### Article

Kyungpook Mathematical Journal 2020; 60(1): 21-43

**Published online** March 31, 2020

Copyright © Kyungpook Mathematical Journal.

### The Leavitt Path Algebras of Ultragraphs

Mostafa Imanfar and Abdolrasoul Pourabbas∗, Hossein Larki

Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, 15914 Tehran, Iran e-mail : m.imanfar@aut.ac.ir and arpabbas@aut.ac.ir

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Iran e-mail : h.larki@scu.ac.ir

### Abstract

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

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 analogy of Exel-Laca algebras.

**Keywords**: ultragraph *C*^{∗,}-algebra, Leavitt path algebra, Exel-Laca algebra.

### 1. Introduction

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

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 ^{*}-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 ^{*}-algebra ^{*}(_{E}^{*}-algebras and Exel-Laca algebras are differ from each other.

To study both graph ^{*}-algebras and Exel-Laca algebras under one theory, Tomforde [15] introduced the notion of an ultragraph and its associated ^{*}-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 ^{*}-algebra of is isomorphic to the Exel-Laca algebra of ^{*}-algebra can be considered as an ultragraph ^{*}-algebra, whereas there is an ultragraph ^{*}-algebra which is not a graph ^{*}-algebra nor an Exel-Laca algebra.

Recently many authors have discussed the algebraic versions of matrix and graph ^{*}-algebras. For example, in [3] the algebraic analogue of the Cuntz-Krieger algebra , denoted by , was studied for finite matrix _{K}^{*}-algebra ^{*}(_{ℂ}(^{*}-algebra ^{*}(

The purpose of this paper is to define the algebraic versions of ultragraphs ^{*}-algebras and Exel-Laca algebras. For an ultragraph and unital commutative ring

Every Leavitt path algebra of a directed graph can be embedded as an subalgebra in a unital ultragraph Leavitt path algebra. Also, the ultragraph Leavitt path algebra is isomorphic to a dense *-subalgebra of .

By using the definition of ultragraph Leavitt path algebras, we give an algebraic 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 of an ultragraph over a unital commutative ring

By applying the graded-uniqueness theorem in Section 3, we give a complete description of basic graded ideals of in terms of admissible pairs in . In Section 4, we use the Cuntz-Krieger uniqueness theorem to show that an ultragraph satisfies Condition (K) if and only if every basic ideal in is graded.

In Section 5, we generalize the algebraic Cuntz-Krieger algebra of [3], denoted by ℰℒ_{A}_{A}

### 2. Leavitt Path Algebras

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

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 ^{0}, a set of edges , the source map and the range map , where ℘(^{0}) is the collection of all subsets of ^{0}. Throughout this work, ultragraph will be assumed to be countable in the sense that both ^{0} and are countable.

For a set ^{0}) generated by .

A _{1}_{2} · · · _{n}^{*} is the

**Definition 2.1**

Let be an ultragraph and let _{A}, s_{e}, s_{e}_{*} : and } of elements in

p _{∅︀}= 0,p =_{A}p_{B}p and_{A∩B}p =_{A∪B}p +_{A}p −_{B}p for all ;_{A∩B}and for all ;

for all

e , ;for every vertex

v with$0<\mid {s}_{\mathcal{G}}^{-1}(v)\mid <\infty $ ,

where _{v}_{{}_{v}_{}}. The _{R}

We say that the Leavitt -family {_{R}_{A}_{A}_{e}_{e}_{e}_{*}) = _{e}_{*} for every and every . The

### 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 be an ultragraph. A subcollection is called

implies for all

e ∈ .A ∪B ∈H for allA, B ∈H .A ∈H , andB ⊆A , implyB ∈H .

Also, is called ^{0} with

For a saturated hereditary subcollection , the

An _{H}

In order to define the quotient of ultragraphs we need to recall and introduce some notations from [13, Section 3]. Let (_{H} S^{0} := ^{0} ∪ {_{H} S

for every , respectively. We write for the algebra generated by the sets {

**Lemma 2.3. ([13, Lemma 3.5])**

We usually denote [^{0}. For , we write [_{A}_{∈}_{H} A

**Definition 2.4. ([13, Definition 3.6])**

Let (

and and are the maps defined by and

**Lemma 2.5**

**Proof**

We denote by

### Remark 2.6

If , then

For example we have

and

Furthermore, it follows from Lemma 2.3 that .

### Remark 2.7

The hereditary property of

Similar to ultragraphs, a _{1}_{2} · · · _{n}_{i}_{+1}) ⊆ _{i}^{*} := [

Using Theorem 3.4, we define the Leavitt path algebra of a quotient ultragraph which is corresponding to the quotient

### Definition 2.8

Let be a quotient ultragraph and let

q _{[∅︀]}= 0,q _{[}_{A}_{]}q _{[}_{B}_{]}=q _{[}_{A}_{]}_{∩}_{[}_{B}_{]}andq _{[}_{A}_{]}_{∪}_{[}_{B}_{]}=q _{[}_{A}_{]}+q _{[}_{B}_{]}−q _{[}_{A}_{]}_{∩}_{[}_{B}_{]};q _{s}_{(}_{e}_{)}t =_{e}t _{e}q_{r}_{(}_{e}_{)}=t and_{e}q _{r}_{(}_{e}_{)}t _{e}_{*}=t _{e}_{*}q _{s}_{(}_{e}_{)}=t _{e}_{*};t _{e}_{*}t =_{f}δ _{e, f}q_{r}_{(}_{e}_{)};q _{[}_{v}_{]}= ∑_{s}_{(}_{e}_{)=[}_{v}_{]}t _{e}t_{e}_{*}for every [v ] ∈ Φ(G ^{0}) with 0< |s ^{−1}([v ])|< ∞.

The _{R}

If (

Let _{1}_{2} · · ·_{n}

Now, we show that for every quotient ultragraph , there exists a universal Leavitt -family. Suppose that and

{[∅︀], [

A ][B ] − [A ∩B ], },{

e –s (e )e, e –er (e ),e ^{*}–e ^{*}s (e ), },{

e ^{*}f –δ (_{e, f}re ) :e , },{

v – ∑_{s}_{(}_{e}_{)=[}_{v}_{]}ee ^{*}: 0< |s ^{−1}([v ])|< ∞}.

If is the quotient map, then it is easy to check that the collection { } is a Leavitt -family. For our convenience, we denote _{[}_{A}_{]} := _{e}_{e}_{*} := ^{*}) for every and , and we show that the Leavitt -family {

Assume that {_{[}_{A}_{]}, _{e}^{*}) = _{e}_{*}, then by [4, Proposition 2.6], there is an _{f}_{X}_{f}_{f}_{[}_{A}_{]}) = _{[}_{A}_{]}, _{e}_{e}_{e}_{*}) = _{e}_{*}.

From now on we denote the universal Leavitt -family and -family by {

### Theorem 2.9

**Proof**

If _{α}q_{[}_{A}_{]}_{β}_{*}, , then

which proves the first statement. Let and consider as defined before. If we define a degree map ^{*}) = −1 for every and , then by [4, Proposition 2.7], is a ℤ-graded ring with the grading

### Theorem 2.10

_{A} of_{e}_{e}_{*}≠ 0

**Proof**

By the universality, it suffices to generate a Leavitt -family {_{A}_{e}^{0}, let

where ⊕ _{A}_{v}_{∈}_{A} Z_{v}_{v}_{∈}_{A} Z_{v}_{v}_{∈}_{G}_{0}_{v}_{A}, S_{e}, S_{e}_{*} to homomorphisms _{A}, s̃_{e}, s̃_{e}_{*} ∈ Hom_{R}_{R}_{A}

Note that we cannot follow the argument of Theorem 2.10 to show that _{[}_{A}_{]} ≠ 0. For example, suppose that is the ultragraph

and let _{1}, _{2}, . . .}, which is a hereditary and saturated subcollection of . If we consider the quotient ultragraph , then {[∅︀] ≠ [_{r}_{(}_{e}_{)} : ⊕ _{[}_{v}_{]⊆}_{r}_{(}_{e}_{)}_{[}_{v}_{]} → ⊕ _{[}_{v}_{]⊆}_{r}_{(}_{e}_{)}_{[}_{v}_{]} as in the proof of Theorem 2.10. In Section 3 we will solve this problem.

### Remark 2.11

Every directed graph ^{0}, ^{1}, ^{0} := ^{0}, and the map is defined by for every . In this case, the algebra is the collection of all finite subsets of ^{0}. The Leavitt path algebra _{R}

### Lemma 2.12

**Proof**

If , then the Relations of Definition 2.1 imply that _{G}_{0} is a unit for .

Conversely, suppose that is unital and write

where _{k}^{0}

this contradicts Theorem 2.10 and it follows , as desired.

We note that, for directed graph _{R}^{0} is finite. If ^{0} is infinite, then we can define an ultragraph associated to _{R}^{0} = ^{0} ∪ {_{e}_{e}_{v}_{v}^{1} and ^{0}, respectively. It is straightforward to see that {

### 2.3. Uniqueness Theorems

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

Let be a quotient ultragraph. We prove the graded and Cuntz-Krieger uniqueness theorems for and . 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 [^{0}) is called a ^{−1}([^{−1}([_{sg}(^{0}).

Let be a finite subset and write ^{0} := _{sg}(^{0}) and . Following [13], we construct a finite graph _{F}_{1}, . . . , _{n}^{n} ^{n}_{ω}_{i=1}_{i}_{ω}_{j=0}_{j}_{[}_{v}_{]∈}_{F}_{0} [

and

Now we define the finite graph

with _{F}_{F}_{F}_{F}_{F}_{F}

### Lemma 2.13

For every finite set ,the elements $$\begin{array}{lll}{P}_{e}:{t}_{e}{t}_{e*},\hfill & {P}_{[v]}:={q}_{[v]}(1-\sum _{e\in {F}^{1}}{t}_{e}{t}_{e*}),\hfill & {P}_{\omega}:={q}_{R(\omega )}(1-\sum _{e\in {F}^{1}}{t}_{e}{t}_{e*}),\hfill \\ {S}_{(e,f)}:={t}_{e}{P}_{f},\hfill & {S}_{(e,[v])}:={t}_{e}{P}_{[v]},\hfill & {S}_{(e,\omega )}:={t}_{e}{P}_{\omega},\hfill \\ {S}_{(e,f)*}:={P}_{f}{t}_{e*},\hfill & {S}_{(e,[v])*}:={P}_{[v]}{t}_{e*},\hfill & {S}_{(e,\omega )*}:={P}_{\omega}{t}_{e*},\hfill \end{array}$$ form a Leavitt E _{F}-family which generates the subalgebra ofgenerated by {q _{[}_{v}_{]},t _{e}, t_{e}_{*}: [v ] ∈F ^{0},e ∈F ^{1}}.For r ∈R {0},if rq _{[}_{A}_{]}≠ 0for all [A ] ≠ [∅︀]in ,then rP ≠ 0_{z}for all $z\in {E}_{F}^{0}$ . In this case, we have $${L}_{R}({E}_{F})\cong {L}_{R}(S,P)=\text{Alg}\{{q}_{[v]},{t}_{e},{t}_{e*}\in {L}_{R}(\mathcal{G}/(H,S)):[v]\in {F}^{0},e\in {F}^{1}\}.$$

**Proof**

The statement (i) follows from the fact that {

For (ii), fix _{[}_{A}_{]} ≠ 0 for every , then _{e}_{e}_{*} ≠ 0 for every edge _{e}^{1}. Let [^{0}. If [_{[}_{v}_{]} = _{[}_{v}_{]} ≠ 0. If [_{[}_{v}_{]}_{f}_{[}_{v}_{]}_{f}_{f}_{[}_{v}_{]} ≠ 0 for all [^{0}. Moreover, for each _{F}_{[}_{v}_{]}(_{ω}_{[}_{v}_{]} ≠ 0 and in the later, _{f}_{*} (_{ω}_{f}_{*} ≠ 0. Thus all _{ω}_{z}

Now we show that _{R}_{F}_{R}_{z}_{g}_{g}_{*} as elements in are 0, 1 and −1, respectively. So _{R}

Let {_{R}_{F}_{R}_{F}_{R}_{z}_{z}_{g}_{g}_{g}_{*}) = _{g}_{*} for _{R}_{F}_{R}

### Theorem 2.14. (The graded-uniqueness theorem)

_{[}_{A}_{]}) ≠ 0

**Proof**

Let {_{n}_{n}_{[}_{A}_{]}) ≠ 0 for all and _{n}_{R}_{F}_{n}) → _{n}_{n}_{R}_{F}_{n}) →

Fix _{n}_{z}_{z}_{[}_{A}_{]}) ≠ 0 for all , the elements _{e}_{e}_{*}) are nonzero for every edge _{e}t_{e}_{e}_{e}^{0}. If [_{[}_{v}_{]}) = _{[}_{v}_{]}) ≠ 0. If [_{[}_{v}_{]}_{f}_{[}_{v}_{]}_{f}_{f}_{[}_{v}_{]}) ≠ 0. For every _{F}_{[}_{v}_{]}(_{ω}_{[}_{v}_{]}) ≠ 0 and in the later, _{f}_{*} (_{ω}_{f}_{*}) ≠ 0. Thus _{n}_{z}_{n}

If [_{[}_{v}_{]} = ∑_{s}_{(}_{e}_{)=[}_{v}_{]}_{e}t_{e}_{*}. Furthermore, _{[}_{A}_{]}_{}_{[}_{B}_{]} = _{[}_{A}_{]} – _{[}_{A}_{]}_{[}_{B}_{]} for every . Thus, by Lemma 2.5, is an

and so

### Corollary 2.15

_{A}

### Definition 2.16

A _{1} · · · _{n}_{i}_{i}_{+1} for some 1 ≤ _{n}_{+1} := _{1}. We say that satisfies _{1} · · · _{n}_{i}_{i}_{+1}) for some 1 ≤

### Theorem 2.17. (The Cuntz-Krieger uniqueness theorem)

_{[}_{A}_{]}) ≠ 0

**Proof**

Choose an increasing sequence {_{n}_{n}_{[}_{A}_{]}) ≠ 0 for all and _{n}_{R}_{F}_{n}) → _{n}_{n}_{z}_{F}_{n} satisfy Condition (L). So, the Cuntz-Krieger uniqueness theorem for graphs [18, Theorem 6.5] implies that _{n}

### Corollary 2.18

_{A}

### 3. Basic Graded Ideals

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

In this section, we apply the graded-uniqueness theorem for quotient ultragraphs to investigate the ideal structure of . We would like to consider the ideals of that are reflected in the structure of the ultragraph . For this, we give the following definition of basic ideals.

Let (_{H}

Let

rp ∈_{A}I impliesp ∈_{A}I for andr ∈R {0}.$r{p}_{w}^{{H}_{I}}\in I$ implies${p}_{w}^{{H}_{I}}\in I$ forw ∈B _{H}_{I}andr ∈R {0}.

For an admissible pair (_{(}_{H, S}_{)}.

### Lemma 3.1. (cf. [12, Lemma 3.9])

_{(}_{H, S}_{)}

**Proof**

We denote the right-hand side of the above equality by _{(}_{H, S}_{)}. On the other hand, all generators of _{(}_{H, S}_{)} belong to _{(}_{H, S}_{)} =

Note that the elements _{α}p_{A}s_{β}_{*} and _{(}_{H, S}_{)} is a graded ideal.

To show that _{(}_{H, S}_{)} is a basic ideal, suppose _{A}_{(}_{H, S}_{)} for some and

where _{i}_{j}_{i}, s_{j}

**Step I**

If

Note that _{v}x_{i}_{j}_{i}A_{i}_{i}_{i,}_{1}_{i,}_{2} · · · _{i,}_{|}_{α}_{|}. If _{1,1}, . . . , _{n,}_{1}, _{1,1}, . . . , _{m,}_{1} with . So _{e}_{*} = _{e}_{*} (_{v}_{e}_{*}

**Step II**

If

Let _{A}_{j}_{A}_{A}rp_{A}_{A}x_{j}_{j}_{H}_{j}_{j}_{i}_{v}_{v}p_{A}_{v}x

**Step III**

If

We go toward a contradiction and assume {_{1} := _{1} ∈ _{H}_{1} ∉ _{H}_{1} with and . By Step II, there is a vertex such that {_{2}} ∉ _{1}_{L}_{i, j}_{i}_{j}_{k}_{γ}_{v}_{γ}_{γ}

**Step IV**

If _{A}_{(}_{H, S}_{)}, then

If _{A}_{(}_{H, S}_{)}, as desired.

Now, we show that _{(}_{H, S}_{)} satisfies assertion (2) of the definition of basic ideal. Note that, by Step IV, we have _{I}_{(H, S)} = _{H}

where _{i}_{j}_{i}, s_{j}_{i}_{H}_{1,1}, . . . , _{n,}_{1}, _{1,1}, . . . , _{m,}_{1} such that and . If

### Remark 3.2

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

Let (

### Lemma 3.3. (cf. [13 , Proposition 3.3])

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

### Proof

Suppose that {

then it is straightforward to see that {_{R}_{A}

We show that . For , _{A}_{Ā}_{R}_{H} S

for every . Thus for _{H} S^{−1}(^{−1}(

Also,

Thus, by Remark 2.6, _{A}_{R}_{e}, S̃_{e}_{*} ∈ _{R}

### Theorem 3.4. (cf. [12 , Theorem 3.10])

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

For any admissible pair (H, S )in ,we have .The map (H, S ) ↦I _{(}_{H, S}_{)}is a bijection from the set of all admissible pairs of to the set of all graded basic ideals of .

### Proof

(1) Let {

It can be shown that the family is a Leavitt -family that generates . Furthermore, by Remark 3.2, _{[}_{A}_{]} ≠ 0 for all and

Now we use the universal property of to get an _{e}_{e}_{e}_{*}) = _{e}_{*} and _{[}_{A}_{]}) = _{[}_{A}_{]} ≠ 0 for and _{(}_{H, S}_{)} is a graded ideal, the quotient is graded. Moreover, the elements _{[}_{A}_{]}, _{e}_{e}_{*} have degrees 0, 1 and −1 in , respectively and thus

(2) The injectivity of the map (_{(}_{H, S}_{)} is a consequence of Remark 3.2. To see that it is onto, let _{(}_{H}_{I}, _{S}_{I)} ⊆ _{I}, S_{I}_{I}, S_{I}_{A}_{H}_{I}_{I}

We show that _{[}_{A}_{]}) = _{[}_{A}_{]}) = _{A}_{I}_{A}_{B}_{[}_{A}_{]}) ≠ 0. If _{H}_{I}_{I}_{[}_{w′}_{]}) ≠ 0. In view of Remark 2.10, we deduce that _{[}_{A}_{]}) ≠ 0 for every . Furthermore, _{(}_{H}_{I}, _{S}_{I)}.

As we have seen in the proof of Theorem 3.4, if (_{[}_{A}_{]} ≠ 0 for all and

### Proposition 3.5

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

_{[}_{A}_{]} ≠ 0

### 4. Condition (K)

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

In this section we recall Condition (K) for ultragraphs and we consider the ultragraph that satisfy Condition (K) to describe that all basic ideals of are graded.

Let be an ultragraph and let ^{0}. A _{1}_{2} · · · _{n}

### Definition 4.1. ([9, Section 7])

An ultragraph satisfies ^{0} is either the base of no first-return path or it is the base of at least two first-return paths.

Let be a quotient ultragraph. By rewriting Definition 2.2 for , one can define the hereditary property for the subcollections of . More precisely, a subcollection is called

s (e ) ∈K impliesr (e ) ∈K for all .[

A ] ∪ [B ] ∈K for all [A ], [B ] ∈K .[

A ] ∈K , and [B ] ⊆ [A ], imply [B ] ∈K .

For any hereditary subcollection , the ideal _{K}_{[}_{A}_{]} : [

**Lemma 4.2**

_{[}_{A}_{]} ∉

**Proof**

Suppose that contains a closed path _{1}_{2} · · · _{n}_{i}_{i}_{+1}) for 1 ≤ _{n}_{+1} := _{1}. Thus we can assume that _{i}_{j}_{i}_{i}_{e}_{i}, _{[}_{v}_{i]} : 1 ≤

**Claim 1**

The subalgebra _{R}

**Proof of Claim 1**

Set _{w}_{i} := _{[}_{v}_{i]}, _{f}_{i} := _{e}_{i} and _{R}_{R}_{f}_{i}) = _{e}_{i}, _{w}_{i}) = _{[}_{v}_{i]} for all _{w}_{i}) = _{[}_{v}_{i]} ≠ 0 for all _{R}

By [18, Lemma 7.16], _{R}

### Claim 2

_{[}_{A}_{]} ∉

**Proof of Claim 2**

We show that _{w}_{i} ∉ _{w}_{j} ∈ _{w}_{j} ∈ _{j}_{i}_{R}

Suppose that there exist _{[}_{A}_{]} ∈

of , then _{k}_{k}_{k}_{1}], . . . , [_{n}

which implies that _{j}_{[}_{v}_{j]} = _{[}_{v}_{j]}_{[}_{A}_{]} ∈ _{w}_{j} = ^{−1}(_{[}_{v}_{j]}) ∈

Let

and set _{i}_{i}_{+1}),

### Claim 3

If _{[}_{v}_{]} ∈ _{K}

**Proof of Claim 3**

Suppose that

where _{i}_{r}_{(}_{e}_{)} ∈ _{K}_{[}_{w}_{]} ∈ _{K}_{1} · · · _{L}_{i}_{i}_{k}_{η}_{[}_{v}_{]})_{η}_{η}

### Claim 4

_{K}π_{K}

**Proof of Claim 4**

Suppose that _{n}_{[}_{A}_{]}_{[}_{A}_{]} = _{[}_{A}_{]}_{[}_{A}_{]} ∈ _{n}_{n}_{1}_{n}_{1}_{n}_{1}_{n}_{1} + _{n}_{m}_{n}_{m}_{n}_{m}_{n}_{m}, where _{n}_{k} ∈ _{n}_{k}, _{n}_{k} ∈ _{K}_{n}_{k}_{n}_{k}_{n}_{k} ∈ _{n}_{k}. Thus

Since _{[}_{A}_{]}_{n}_{k}, _{n}_{k}_{[}_{A}_{]} ∈ _{n}_{k}_{[}_{A}_{]}_{n}_{k}_{n}_{k}_{n}_{k}_{[}_{A}_{]} ∈ _{n}

Finally, we show that _{[}_{B}_{]} ∉ _{[}_{B}_{]} ∈ _{[}_{B}_{]} ∈ _{K}_{[}_{v}_{]} ∈ _{K}_{i}_{[}_{v}_{i]} = _{α}_{*}_{[}_{v}_{]}_{α}q_{[}_{v}_{i]} ∈ _{[}_{v}_{i]} in terms of its representation in _{K}π_{K}_{[}_{v}_{i]} ∈ _{[}_{B}_{]} ∉

### Theorem 4.3. (cf. [12 , Theorem 3.18], [18 , Theorem 7.17])

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

### Proof

Suppose that satisfies Condition (K). If

such that _{[}_{A}_{]}) ≠ 0 for all and all _{(}_{H}_{I}, _{S}_{I)}. It follows from Lemma 3.1 that

Conversely, if does not satisfy Condition (K), then [13, Proposition 6.2] and Lemma 4.2 imply that there exist _{[}_{A}_{]} ∉ ^{−1}(_{A}

for _{H} S

### Corollary 4.4

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

_{(}_{H, S}_{)}

### 5. Exel-Laca R -algebras

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

The Exel-Laca algebras are generated by partial isometries whose relations are determined by a countable {0, 1}-valued matrix

### Definition 5.1

Let _{i}_{j}

By [15, Theorem 4.5], the Exel-Laca algebra is canonically isomorphic to . Motivated by this fact we give the following definition.

### Definition 5.2

Let _{A}

### Example 5.3

Let _{R},_{R}_{1}, _{1}, . . . , _{n}, y_{n}

x =_{i}y_{i}x_{i}x and_{i}y =_{i}x_{i}y_{i}y for all_{i}i ,yix = 0 for all_{j}i ≠j ,${y}_{i}{x}_{i}={\sum}_{j=1}^{n}A(i,j){x}_{j}{y}_{j}$ for alli ,${\sum}_{j=1}^{n}{x}_{j}{y}_{j}=1$ .

The Cuntz-Krieger _{B}_{v}_{i∈}_{B} x_{i}y_{i}_{i}_{i}_{i}_{*} := _{i}

On the other hand, if {_{i}_{i}_{i}_{i}_{*}, 1 ≤

Note that for any {0, 1}-valued matrix ^{0} and ^{−1}(_{e}_{e}^{−1}(^{0} := {_{e}^{0} and ^{−1}(_{e}

### Remark 5.4

We notice that each vertex _{e}

### Lemma 5.5

**Proof**

Let {^{0} we have . Define

By Remark 5.4, for each , the idempotent _{A}_{e}P_{r}_{(}_{e}_{)} = _{e}q_{r̃}_{(}_{e}_{)} = _{e}_{e}

Thus there is an _{e}_{e}_{e}_{*}) = _{e}_{*} and for ^{0}, and _{A}_{e}^{0}, we have (^{−1}(_{e}_{e}S_{e}_{*} = _{e}t_{e}_{*} = _{v}_{e}. Therefore,

### Definition 5.6. ([15, Definition 2.4])

Let be an ultragraph. The edge matrix of an ultragraph is the matrix given by

We can check that, if is an ultragraph with no singular vertices, then . So, by Lemma 5.5 we have the following.

### Theorem 5.7

### Example 5.8

Suppose that ^{0} = {^{1} = {_{1}, . . . ._{n}_{R}_{R}_{n}_{×}_{n}_{E}_{A}_{R}

### Remark 5.9

Theorem 5.7 shows that the Leavitt path algebras of ultragraphs with no singular vertices are precisely the Exel-Laca

### Remark 5.10

We know that for every directed graph _{ℂ}(^{*}-algebra ^{*}(^{*}-algebra introduced in [15]. In particular, for any countable {0, 1}-valued matrix _{A}

Last part of this section is an example to emphasize that the class of the Leavitt path algebras of ultragraphs is strictly larger than the class of the Leavitt path algebras of directed graphs as well as the class of algebraic Exel-Laca algebras.

### Example 5.11

Let _{2} and let be the ultragraph

with one ultraedge _{0} and _{1}, _{2}, . . .}. Note that satisfies Condition (K) and so all ideals of are basic and graded by Theorem 4.3. Moreover, is unital, because by Lemma 2.12 . We claim that the algebra is not isomorphic to the Leavitt path algebra of a graph. Suppose on the contrary that there is a such graph _{ℤ2} (_{α}_{ℤ2} (_{ℤ2} (_{ℤ2} (_{ℤ2} (^{0} must be finite and hence _{ℤ2} (_{{}_{v}_{n}} for

Now assume for some matrix _{{}_{v}_{1}} in . Since _{1} is a sink, we have

and so |_{{}_{v}_{1}}| _{{}_{v}_{1}}) in includes infinitely many elements, which is a contradiction.

### Acknowledgements

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

The authors are grateful to the referee for carefully reading the paper, pointing out a number of misprints and for some helpful comments.

### References

- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References

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