pISSN 1225-6951 eISSN 0454-8124 c
Kyungpook Mathematical Journal
The Leavitt Path Algebras of Ultragraphs
Mostafa Imanfar and Abdolrasoul Pourabbas∗
Faculty of Mathematics and Computer Science, Amirkabir University of Technol- ogy, 424 Hafez Avenue, 15914 Tehran, Iran
e-mail : m.imanfar@aut.ac.ir and arpabbas@aut.ac.ir Hossein Larki
Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Iran
e-mail : h.larki@scu.ac.ir
Abstract. We introduce the Leavitt path algebras of ultragraphs and we characterize their ideal structures. We then use this notion to introduce and study the algebraic anal- ogy of Exel-Laca algebras.
1. Introduction
The Cuntz-Krieger algebras were introduced and studied in [6] for binary-valued matrices with finite index. Two immediate and important extensions of the Cuntz- Krieger algebras are: (1) the class of C∗-algebras associated to (directed) graphs [5, 8, 10, 11] and (2) the Exel-Laca algebras of infinite matrices with {0, 1}-entries [7]. It is shown in [8] that if E is a graph with no sinks and sources, then the C∗-algebra C∗(E) is canonically isomorphic to the Exel-Laca algebraOAE, where AEis the edge matrix of E. However, the class of graph C∗-algebras and Exel-Laca algebras are differ from each other.
To study both graph C∗-algebras and Exel-Laca algebras under one theory, Tomforde [15] introduced the notion of an ultragraph and its associated C∗-algebra.
Briefly, an ultragraph is a directed graph which allows the range of each edge to be a nonempty set of vertices rather than a singleton vertex. We see in [15] that for each binary-valued matrix A there exists an ultragraphGAso that the C∗-algebra ofGA
is isomorphic to the Exel-Laca algebra of A. Furthermore, every graph C∗-algebra can be considered as an ultragraph C∗-algebra, whereas there is an ultragraph C∗-
* Corresponding Author.
Received November 5, 2017; revised October 13, 2019; accepted November 18, 2019.
2010 Mathematics Subject Classification: 16W50, 46L55..
Key words and phrases: ultragraph C∗-algebra, Leavitt path algebra, Exel-Laca algebra.
21
algebra which is not a graph C∗-algebra nor an Exel-Laca algebra.
Recently many authors have discussed the algebraic versions of matrix and graph C∗-algebras. For example, in [3] the algebraic analogue of the Cuntz-Krieger algebra OA, denoted by CKA(K), was studied for finite matrix A, where K is a field. Also the Leavitt path algebra LK(E) of directed graph E was introduced in [1, 2] as the algebraic version of graph C∗-algebra C∗(E). The class of Leavitt path algebras includes naturally the algebras CKA(K) of [3] as well as the well-known Leavitt algebras L(1, n) of [14]. More recently, Tomforde defined a new version of Leavitt path algebras with coefficients in a unital commutative ring [18]. In the case R = C, the Leavitt path algebra LC(E) is isomorphic to a dense ∗-subalgebra of the graph C∗-algebra C∗(E) [17].
The purpose of this paper is to define the algebraic versions of ultragraphs C∗-algebras and Exel-Laca algebras. For an ultragraphG and unital commutative ring R, we define the Leavitt path algebra LR(G). To study the ideal structure of LR(G), we use the notion of quotient ultragraphs from [13]. Given an admissible pair (H, S) in G, we define the Leavitt path algebra LR(G/(H, S)) associated to the quotient ultragraph G/(H, S) and we prove two kinds of uniqueness theorems, namely the Cuntz-Krieger and the graded-uniqueness theorems, for LR(G/(H, S)).
Next we apply these uniqueness theorems to analyze the ideal structure of LR(G).
Although the construction of Leavitt path algebra of ultragraph will be similar to that of ordinary graph, we see in Sections 3 and 4 that the analysis of its structure is more complicated. The aim of the definition of ultragraph Leavitt path algebras can be summarized as follows:
• Every Leavitt path algebra of a directed graph can be embedded as an sub- algebra in a unital ultragraph Leavitt path algebra. Also, the ultragraph Leavitt path algebra LC(G) is isomorphic to a dense ∗-subalgebra of C∗(G).
• By using the definition of ultragraph Leavitt path algebras, we give an alge- braic version of Exel-Laca algebras.
• The class of ultragraph Leavitt path algebras is strictly larger than the class of Leavitt path algebras of directed graphs.
The article is organized as follows. We define in Section 2 the Leavitt path algebra LR(G) of an ultragraph G over a unital commutative ring R. We continue by considering the definition of quotient ultragraphs of [13]. For any admissible pair (H, S) in an ultragraphG, we associate the Leavitt path algebra LR G/(H, S)
to the quotient ultragraph G/(H, S) and we see that the Leavitt path algebras LR(G) and LR(G/(H, S) have a similar behavior in their structure. Next, we prove versions of the graded and Cuntz-Krieger uniqueness theorems for LR G/(H, S)
by approximating LR G/(H, S) with R-algebras of finite graphs.
By applying the graded-uniqueness theorem in Section 3, we give a complete description of basic graded ideals of LR(G) in terms of admissible pairs in G. In Section 4, we use the Cuntz-Krieger uniqueness theorem to show that an ultragraph G satisfies Condition (K) if and only if every basic ideal in LR(G) is graded.
In Section 5, we generalize the algebraic Cuntz-Krieger algebraCKA(K) of [3], denoted by ELA(R), for every infinite matrix A with entries in {0, 1} and every unital commutative ring R. In the case R = C, the Exel-Laca C-algebra ELA(C) is isomorphic to dense ∗-subalgebra ofOA. We prove that the class of Leavitt path algebras of ultragraphs contains the Leavitt path algebras as well as the algebraic Exel-Laca algebras. Furthermore, we give an ultragraph G such that the Leavitt path algebra LR(G) is neither a Leavitt path algebra of graph nor an Exel-Laca R-algebra.
2. Leavitt Path Algebras
In this section, we follow the standard constructions of [1] and [15] to define the Leavitt path algebra of an ultragraph. Since the quotient of ultragraph is not an ultragraph, we will have to define the Leavitt path algebras of quotient ultragraphs and prove the uniqueness theorems for them to characterize the ideal structure in Section 3.
2.1. Ultragraphs
Recall from [15] that an ultragraph G = (G0,G1, rG, sG) consists of a set of vertices G0, a set of edges G1, the source map sG : G1 → G0 and the range map rG:G1→P(G0)\{∅}, whereP(G0) is the collection of all subsets of G0. Throughout this work, ultragraphG will be assumed to be countable in the sense that both G0 andG1 are countable.
For a set X, a subcollection C of P(X) is said to be lattice if ∅ ∈ C and it is closed under the set operations ∩ and ∪. An algebra is a lattice C such that A \ B ∈ C for all A, B ∈ C. If G is an ultragraph, we write G0 for the algebra in P(G0) generated by{v}, rG(e) : v ∈ G0, e ∈G1 .
A path in ultragraphG is a sequence α = e1e2· · · en of edges with sG(ei+1) ∈ rG(ei) for 1 ≤ i ≤ n − 1 and we say that the path α has length |α| := n. We write Gn for the set of all paths of length n and Path(G) := S∞n=0Gn for the set of finite paths. We may extend the maps rG and sGon Path(G) by setting rG(α) := rG(en) and sG(α) := sG(e1) for |α| ≥ 1 and rG(A) = sG(A) := A for A ∈G0. For every edge e, We say that e∗ is the ghost edge associated to e. The following definition is the algebraic version of [15, Definition 2.7].
Definition 2.1. LetG be an ultragraph and let R be a unital commutative ring.
A Leavitt G-family in an R-algebra A is a set {pA, se, se∗ : A ∈ G0and e ∈ G1} of elements in A such that
(1) p∅= 0, pApB= pA∩B and pA∪B= pA+ pB− pA∩B for all A, B ∈G0; (2) psG(e)se= seprG(e)= seand prG(e)se∗ = se∗psG(e)= se∗ for all e ∈G1; (3) se∗sf = δe,fprG(e) for all e, f ∈G1;
(4) pv=P
sG(e)=vsese∗ for every vertex v with 0 < |s−1G (v)| < ∞,
where pv denotes p{v}. The R-algebra generated by the LeavittG-family {s, p} is denoted by LR(s, p).
We say that the Leavitt G-family {s, p} is universal, if B is an R-algebra and {S, P } is a Leavitt G-family in B, then there exists an algebra homomorphism φ : LR(s, p) → B such that φ(pA) = PA, φ(se) = Se and φ(se∗) = Se∗ for every A ∈ G0 and every e ∈ G1. The Leavitt path algebra of G with coefficients in R, denoted by LR(G), is the (unique up to isomorphism) R-algebra generated by a universal LeavittG-family.
2.2. Quotient Ultragraphs
We will use the notion of quotient ultragraphs and we generalize the definition of Leavitt path algebras for quotient ultragraphs.
Definition 2.2.([16, Definition 3.1]) Let G = (G0,G1, rG, sG) be an ultragraph. A subcollection H ⊆G0 is called hereditary if satisfying the following conditions:
(1) {sG(e)} ∈ H implies rG(e) ∈ H for all e ∈G1. (2) A ∪ B ∈ H for all A, B ∈ H.
(3) A ∈ H, B ∈G0and B ⊆ A, imply B ∈ H.
Also, H ⊆G0 is called saturated if for any v ∈ G0 with 0 < |s−1G (v)| < ∞, we have
rG(e) : e ∈G1 and sG(e) = v ⊆ H implies {v} ∈ H.
For a saturated hereditary subcollection H ⊆G0, the breaking vertices of H is denoted by
BH :=n
v ∈ G0: s−1G (v)
= ∞ but 0 <
s−1G (v) ∩ {e : rG(e) /∈ H}
< ∞o . An admissible pair in G is a pair (H, S) of a saturated hereditary set H ⊆ G0 and some S ⊆ BH.
In order to define the quotient of ultragraphs we need to recall and introduce some notations from [13, Section 3]. Let (H, S) be an admissible pair in G. For each A ∈ G0, we denote A := A ∪ {w0 : w ∈ A ∩ (BH \ S)}, where w0 denotes another copy of w. Consider the ultragraph G := (G0,G1, r, s), where G1 := G1, G0:= G0∪ {w0 : w ∈ BH\ S} and the maps r, s are defined by r(e) := rG(e) and
s(e) :=
sG(e)0 if sG(e) ∈ BH\ S and rG(e) ∈ H, sG(e) otherwise,
for every e ∈G1, respectively. We write G0 for the algebra generated by the sets {v}, {w0} and r(e).
Lemma 2.3.([13, Lemma 3.5]) Let (H, S) be an admissible pair in an ultragraphG and let ∼ be a relation onG0defined by A ∼ B if and only if there exists V ∈ H such that A ∪ V = B ∪ V . Then ∼ is an equivalence relation on G0 and the operations
[A] ∪ [B] := [A ∪ B], [A] ∩ [B] := [A ∩ B] and [A] \ [B] := [A \ B]
are well-defined on the equivalence classes {[A] : A ∈G0}.
We usually denote [v] instead of [{v}] for every v ∈ G0. For A, B ∈G0, we write [A] ⊆ [B] whenever [A] ∩ [B] = [A]. The set S
A∈HA is denoted byS H.
Definition 2.4.([13, Definition 3.6]) Let (H, S) be an admissible pair in G. The quotient ultragraph ofG by (H, S) is the quadruple G/(H, S) := (Φ(G0), Φ(G1), r, s), where
Φ(G0) :=[v] : v ∈ G0\[
H ∪ [w0] : w ∈ BH\ S , Φ(G1) :=e ∈G1: rG(e) /∈ H ,
and s : Φ(G1) → Φ(G0) and r : Φ(G1) → {[A] : A ∈ G0} are the maps defined by s(e) := [sG(e)] and r(e) := [rG(e)] for every e ∈ Φ(G1), respectively.
Lemma 2.5. IfG/(H, S) is a quotient ultragraph, then
Φ(G0) =
k [
j=1 nj
\
i=1
Ai,j\ Bi,j: Ai,j, Bi,j∈ Φ(G0) ∪r(e) : e ∈ Φ(G1)
,
where Φ(G0) is the smallest algebra in {[A] : A ∈G0} containing n
[v], [w0] : v ∈ G0\[
H, w ∈ BH\ So
∪n
r(e) : e ∈ Φ(G1)o .
Proof. We denote by X the right hand side of the above equality. It is clear that X ⊆ Φ(G0), because Φ(G0) is an algebra generated by the elements [v],[w0] and r(e).
For the reverse inclusion, we note that X is a lattice. Furthermore, one can show that X is closed under the operation \. Thus X is an algebra contains [v],[w0] and
r(e) and consequently Φ(G0) ⊆ X. 2
Remark 2.6. If A, B ∈G0, then A ∪ B = A∪B, A ∩ B = A∩B and A \ B = A\B.
Thus, by applying an analogous lemma of Lemma 2.5 forG0andG0, we deduce that A ∈G0 for all A ∈G0. One can see that
G0=A ∪ F1∪ F2: A ∈G0, F1and F2 are finite subsets of G0
and {w0 : w ∈ BH}, respectively . For example we have
A \ {v} = A \ {v} ∪
{v} \ {v} ∩ A \ A ,
and
A \ {w0} = A \ {w} ∪
A ∩ {w} .
Furthermore, it follows from Lemma 2.3 that Φ(G0) =[A] : A ∈G0 .
Remark 2.7. The hereditary property of H and Remark 2.6 imply that A ∼ B if and only if both A \ B and B \ A belong to H.
Similar to ultragraphs, a path inG/(H, S) is a sequence α := e1e2· · · enof edges in Φ(G1) such that s(ei+1) ⊆ r(ei) for 1 ≤ i ≤ n − 1. We say the path α has length |α| := n and we consider the elements in Φ(G0) to be paths of length zero.
We denote by Path(G/(H, S)), the union of paths with finite length. The maps r and s can be naturally extended on Path(G/(H, S)). Let Φ(G1)∗ be the set of ghost edges {e∗ : e ∈ Φ(G1)}. We also define the ghost path α∗:= e∗ne∗n−1· · · e∗1 for every α = e1e2· · · en ∈ Path(G/(H, S)) \ Φ(G0) and [A]∗:= [A] for every [A] ∈ Φ(G0).
Using Theorem 3.4, we define the Leavitt path algebra of a quotient ultragraph G/(H, S) which is corresponding to the quotient R-algebra LR(G)/I(H,S). We use this concept to characterize the ideal structure of LR(G) in Section 3. The following definition is the algebraic version of [13, Definition 3.8].
Definition 2.8. Let G/(H, S) be a quotient ultragraph and let R be a unital commutative ring. A LeavittG/(H, S)-family in an R-algebra A is a set {q[A], te, te∗: [A] ∈ Φ(G0) and e ∈ Φ(G1)} of elements in A such that
(1) q[∅]= 0, q[A]q[B]= q[A]∩[B] and q[A]∪[B]= q[A]+ q[B]− q[A]∩[B]; (2) qs(e)te= teqr(e)= te and qr(e)te∗= te∗qs(e)= te∗;
(3) te∗tf = δe,fqr(e); (4) q[v]=P
s(e)=[v]tete∗ for every [v] ∈ Φ(G0) with 0 < |s−1([v])| < ∞.
The R-algebra generated by the Leavitt G/(H, S)-family {t, q} is denoted by LR(t, q). The Leavitt path algebra of G/(H, S) with coefficients in R, denoted by LR(G/(H, S)), is the (unique up to isomorphism) R-algebra generated by a univer- sal Leavitt G/(H, S)-family (the definition of universal Leavitt G/(H, S)-family is similar to ultragraph case).
If (H, S) = (∅, ∅), then we have [A] = {A} for each A ∈ Φ(G0). In this case, every LeavittG/(∅, ∅)-family is a Leavitt G-family and vice versa. So, we can consider the ultragraph Leavitt path algebra LR(G) as LR(G/(∅, ∅)).
Let R be a unital commutative ring. For a nonempty set X, we write w(X) for the set of words w := w1w2· · · wn from the alphabet X. The free R-algebra generated by X is denoted by FR(w(X)). For definition of the free R-algebra we refer the reader to [4, Section 2.3].
Now, we show that for every quotient ultragraph G/(H, S), there exists a uni- versal LeavittG/(H, S)-family. Suppose that X := Φ(G0) ∪ Φ(G1) ∪ Φ(G1)∗ and I is the ideal of the free R-algebra FR(w(X)) generated by the union of the following sets:
(1) [∅], [A][B] − [A ∩ B], [A ∪ B] + [A ∩ B] − ([A] + [B]) : [A], [B] ∈ Φ(G0) , (2) e − s(e)e, e − er(e), e∗− e∗s(e), e∗− r(e)e∗: e ∈ Φ(G1) ,
(3) e∗f − δe,fr(e) : e, f ∈ Φ(G1) ,
(4) v − Ps(e)=[v]ee∗: 0 < |s−1([v])| < ∞ .
If π : FR(w(X)) → FR(w(X))/I is the quotient map, then it is easy to check that the collection {π([A]), π(e), π(e∗) : [A] ∈ Φ(G0), e ∈ Φ(G1)} is a Leavitt G/(H, S)- family. For our convenience, we denote q[A]:= π(A), te:= π(e) and te∗ := π(e∗) for every [A] ∈ Φ(G0) and e ∈ Φ(G1), and we show that the LeavittG-family {t, q} has the universal property.
Assume that {T, Q} is a LeavittG/(H, S)-family in an R-algebra A. If we define f : X → A by f ([A]) = P[A], f (e) = Te and f (e∗) = Te∗, then by [4, Proposition 2.6], there is an R-algebra homomorphism φf: FR(w(X)) → A such that φf
X= f . Since {T, Q} is a LeavittG/(H, S)-family, φf vanishes on the generator of I. Hence, we can define an R-algebra homomorphism φ : FR(w(X))/I → A by φ π(x) = φf(x) for every x ∈ FR(w(X)). Furthermore, we have φ(q[A]) = Q[A], φ(te) = Te
and φ(te∗) = Te∗.
From now on we denote the universal LeavittG-family and G/(H, S)-family by {s, p} and {t, q}, respectively. So, we suppose that {s, p} and {t, q} are the canonical generators of LR(G) and LR(G/(H, S)), respectively.
Theorem 2.9. LetG/(H, S) be a quotient ultragraph and let R be a unital commu- tative ring. Then LR(G/(H, S)) is of the form
spanRtαq[A]tβ∗∈ LR(G/(H, S)) : α, β ∈ Path G/(H, S), r(α) ∩ [A] ∩ r(β) 6= [∅] . Furthermore, LR(G/(H, S)) is a Z-graded ring by the grading
LR(G/(H, S))n= spanRtαq[A]tβ∗∈ LR(G/(H, S)) : |α| − |β| = n (n ∈ Z).
Proof. If tαq[A]tβ∗, tµq[B]tν∗∈ LR(G/(H, S)), then
(tαq[A]tβ∗)(tµq[B]tν∗) =
tαµ0q[B]tν∗ if µ = βµ0 and s(µ0) ⊆ A ∩ r(α), tαq[A]∩r(β)∩[B]tν∗ if µ = β,
tαq[A]t(νβ0)∗ if β = µβ0 and s(β0) ⊆ B ∩ r(ν),
0 otherwise,
which proves the first statement. Let X := Φ(G0) ∪ {e, e∗: e ∈ Φ(G1)} and consider LR(G/(H, S)) = FR(w(X))/I as defined before. If we define a degree map d : X → Z by d([A]) = 0, d(e) = 1 and d(e∗) = −1 for every [A] ∈ Φ(G0) and e ∈ Φ(G1), then by [4, Proposition 2.7], LR(G/(H, S)) is a Z-graded ring with the grading
LR(G/(H, S))n= spanRtαq[A]tβ∗∈ LR(G/(H, S)) : |α| − |β| = n . 2
Theorem 2.10. Let G be an ultragraph and let R be a unital commutative ring.
Then for all nonempty sets A ∈G0 and all r ∈ R \ {0}, the elements rpA of LR(G) are nonzero. In particular, rse6= 0 and rse∗ 6= 0 for every A ∈G0\ {∅} and every r ∈ R \ {0}.
Proof. By the universality, it suffices to generate a LeavittG-family {˜s, ˜p} in which r ˜pA 6= 0 for every nonempty set A ∈G0 and every r ∈ R \ {0}. For each e ∈G1, we define a disjoint copy Ze:=L R, where the direct sum is taken over countably many copies of R and for each v ∈ G0, let
Zv:=
L
s(e)=v
Ze if |s−1G (v)| 6= 0,
L R if |s−1G (v)| = 0,
where L R is another disjoint copy for each v. For every ∅ 6= A ∈ G0, define PA :L
v∈AZv → L
v∈AZv to be the identity map. Also, for each e ∈ G1 choose an isomorphism Se : L
v∈rG(e)Zv → Ze and let Se∗ := S−1e : Ze → L
v∈rG(e)Zv. Now if Z :=L
v∈G0Zv, then we naturally extend all PA, Se, Se∗to homomorphisms
˜
pA, ˜se, ˜se∗∈ HomR(Z, Z), respectively by setting the yet undefined components to zero. It is straightforward to verify that {˜s, ˜p} is a LeavittG-family in HomR(Z, Z) such that r ˜pA6= 0 for every ∅ 6= A ∈G0 and every r ∈ R \ {0}. 2 Note that we cannot follow the argument of Theorem 2.10 to show that rq[A]6=
0. For example, suppose thatG is the ultragraph
e v1
e v2
e v3 e
. . .
v0
and let H be the collection of all finite subsets of {v1, v2, . . .}, which is a hereditary and saturated subcollection ofG0. If we consider the quotient ultragraphG/(H, ∅), then {[∅] 6= [v] : [v] ⊆ r(e)} = ∅. So we can not define the idempotent qr(e) : L
[v]⊆r(e)Z[v] → L
[v]⊆r(e)Z[v] as in the proof of Theorem 2.10. In Section 3 we will solve this problem.
Remark 2.11. Every directed graph E = (E0, E1, r, s) can be considered as an ultragraphG = (G0,G1, rG, sG), where G0:= E0,G1:= E1 and the map rG:G1→ P(G0) \ {∅} is defined by rG(e) = {r(e)} for every e ∈G1. In this case, the algebra G0 is the collection of all finite subsets of G0. The Leavitt path algebra LR(E) is naturally isomorphic to LR(G) (see [1, 18] for more details about the Leavitt path algebras of directed graphs). So the class of ultragraph Leavitt path algebras contains the class of Leavitt path algebras of directed graphs.
Lemma 2.12. LetG be an ultragraph and let R be a unital commutative ring. Then LR(G) is unital if and only if G0∈G0 and in this case 1LR(G)= pG0.
Proof. If G0∈G0, then the Relations of Definition 2.1 imply that pG0 is a unit for LR(G).
Conversely, suppose that LR(G) is unital and write
1LR(G) =
n
X
k=1
rksαkpAksβ∗
k,
where rk ∈ R, Ak ∈ G0 and αk, βk ∈ Path(G). Let A := Snk=1s(αk) ∈ G0. If G0∈/G0, then we can choose an element v ∈ G0\ A and derive
pv= pv· 1LR(G)=
n
X
k=1
rkpvsαkpAksβ∗
k =
n
X
k=1
rkpvpsG(αk)sαkpAksβ∗
k= 0, this contradicts Theorem 2.10 and it follows G0∈G0, as desired. 2
We note that, for directed graph E, the algebra LR(E) is unital if and only if E0 is finite. If E0 is infinite, then we can define an ultragraph G associated to E such that LR(G) is unital and LR(E) is an embedding subalgebra of LR(G). More precisely, Consider the ultragraphG = (G0,G1, rG, sG) where G0= E0∪ {v}, G1= E1∪ {e}, sG(e) = v and rG(e) = E0. Since rG(e) ∪ {v} = G0∈G0, by Lemma 2.12, LR(G) is unital. Define Se= seand Pv = pv for e ∈ E1 and v ∈ E0, respectively.
It is straightforward to see that {S, P } is a Leavitt E-family for directed graph E.
The result now follows by [18, Theorem 5.3].
2.3. Uniqueness Theorems
LetG/(H, S) be a quotient ultragraph. We prove the graded and Cuntz-Krieger uniqueness theorems for LR(G/(H, S)) and LR(G). We do this by approximating the Leavitt path algebras of quotient ultragraphs with the Leavitt path algebras of finite graphs. Our proof in this section is standard (see [13, Section 4]), and we give the details for simplicity of further results of the paper.
A vertex [v] ∈ Φ(G0) is called a sink if s−1([v]) = ∅ and [v] is called an infinite emitter if |s−1([v])| = ∞. A singular vertex is a vertex that is either a sink or an infinite emitter. The set of all singular vertices is denoted by Φsg(G0).
Let F ⊆ Φsg(G0) ∪ Φ(G1) be a finite subset and write F0 := F ∩ Φsg(G0) and F1 := F ∩ Φ(G1) = {e1, . . . , en}. Following [13], we construct a finite graph EF
as follows. First, for every ω = (ω1, . . . , ωn) ∈ {0, 1}n\ {0n}, we define r(ω) :=
T
ωi=1r(ei) \S
ωj=0r(ej) and R(ω) := r(ω) \S
[v]∈F0[v] which belong to Φ(G0). We also set
Γ0:= {ω ∈ {0, 1}n\ {0n} : there are vertices [v1], . . . , [vm] such that R(ω) =Sm
i=1[vi] and ∅ 6= s−1([vi]) ⊆ F1for 1 ≤ i ≤ m}, and
ΓF := {ω ∈ {0, 1}n\ {0n} : R(ω) 6= [∅] and ω /∈ Γ0} .
Now we define the finite graph EF = (EF0, EF1, rF, sF), where EF0 :=F0∪ F1∪ ΓF,
EF1 :=(e, f ) ∈ F1× F1: s(f ) ⊆ r(e)
∪(e, [v]) ∈ F1× F0: [v] ⊆ r(e)
∪(e, ω) ∈ F1× ΓF : ωi= 1 whenever e = ei ,
with sF(e, f ) = sF(e, [v]) = sF(e, ω) = e and rF(e, f ) = f , rF(e, [v]) = [v], rF(e, ω) = ω.
Lemma 2.13. LetG/(H, S) be a quotient ultragraph and let R be a unital commu- tative ring. Then we have the following assertion:
(i) For every finite set F ⊆ Φsg(G0) ∪ Φ(G1), the elements Pe:= tete∗, P[v]:= q[v](1 − P
e∈F1
tete∗), Pω:= qR(ω)(1 − P
e∈F1
tete∗), S(e,f ):= tePf, S(e,[v]):= teP[v], S(e,ω) := tePω,
S(e,f )∗:= Pfte∗, S(e,[v])∗:= P[v]te∗, S(e,ω)∗:= Pωte∗,
form a Leavitt EF-family which generates the subalgebra of LR(G/(H, S)) generated by q[v], te, te∗: [v] ∈ F0, e ∈ F1 .
(ii) For r ∈ R \ {0}, if rq[A]6= 0 for all [A] 6= [∅] in Φ(G0), then rPz6= 0 for all z ∈ EF0. In this case, we have
LR(EF) ∼= LR(S, P ) = Alg{q[v], te, te∗∈ LR(G/(H, S)) : [v] ∈ F0, e ∈ F1}.
Proof. The statement (i) follows from the fact that {t, q} is a Leavitt G/(H, S)- family, or see the similar [13, Proposition 4.2].
For (ii), fix r ∈ R \ {0}. If rq[A]6= 0 for every [A] ∈ Φ(G0) \ {[∅]}, then rte6= 0 and rte∗6= 0 for every edge e. Thus rPe6= 0 for every edge e ∈ F1. Let [v] ∈ F0. If [v] is a sink, then rP[v] = rq[v] 6= 0. If [v] is an infinite emitter, then there is f ∈ Φ(G0)\F1such that s(f ) = [v]. In this case, we have rP[v]tf = rq[v]tf = tf 6= 0.
Therefore rP[v] 6= 0 for all [v] ∈ F0. Moreover, for each ω ∈ ΓF, there is a vertex [v] ⊆ R(ω) such that either [v] is a sink or there is an edge f ∈ Φ(G1) \ F1 with s(f ) = [v]. In the former case, we have q[v](rPω) = rq[v] 6= 0 and in the later, tf∗(rPω) = rtf∗ 6= 0. Thus all rPω are nonzero. Consequently, rPz 6= 0 for every vertex z ∈ EF0.
Now we show that LR(EF) ∼= LR(S, P ). Note that for z ∈ EF0 and g ∈ EF1, the degree of Pz, Sg and Sg∗ as elements in LR(G/(H, S)) are 0, 1 and -1, respectively.
So LR(S, P ) is a graded subalgebra of LR(G/(H, S)) with the grading LR(S, P )n:= LR(S, P ) ∩ LR(G/(H, S))n.
Let {˜s, ˜p} be the canonical generators of LR(EF). By the universal property, there is an R-algebra homomorphism π : LR(EF) → LR(S, P ) such that π(r ˜pz) = rPz6= 0,
π(˜sg) = Sgand π(˜sg∗) = Sg∗for z ∈ EF0, g ∈ EF1 and r ∈ R \ {0}. Since π preserves the degree of generators, the graded-uniqueness theorem for graphs [18, Theorem 5.3] implies that π is injective. As π is also surjective, we conclude that LR(EF) is
isomorphic to LR(S, P ) as R-algebras. 2
Theorem 2.14.(The graded-uniqueness theorem) Let G/(H, S) be a quotient ul- tragraph and let R be a unital commutative ring. If T is a Z-graded ring and π : LR(G/(H, S)) → T is a graded ring homomorphism with π(rq[A]) 6= 0 for all [A] 6= [∅] in Φ(G0) and r ∈ R \ {0}, then π is injective.
Proof. Let {Fn} be an increasing sequence of finite subsets of Φsg(G0) ∪ Φ(G1) such that ∪∞n=1Fn = Φsg(G0) ∪ Φ(G1). For each n, the graded subalgebra of LR(G/(H, S)) generated by {q[v], te, te∗ : [v] ∈ Fn0and e ∈ Fn1} is denoted by Xn. Since π(rq[A]) 6= 0 for all [A] ∈ Φ(G0) \ {[∅]} and r ∈ R \ {0}, by Lemma 2.13, there is an graded isomorphism πn : LR(EFn) → Xn. Thus π ◦ πn : LR(EFn) → T is a graded homomorphism.
Fix r ∈ R \ {0}. We show that π ◦ πn(r ˜pz) = π(rPz) 6= 0 for all z ∈ E0F
n. Since π(rq[A]) 6= 0 for all [A] ∈ Φ(G0) \ {[∅]}, the elements π(rte) and π(rte∗) are nonzero for every edge e. So π(rPete) = π(rte) 6= 0 and thus π(rPe) 6= 0.
Let [v] ∈ F0. If [v] is a sink, then π(rP[v]) = π(rq[v]) 6= 0. If [v] is an infinite emitter, then there is f ∈ Φ(G0) \ F1 such that s(f ) = [v]. In this case, we have π(rP[v]tf) = π(rq[v]tf) = π(tf) 6= 0, hence π(rP[v]) 6= 0. For every ω ∈ ΓF, there is a vertex [v] ⊆ R(ω) such that either [v] is a sink or there is an edge f ∈ Φ(G1) \ F1 with s(f ) = [v]. In the former case, we have π(q[v](rPω)) = π(rq[v]) 6= 0 and in the later, π(tf∗(rPω)) = π(rtf∗) 6= 0. Thus π ◦ πn(r ˜pz) 6= 0 for every z ∈ EF0
n and r ∈ R \ {0}. Hence, we may apply the graded-uniqueness theorem for graphs [18, Theorem 5.3] to obtain the injectivity of π ◦ πn.
If [v] is a non-singular vertex, then we have q[v] = P
s(e)=[v]tete∗. Further- more, q[A]\[B] = q[A]− q[A]q[B] for every [A], [B] ∈ Φ(G0). Thus, by Lemma 2.5, LR(G/(H, S)) is an R-algebra generated by
q[v], te, te∗: [v] ∈ Φsg(G0) and e ∈ Φ(G1) ,
and so ∪∞n=1Xn = LR(G/(H, S)). It follows that π is injective on LR(G/(H, S)), as
desired. 2
Corollary 2.15. Let G be an ultragraph, R a unital commutative ring and T a Z- graded ring. If π : LR(G) → T is a graded ring homomorphism such that π(rpA) 6= 0 for all A ∈G0\ {∅} and r ∈ R \ {0}, then π is injective.
Definition 2.16. A loop in a quotient ultragraphG/(H, S) is a path α with |α| ≥ 1 and s(α) ⊆ r(α). An exit for a loop α1· · · αnis an edge f ∈ Φ(G1) with the property that s(f ) ⊆ r(αi) but f 6= αi+1 for some 1 ≤ i ≤ n, where αn+1 := α1. We say that G/(H, S) satisfies Condition (L) if every loop α := α1· · · αn in G/(H, S) has an exit, or r(αi) 6= s(αi+1) for some 1 ≤ i ≤ n.
Theorem 2.17.(The Cuntz-Krieger uniqueness theorem) LetG/(H, S) be a quotient ultragraph satisfying Condition (L) and let R be a unital commutative ring. If T is
a ring and π : LR(G/(H, S)) → T is a ring homomorphism such that π(rq[A]) 6= 0 for every [A] ∈ Φ(G0) \ {[∅]} and r ∈ R \ {0}, then π is injective.
Proof. Choose an increasing sequence {Fn} of finite subsets of Φsg(G0) ∪ Φ(G1) such that ∪∞n=1Fn = Φsg(G0) ∪ Φ(G1). Let Xn be the subalgebras of LR(G/(H, S)) as in Theorem 2.14. Since π(rq[A]) 6= 0 for all [A] ∈ Φ(G0) \ {[∅]} and r ∈ R \ {0}, by Lemma 2.13, there exists an isomorphism πn : LR(EFn) → Xn for each n ∈ N.
Furthermore, π ◦ πn(r ˜pz) 6= 0 for every z ∈ EF0n and r ∈ R \ {0}. SinceG/(H, S) satisfies Condition (L), By [13, Lemma 4.8], all finite graphs EFn satisfy Condition (L). So, the Cuntz-Krieger uniqueness theorem for graphs [18, Theorem 6.5] implies that π ◦ πn is injective for n ≥ 1. Now by the fact ∪∞n=1Xn = LR(G/(H, S)), we
conclude that π is an injective homomorphism. 2
Corollary 2.18. LetG be an ultragraph satisfying Condition (L), R a unital com- mutative ring and T a ring. If π : LR(G) → T is a ring homomorphism such that π(rpA) 6= 0 for all A ∈G0\ {∅} and r ∈ R \ {0}, then π is injective.
3. Basic Graded Ideals
In this section, we apply the graded-uniqueness theorem for quotient ultragraphs to investigate the ideal structure of LR(G). We would like to consider the ideals of LR(G) that are reflected in the structure of the ultragraph G. For this, we give the following definition of basic ideals.
Let (H, S) be an admissible pair in an ultragraphG. For any w ∈ BH, we define the gap idempotent
pHw := pw− X
s(e)=w, r(e) /∈H
sese∗.
Let I be an ideal in LR(G). We write HI := {A ∈G0 : pA ∈ I}, which is a saturated hereditary subcollection ofG0. Also, we set SI := {w ∈ BHI : pHwI ∈ I}.
We say that the ideal I is basic if the following conditions hold:
(1) rpA∈ I implies pA∈ I for A ∈G0and r ∈ R \ {0}.
(2) rpHwI ∈ I implies pHwI ∈ I for w ∈ BHI and r ∈ R \ {0}.
For an admissible pair (H, S) inG, the (two-sided) ideal of LR(G) generated by the idempotents {pA: A ∈ H} ∪pHw : w ∈ S is denoted by I(H,S).
Lemma 3.1.(cf. [12, Lemma 3.9]) If (H, S) is an admissible pair in ultragraphG, then
I(H,S):= spanRsαpAsβ∗, sµpHwsν∗∈ LR(G) : A ∈ H and w ∈ S and I(H,S)is a graded basic ideal of LR(G).
Proof. We denote the right-hand side of the above equality by J . The hereditary property of H implies that J is an ideal of LR(G) being contained in I(H,S). On the other hand, all generators of I(H,S) belong to J and so we have I(H,S) = J .
Note that the elements sαpAsβ∗ and sµpHwsν∗are homogeneous elements of degrees
|α| − |β| and |µ| − |ν|, respectively. Thus I(H,S)is a graded ideal.
To show that I(H,S) is a basic ideal, suppose rpA∈ I(H,S)for some A ∈G0and r ∈ R \ {0} and write
rpA= x :=
n
X
i=1
risαipAisβ∗
i +
m
X
j=1
sjsµjpHw
jsν∗
j,
where Ai∈ H, wj ∈ S and ri, sj∈ R for all i, j. We first show assertion (1) of the definition of basic ideal in several steps.
Step I: If A = {v} /∈ H and v /∈ S, then 0 < |s−1G (v)| < ∞.
Note that pvx = x, so the assumption v /∈ S H ∪ S and the hereditarity of H imply that we may assume that |αi|, |µj| ≥ 1 and sG(αi) = sG(µj) = v for every i, j (because a sum like P riAi could be zero). Hence v is not a sink. Set αi = αi,1αi,2· · · αi,|α|. If |s−1G (v)| = ∞, then there is an edge e 6= α1,1, . . . , αn,1, µ1,1, . . . , µm,1 with sG(e) = v. So rse∗ = se∗(rpv) = se∗x = 0, contradicting Theorem 2.3. Thus 0 < |s−1G (v)| < ∞.
Step II: If A /∈ H, then there exists v ∈ A such that {v} /∈ H.
Let rpA = x. Assume first that |µj| = 0 for some j. Then, since rpA = pArpA = pAx = x, wj ∈ A. As wj ∈ BH we deduce that {wj} /∈ H. So let |µj| ≥ 1 for all j. If |αi| = 0 for some i then we let sG(αi) = Ai. In this case, we have A ⊆ ∪isG(αi) ∪jsG(µj) (if v ∈ A \ ∪isG(αi) ∪jsG(µj), then rpv= rpvpA= pvx = 0, a contradiction). Thus A = ∪i sG(αi) ∩ A ∪ ∪j sG(µj) ∩ A. Suppose that
{v} : v ∈ A ⊆ H. Since H is hereditary, A ∈ H, which is impossible. Hence there is a vertex v ∈ A such that {v} /∈ H.
Step III: If A = {v}, then {v} ∈ H.
We go toward a contradiction and assume {v} /∈ H. Set v1 := v. If v1 ∈ BH, then there is an edge e1∈G1 such that sG(e1) = v1 and rG(e1) /∈ H. If v1 ∈ B/ H, we have 0 < |s−1G (v)| < ∞ by Step I. The saturation of H gives an edge e1 with sG(e1) = v1 and rG(e1) /∈ H. By Step II, there is a vertex v2 ∈ rG(e1) such that {v2} /∈ H. We may repeat the argument to choose a path γ = γ1. . . γL
for L = maxi,j|βi|, |νj| + 1, such that sG(γ) = v and sG(γk), rG(γk) /∈ H for all 1 ≤ k ≤ L. Note that sG(γk) /∈ Aiand so pAipsG(γk)= 0 for all i, k. Moreover, since r(γk) /∈ H we have pHwjsγk = 0 for all j, k. It follows that rsγ = (rpv)sγ = xsγ = 0, a contradiction. Therefore {v} ∈ H.
Step IV: If rpA∈ I(H,S), then A ∈ H.
If A /∈ H, then by Step II there is a vertex v ∈ A such that {v} /∈ H, which contradicts the Step III. Hence A ∈ H and consequently pA∈ I(H,S), as desired.
Now, we show that I(H,S)satisfies assertion (2) of the definition of basic ideal.
Note that, by Step IV, we have HI(H,S) = H. Let w ∈ BH, r ∈ R \ {0} and
rpHw ∈ I(H,S). Using the first part of the lemma, write
rpHw = x :=
n
X
i=1
risαipAisβ∗i +
m
X
j=1
sjsµjpHwjsνj∗,
where Ai ∈ H, wj ∈ S and ri, sj ∈ R for all i, j. Since rpHw = pHwrpHw = pHwx = x and pHwAi = 0, We may assume that |αi| ≥ 1 for all i. As w ∈ BH, we can choose an edge e 6= α1,1, . . . , αn,1, µ1,1, . . . , µm,1 such that sG(e) = w and rG(e) ∈ H. If w /∈ S, then rse∗= se∗(rpHw) = se∗x = 0 which is a contradiction. Therefore w ∈ S
and pHw ∈ I(H,S). 2
Remark 3.2. Let (H, S) be an admissible pair in ultragraphG and let r ∈ R \ {0}.
The argument of Lemma 3.1 implies that
H =A ∈G0: rpA∈ I(H,S) and S = w ∈ BH: rpHw ∈ I(H,S) .
Lemma 3.3.(cf. [13, Proposition 3.3]) LetG be an ultragraph and let R be a unital commutative ring. If (H, S) is an admissible pair in G, then LR(G) ∼= LR(G).
Proof. Suppose that { ˜S, ˜P } is a universal LeavittG-family. If we define
(3.1)
PA:= ˜PA for A ∈G0, Se:= ˜Se for e ∈G1, Se∗:= ˜Se∗ for e ∈G1,
then it is straightforward to see that {S, P } is a LeavittG-family in LR(G). Note that LR(S, P ) inherits the graded structure of LR(G). Since rPA6= 0 for all A ∈G0\ {∅}
and r ∈ R \ {0}, by Corollary 2.15, LR(G) ∼= LR(S, P ).
We show that LR(S, P ) = LR(G). For A ∈ G0, ˜PA = PA ∈ LR(S, P ). If v /∈ BH\ S, then ˜P{v}= ˜Pv = Pv∈ LR(S, P ). We note that
s(e) =
sG(e)0 if sG(e) ∈ BH\ S and rG(e) ∈ H, sG(e) otherwise,
for every e ∈G1. Thus for w ∈ BH\ S, we have 0 < |s−1(w)| < ∞ and rG(e) /∈ H for e ∈ s−1(w). Hence
P˜w= X
s(e)=w
S˜eS˜e∗= X
sG(e)=w, rG(e) /∈H
SeSe∗ ∈ LR(S, P ).
Also,
P˜w0 = ˜P{w,w0}− ˜Pw= Pw− X
sG(e)=w, rG(e) /∈H
SeSe∗= PwH∈ LR(S, P ).
Thus, by Remark 2.6, ˜PA ∈ LR(S, P ) for all A ∈G. Since ˜Se, ˜Se∗ ∈ LR(S, P ), we deduce that LR(S, P ) = LR(G). Consequently, LR(G) ∼= LR(G). 2
Theorem 3.4.(cf. [12, Theorem 3.10]) LetG be an ultragraph and let R be a unital commutative ring. Then
(1) For any admissible pair (H, S) inG, we have LR(G/(H, S)) ∼= LR(G)/I(H,S). (2) The map (H, S) 7→ I(H,S)is a bijection from the set of all admissible pairs of
G to the set of all graded basic ideals of LR(G).
Proof. (1) Let { ˜S, ˜P } be a universal Leavitt G-family and let LR(G) = LR(S, P ), where {S, P } is the LeavittG-family as defined in 3.1. Define
(3.2)
Q[A]:= ˜PA+ I(H,S) for A ∈G0, Te:= ˜Se+ I(H,S) for e ∈ Φ(G1), Te∗:= ˜Se∗+ I(H,S) for e ∈ Φ(G1).
It can be shown that the family {Q[A], Te, Te∗ : [A] ∈ Φ(G0), e ∈ Φ(G1)} is a Leavitt G/(H, S)-family that generates LR(G)/I(H,S). Furthermore, by Remark 3.2, rQ[A]6=
0 for all [A] ∈ Φ(G0) \ {[∅]} and r ∈ R \ {0}.
Now we use the universal property of LR(G/(H, S)) to get an R-homomorphism π : LR(G/(H, S)) → LR(G)/I(H,S)such that π(te) = Te, π(te∗) = Te∗and π(rq[A]) = rQ[A] 6= 0 for [A] ∈ Φ(G0) \ {[∅]}, e ∈ Φ(G1) and r ∈ R \ {0}. Since I(H,S) is a graded ideal, the quotient LR(G)/I(H,S) is graded. Moreover, the elements Q[A], Te and Te∗ have degrees 0, 1 and -1 in LR(G)/I(H,S), respectively and thus π is a graded homomorphism. It follows from the graded-uniqueness Theorem 2.14 that π is injective. Since LR(G)/I(H,S) is generated by { Q[A], Te, Te∗ : [A] ∈ Φ(G0), e ∈ Φ(G1)}, we deduce that π is also surjective. Hence LR(G/(H, S)) ∼= LR(G)/I(H,S)∼= LR(G)/I(H,S).
(2) The injectivity of the map (H, S) 7→ I(H,S) is a consequence of Remark 3.2.
To see that it is onto, let I be a graded basic ideal in LR(G). Then I(HI,SI)⊆ I.
Consider the ultragraph G with respect to admissible pair (HI, SI). Since I is a graded ideal, the quotient ring LR(G)/I is graded. Let π : LR(G/(HI, SI)) ∼= LR(G)/I(HI,SI)→ LR(G)/I be the quotient map. For (HI, SI), consider { ˜S, ˜P } and {T, Q} as defined in Equations 3.1 and 3.2, respectively. Since I is basic, we have rpA∈ I and rp/ HwI ∈ I for A ∈/ G0\ HI, w ∈ BHI\ SI and r ∈ R \ {0}.
We show that π(rq[A]) = π(rQ[A]) = r ˜PA+ I 6= 0 for all [A] ∈ Φ(G0) \ {[∅]} and r ∈ R \ {0}. Fix r ∈ R \ {0}. We know that Φ(G0) = [A] : A ∈G0 . Let A = B for some B ∈ G0. If [A] 6= [∅], then B /∈ HI and thus r ˜PA = rpB ∈ I. Therefore/ π(rq[A]) 6= 0. If w ∈ BHI \ SI, then r ˜Pw0 = rpHwI ∈ I. Hence π(rq/ [w0]) 6= 0. In view of Remark 2.10, we deduce that π(rq[A]) 6= 0 for every [A] ∈ Φ(G0) \ {[∅]}.
Furthermore, π is a graded homomorphism. It follows from Theorem 2.14 that the quotient map π is injective. Hence I = I(HI,SI). 2 As we have seen in the proof of Theorem 3.4, if (H, S) is an admissible pair inG, then there is a LeavittG/(H, S)-family {T, Q} in LR(G)/I(H,S)such that rQ[A]6= 0 for all [A] ∈ Φ(G0) \ {[∅]} and r ∈ R \ {0}. So we have the following proposition.
Proposition 3.5. Let (H, S) be an admissible pair in ultragraph G and let R be a unital commutative ring. If {t, q} is the universal Leavitt G/(H, S)-family, then rq[A]6= 0 for every [A] ∈ Φ(G0) \ {[∅]} and every r ∈ R \ {0}.
4. Condition (K)
In this section we recall Condition (K) for ultragraphs and we consider the ultragraph G that satisfy Condition (K) to describe that all basic ideals of LR(G) are graded.
Let G = (G0,G1, rG, sG) be an ultragraph and let v ∈ G0. A first-return path based at v in G is a loop α = e1e2· · · en such that sG(α) = v and sG(ei) 6= v for i = 2, 3, . . . , n.
Definition 4.1.([9, Section 7]) An ultragraph G satisfies Condition (K) if every vertex in G0 is either the base of no first-return path or it is the base of at least two first-return paths.
LetG/(H, S) be a quotient ultragraph. By rewriting Definition 2.2 for G/(H, S), one can define the hereditary property for the subcollections of Φ(G0). More pre- cisely, a subcollection K ⊆ Φ(G0) is called hereditary if satisfying the following conditions:
(1) s(e) ∈ K implies r(e) ∈ K for all e ∈ Φ(G1).
(2) [A] ∪ [B] ∈ K for all [A], [B] ∈ K.
(3) [A] ∈ K, [B] ∈ Φ(G0) and [B] ⊆ [A], imply [B] ∈ K.
For any hereditary subcollection K ⊆ Φ(G0), the ideal IK inG/(H, S) generated by {q[A]: [A] ∈ K} is equal to
spanRtαq[A]tβ∗∈ LR(G/(H, S)) : α, β ∈ Path(G/(H, S)) and [A] ∈ K .
Lemma 4.2. LetG/(H, S) be a quotient ultragraph and let R be a unital commuta- tive ring. IfG/(H, S) does not satisfy Condition (L), then LR(G/(H, S)) contains a non-graded ideal I such that rq[A]∈ I for all [A] ∈ Φ(/ G0) \ {[∅]} and r ∈ R \ {0}.
Proof. Suppose thatG/(H, S) contains a closed path γ := e1e2· · · en without exits and r(ei) = s(ei+1) for 1 ≤ i ≤ n where en+1 := e1. Thus we can assume that s(ei) 6= s(ej) for all i, j. Let s(ei) = [vi] for 1 ≤ i ≤ n and let X be the subalgebra of LR G/(H, S) generated by {tei, te∗i, q[vi] : 1 ≤ i ≤ n}.
Claim 1: The subalgebra X is isomorphic to the Leavitt path algebra LR(E), where E is the graph containing a single simple closed path of length n, that is,
E0= {w1, . . . , wn}, E1= {f1, . . . , fn},
r(fi) = s(fi+1) = wi+1 for 1 ≤ i ≤ n where wn+1:= w1.
