Let be an ultragraph and let R be a unital commutative ring. A Leavitt-family in an R-algebra A is a set {p_A, s_e, s_e^* : and } of elements in A such that

• p_∅︀ = 0, p_Ap_B = p_A∩B and p_A∪B = p_A + p_B − p_A∩B for all ;

• and for all ;

• for all e, ;

• for every vertex v with 0<∣sG-1(v)∣<∞,

where p_v denotes p_{v}. The R-algebra generated by the Leavitt -family {s, p} is denoted by L_R(s, p).

We say that the Leavitt -family {s, p} is universal, if B is an R-algebra and {S, P} is a Leavitt -family in B, then there exists an algebra homomorphism φ : L_R(s, p) → B such that φ(p_A) = P_A, φ(s_e) = S_e and φ(s_e^*) = S_e^* for every and every . The Leavitt path algebra ofwith coefficients in R, denoted by , is the (unique up to isomorphism) R-algebra generated by a universal Leavitt -family.

Let be an ultragraph. A subcollection is called hereditary if satisfying the following conditions:

• implies for all e ∈ .

• A ∪ B ∈ H for all A, B ∈ H.

• A ∈ H, and B ⊆ A, imply B ∈ H.

Also, is called saturated if for any v ∈ G⁰ with 0<∣sG-1(v)∣<∞, we have

For a saturated hereditary subcollection , the breaking vertices of H is denoted by

An admissible pair in is a pair (H, S) of a saturated hereditary set and some S ⊆ B_H.

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 . For each , we denote Ā := A ∪ {w′ : w ∈ A ∩ (B_H S)}, where w′ denotes another copy of w. Consider the ultragraph , where , Ḡ⁰ := G⁰ ∪ {w′ : w ∈ B_H S} and the maps r̄, s̄ are defined by r¯(e):=rG(e)¯ and

for every , respectively. We write for the algebra generated by the sets {v}, {w′} and r(e).

Let (H, S) be an admissible pair in an ultragraphand let ~ be a relation ondefined by A ~ B if and only if there exists V ∈ H such that A ∪ V = B ∪ V. Then ~ is an equivalence relation onand the operations

are well-defined on the equivalence classes.

We usually denote [v] instead of [{v}] for every v ∈ Ḡ⁰. For , we write [A] ⊆ [B] whenever [A] ∩ [B] = [A]. The set ∪_A∈H A is denoted by ∪H.

Let (H, S) be an admissible pair in . The quotient ultragraph ofby (H, S) is the quadruple , where

and and are the maps defined by and r(e):=[rG(e)¯] for every , respectively.

Ifis a quotient ultragraph, then

whereis the smallest algebra incontaining

Remark 2.6

If , then A∪B¯=A¯∪B¯,A∩B¯=A¯∩B¯ and AB¯=A¯B¯. Thus, by applying an analogous lemma of Lemma 2.5 for and , we deduce that for all . One can see that

For example we have

and

Furthermore, it follows from Lemma 2.3 that .

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 in is a sequence α := e₁e₂ · · · e_n of edges in such that s(e_i+1) ⊆ r(e_i) for 1 ≤ i ≤ n − 1. We say the path α has length |α| := n and we consider the elements in to be paths of length zero. We denote by , the union of paths with finite length. The maps r and s can be naturally extended on . Let be the set of ghost edges. We also define the ghost pathα*:=en*en-1*⋯e1* for every and [A]^* := [A] for every .

Using Theorem 3.4, we define the Leavitt path algebra of a quotient ultragraph which is corresponding to the quotient R-algebra . We use this concept to characterize the ideal structure of in Section 3. The following definition is the algebraic version of [13, Definition 3.8].

Definition 2.8

Let be a quotient ultragraph and let R be a unital commutative ring. A Leavitt-family in an R-algebra A is a set and of elements in A such that

• q_[∅︀] = 0, q_[A]q_[B] = q_[A]∩[B] and q_[A]∪[B] = q_[A] + q_[B] − q_[A]∩[B];

• q_s(e)t_e = t_eq_r(e) = t_e and 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_et_e^* for every [v] ∈ Φ(G⁰) with 0 < |s⁻¹([v])| < ∞.

The R-algebra generated by the Leavitt -family {t, q} is denoted by L_R(t, q). The Leavitt path algebra ofwith coefficients in R, denoted by , is the (unique up to isomorphism) R-algebra generated by a universal Leavitt -family (the definition of universal Leavitt -family is similar to ultragraph case).

If (H, S) = (∅︀, ∅︀), then we have [A] = {A} for each . In this case, every Leavitt -family is a Leavitt -family and vice versa. So, we can consider the ultragraph Leavitt path algebra as .

Let R be a unital commutative ring. For a nonempty set X, we write w(X) for the set of words w := w₁w₂ · · ·w_n from the alphabet X. The free R-algebra generated by X is denoted by . For definition of the free R-algebra we refer the reader to [4, Section 2.3].

Now, we show that for every quotient ultragraph , there exists a universal Leavitt -family. Suppose that and I is the ideal of the free R-algebra generated by the union of the following sets:

• {[∅︀], [A][B] − [A ∩ B], },

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

• {e^*f – δ_{e, f} r(e) : 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 q_[A] := π(A), t_e := π(e) and t_e^* := π(e^*) for every and , and we show that the Leavitt -family {t, q} has the universal property.

Assume that {T, Q} is a Leavitt -family in an R-algebra A. If we define f : X → A by f([A]) = P_[A], f(e) = T_e and f(e^*) = T_e^*, then by [4, Proposition 2.6], there is an R-algebra homomorphism such that φ_f | _X = f. Since {T, Q} is a Leavitt -family, φ_f vanishes on the generator of I. Hence, we can define an R-algebra homomorphism by φ(π(x)) = φ_f (x) for every . Furthermore, we have φ(q_[A]) = Q_[A], φ(t_e) = T_e and φ(t_e^*) = T_e^*.

From now on we denote the universal Leavitt -family and -family by {s, p} and {t, q}, respectively. So, we suppose that {s, p} and {t, q} are the canonical generators of and , respectively.

Theorem 2.9

Letbe a quotient ultragraph and let R be a unital commutative ring. Thenis of the form

Furthermore, is a ℤ-graded ring by the grading

Theorem 2.10

Letbe an ultragraph and let R be a unital commutative ring. Then for all nonempty setsand all r ∈ R {0}, the elements rp_A ofare nonzero. In particular, rs_e ≠ 0 and rs_e^*≠ 0 for everyand every r ∈ R {0}.

Remark 2.11

Every directed graph E = (E⁰, E¹, r, s) can be considered as an ultragraph , where G⁰ := E⁰, and the map is defined by for every . In this case, the algebra is the collection of all finite subsets of G⁰. The Leavitt path algebra L_R(E) is naturally isomorphic to (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

Letbe an ultragraph and let R be a unital commutative ring. Thenis unital if and only ifand in this case.