (ii) and (iii) can be applied at most a finite number of times, since the initial automaton
has finite lobes. The resulting automatonC is cactoid, has the idempotent, equality and
separation properties andC A(S1∗US2,w).
(v) The assimilated formD ofC is a finite-lobe opuntoid automaton andD A(S1∗US2,w), by Lemma4.16.
(vi) The direct limitE of the directed system of all automata obtained fromD, by finite applications
of Construction4.17, is a complete opuntoid automaton and we haveE ∼=A(S1∗US2,w), by Lemma4.19.
The following generalises [1, Section 6].
Definition 4.21. LetΓbe an opuntoid graph. Let∆1and∆2be adjacent lobes, colored byi∈ {1,2} and j=3−i, respectively. Then∆2feeds off ∆1if there is a common intersectionvsuch that, for any loopv→yvin∆2, there is a loopv→f vin∆2, for some f ∈E(U)withy≥ f inSj.
Suppose∆2feeds of∆1and letv0denote any intersection of∆1and∆2. By the assimilation property, we have a pathv→uv0, for someu∈U. If we have a loopv0→yv0in
∆2then we have a loop
v→uyu−1 vin∆
2. Thus we have a loopv→f vin∆2, for some f ∈E(U)withuyu−1≥ f inSj. Hence we have a loopv0→u−1f uv0in
∆2, wherey≥u−1f uinSj. Therefore, if∆2feeds of∆1then for any loopv0→yv0in∆
For non-adjacent lobes∆1and∆nofΓ, we say that∆nfeeds off ∆1if there is a sequence of lobes
∆1,∆2, . . . ,∆n, where∆k+1is adjacent to∆kand∆k+1feeds off∆k, for 1≤k≤n−1.
LetΓ0be a subopuntoid subgraph ofΓ. A lobe ofΓthat does not belong toΓ0is calledexternal
toΓ0. An extremal lobe ofΓ0is a called aparasiteif it feeds off the unique lobe ofΓ0to which it is
adjacent. The subgraphΓ0isparasite-freeif it has no parasites. The subgraphΓ0is ahostofΓif it has
finitely many lobes, is parasite-free, and every lobe ofΓthat is external toΓ0feeds off some lobe ofΓ0.
A host of an opuntoid automaton is a host of its underlying graph.
Lemma 4.22. Every finite-lobe opuntoid graph has a host.
Proof. The proof follows from the properties of finite trees, since the lobe graph of an opuntoid graph
is a tree, and is similar to that of [1, Lemma 6.1].
Lemma 4.23. LetΓbe an opuntoid graph. Then a host ofΓis a maximal parasite-free subopuntoid subgraph. IfΓhas more than one host, then every host is a lobe ofΓ. In addition, the unique reduced
lobe path between any two hosts consists entirely of lobes that are hosts.
Proof. The proof follows from the properties of finite trees and is similar to [1, Lemma 6.2].
Lemma 4.24. LetD be a finite-lobe opuntoid automaton with a hostΣ. IfD0is obtained fromD by Construction4.17thenΣis also a host ofD0.
Proof. We adopt the notation of Construction4.17. The automatonD is embedded intoD0, from
Lemma4.18. Letv00 denote its image inD0. From the proof of Lemma4.18, for any loopv00→yv00 in
∆j(v00)we havey≥ f inSj, for some f ∈E(U), where f labels a loop atv00 in∆j(v00). Thus∆j(v00)
feeds off∆i(v00). If∆i(v00)is a lobe ofΣthen, since∆j(v00)feeds off∆i(v00), it follows thatΣis a host
ofD0. Otherwise, the lobe∆i(v00)feeds off some lobe∆ofΣ. In which case, the lobe∆j(v00)also
feeds off∆and soΣis a host ofD0.
Lemma 4.25. LetΓbe a complete opuntoid graph with a hostΣ. LetΓ0be a subopuntoid subgraph of ΓcontainingΣand let v∈V(Γ0). Then the direct limitE of the directed system of all automata
Proof. Ifv1∈V(Γ0)is a bud of∆i(v1)inΓ0, fori∈ {1,2}, then we can apply Construction4.17 to(v,Γ0,v), resulting in an automaton(v0,Γ00,v0). Letv001denote the image ofv1inΓ00and put j=3−i. From Lemma4.18, the automaton(v,Γ0,v)is a subopuntoid subautomaton of(v0,Γ00,v0). From the
workings of Lemma4.18, the languageL[(v001,∆j(v001),v001)]consists of all wordsysuch thaty≥ f inSj, for some f ∈E(U)labeling a loopv1→f v1in∆i(v1).
SinceΓis complete, there is a lobe∆ofΓ, colored by j, containingv1. Since the lobe graph ofΓis a tree andΓ0contains the hostΣ, the lobe ∆feeds off ∆i(v1). SinceΓhas the equality
property, we then haveL[(v1,∆,v1)] =L[(v100,∆j(v001),v001)]. Since the lobes are deterministic, we have
(v1,∆,v1)∼= (v001,∆j(v001),v001). Thus(v0,Γ00,v0)is a subopuntoid subautomaton of(v,Γ,v).
Conversely, any lobe ofΓ\Γ0that is adjacent to a lobe ofΓ0must feed off the lobe ofΓ0and so, by
the equality property, must share a vertex that is a bud ofΓ0. It now follows thatE ∼= (v,Γ,v).
Theorem 4.26. Let U denote a lower bounded inverse subsemigroup of inverse semigroups S1and S2. Then the Sch¨utzenberger automata ofhS1∗US2iare complete opuntoid with a host.
Proof. Steps (i), (ii), (iii), (iv), (v) of Algorithm4.20result in a finite-lobe opuntoid automatonD. By
Lemma4.22, the automatonD has a hostΣ. Then, by step (vi), the direct limitE of the directed system
of all automata obtained fromD, by finite applications of Construction4.17, is complete opuntoid that
is closed, relative tohS1∗US2i.
The automatonD is embedded as a subopuntoid subautomaton into every automaton in the directed
system, by Lemma4.18. Let∆be a lobe ofE that is not a lobe ofD. There is some automatonD0in
the directed system such that∆is a lobe ofD0. By Lemma4.24, the subgraphΣis a host ofD0. Thus
∆feeds off some lobe ofΣand soΣis a host ofE. Hence every Sch¨utzenberger automaton ofS1∗US2
is complete opuntoid with a host.
Lemma 4.27. A host ofA(S1∗US2,w)having more than one lobe is the assimilated form of the
(underlying graph of the) direct limit of the directed system of all obtained fromA(S1∗S2,w0), by finite applications of Construction4.5, for some word w0. A host ofA(S1∗US2,w)with precisely one
lobe is isomorphic to a Sch¨utzenberger graph ofhS1iorhS2i.
Proof. From Corollary4.14, there exists a wordw0such thatw=w0inS1∗US2and the direct limit
applications of Construction4.5, has the separation property, whereA has at most as many lobes
asA(S1∗S2,w). PutA = (α1,Γ1,β1),B= (α2,Γ2,β2)and then letD= (α4,Γ4,β4)denote the assimilated form ofB.
Ifγ1,δ1∈V(Γ1)then we have(γ1,Γ1,δ1)∼=A(S1∗S2,w00), wherew00Dw0inS1∗US2. From Lemma4.7, the direct limit of the directed system of all automata obtained from(γ1,Γ1,δ1), by finite applications of Construction4.5, is isomorphic to(γ2,Γ2,δ2), lettingγ2andδ2denote the respective images ofγ1andδ1inΓ2. Thus, for the purpose of describing the hosts ofA(S1∗US2,w), we can assume thatα1andβ1are any vertices ofΓ1. From Lemma4.8, we can assume that the homomorphism fromA intoBinduces an isomorphism of the lobe trees. From Corollary4.11, we can
assume that no application of Construction4.9can be applied toB, for any choice ofα2andβ2. IfΓ4has a parasite then there is an intersectionvofΓ4such that∆i(v)is an extremal lobe, for some
i∈ {1,2}, and for any loopv→yvin∆
i(v), there exists a loopv→f vin∆i(v), where f ∈E(U)and
y≥ f inSi. Letv0denote the unique intersection ofB that is a preimage ofvand put j=3−i. Since
Bhas the equality property, for any loopv0→yv0in∆i(v0), there exists a loopv0→f v0in∆j(v0),
where f ∈E(U)andy≥ f inSi. Thus we can apply Construction4.9toB, a contradiction. HenceΓ4 is parasite-free. Then, from the proof of Theorem4.26, the graphΓ4is a host ofA(S1∗US2,w).
If Γ4 has precisely one lobe then Γ4∼=Γ2∼=Γ1∼=SΓ(Si,s), for some s∈Si and i∈ {1,2}. Suppose∆is a host ofA(S1∗US2,w)that is adjacent toΓ4. Letvbe an intersection common toΓ4 and∆. We have(v,Γ4,v)∼=A(Si,e), for somee∈E(Si)andi∈ {1,2}. SinceΓ4feeds off∆, we havee∈E(U)and(v,∆,v)∼=A(Sj,e), where j=3−i. The unique reduced lobe path between any
two hosts consists of lobes that feed off each other, by Lemma4.23. Thus ifA(S1∗US2,w)has more than one host then every host is isomorphic toSΓ(S1,f)orSΓ(S2,f), for some f ∈E(U).
We prove some results on homomorphisms of Sch¨utzenberger graphs.
Lemma 4.28. LetΓandΓ0be complete opuntoid graphs that have hosts and letΣbe a host ofΓ. Then every homomorphismΣ→Γ0extends (uniquely) to a homomorphismΓ→Γ0.
Proof. Letv∈V(Σ). The automaton(v,Γ,v)is the direct limit of the directed system of all automata
