Graph of BCI-Algebras

Abstract

We associate a graph to any subset Y of a BCI-algebra X and denote it by G(Y). Then we find the set of all connected components of G(X) and verify the relation between X and G(X), when X is commutative BCI-algebra or G(X) is complete graph or n-star graph. Finally, we attempt to investigate the relation between some operations on graph and some operations on BCI-algebras.

1. Introduction

𝐵𝐶𝐾- and 𝐵𝐶𝐼-algebras are two classes of abstract algebras were introduced by Imai and Iséki [1, 2], in 1966. The notion of 𝐵𝐶𝐾-algebras is originated from two different ways. One of the motivations is from classical and nonclassical propositional logic. Another motivation is based on set theory. It is known that the class of 𝐵𝐶𝐾-algebras is a proper subclass of the class of 𝐵𝐶𝐼-algebras. Many authors studied the graph theory in connection with semigroups and rings. For example, Beck [3] associated to any commutative ring 𝑅 its zero divisors graph 𝐺(𝑅), whose vertices are the zero divisors of 𝑅, with two vertices 𝑎,𝑏 jointed by an edge in case 𝑎𝑏=0. In [4], Jun and Lee defined the notion of zero divisors and quasi-ideals in 𝐵𝐶𝐼-algebra and show that all zero divisors are quasi-ideal. Then, they introduced the concept of associated graph of 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra and verified some properties of this graph and proved that if 𝑋 is a 𝐵𝐶𝐾-algebra, then associated graph of 𝑋 is a connected graph. Moreover, if 𝑋 is a 𝐵𝐶𝐼-algebra and 𝑥∈𝑋 such that 𝑥 is not contained in 𝐵𝐶𝐾-part of 𝑋, then there is not any edge connecting 𝑥 and 𝑦, for any 𝑦∈𝑋. In this paper, we associate new graph to a 𝐵𝐶𝐼-algebras 𝑋 which is denoted it by 𝐺(𝑋). This definition based on branches of 𝑋. If 𝑋 is a 𝐵𝐶𝐾-algebras, then this definition and last definition, which was introduced by Jun and Lee, are the same. Then, for any 𝑎∈𝑃, we defined the concept of 𝑎-divisor, where 𝑃 is a 𝑝-semisimple part of 𝐵𝐶𝐼-algebra 𝑋 and show that it is quasi-ideal of 𝑋. Then, we explain some properties of this graph as mentioned in the abstract.

2. Preliminaries

Definition 2.1 (see [1, 2]). A 𝐵𝐶𝐼-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying the following conditions:(𝐵𝐶𝐼1)((𝑥∗𝑦)∗(𝑥∗𝑧))∗(𝑧∗𝑦)=0;(𝐵𝐶𝐼2)𝑥∗0=𝑥;(𝐵𝐶𝐼3)𝑥∗𝑦=0 and 𝑦∗𝑥=0 imply 𝑦=𝑥.
If 𝑋 is a 𝐵𝐶𝐼-algebra, then for all 𝑥,𝑦∈𝑋 the following hold:
(𝐵𝐶𝐼4)𝑥∗𝑥=0;(𝐵𝐶𝐼5)𝑥∗(𝑥∗(𝑥∗𝑦))=𝑥∗𝑦;(𝐵𝐶𝐼6)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦;(𝐵𝐶𝐼7)𝑥≤𝑦 implies 𝑥∗𝑧≤𝑦∗𝑧 and 𝑧∗𝑦≤𝑧∗𝑥, for any 𝑧∈𝑋. Moreover, the relation ≤ was defined by 𝑥≤𝑦⇔𝑥∗𝑦=0, for any 𝑥,𝑦∈𝑋, is a partial-order on 𝑋, which is called 𝐵𝐶𝐼-ordering of 𝑋. The set 𝐵={𝑥∈𝑋∣0∗𝑥=0} is called 𝐵𝐶𝐾-part of 𝑋. A 𝐵𝐶𝐼-algebra 𝑋 is called a 𝐵𝐶𝐾-algebra if 𝐵=𝑋. A 𝐵𝐶𝐾-chain is a 𝐵𝐶𝐾-algebra such that (𝑋,≤) is a chain, where ≤ is the 𝐵𝐶𝐼-ordering of 𝑋. A nonzero element 𝑎 of 𝐵𝐶𝐾-algebra 𝑋 is called an atom of 𝑋 if 𝑥∗𝑎=0 implies 𝑥=𝑎, for any nonzero element 𝑥∈𝑋. Moreover, 𝑃={𝑥∈𝑋∣0∗(0∗𝑥)=𝑥} is called 𝑝-semisimple part of a 𝐵𝐶𝐼-algebra 𝑋. It is the set of all minimal elements of 𝑋, with respect to the 𝐵𝐶𝐼-ordering of 𝑋. The 𝐵𝐶𝐼-algebra 𝑋 is called a 𝑝-semisimple 𝐵𝐶𝐼-algebra if 𝑃=𝑋. For any 𝑎∈𝑃, we use the notation 𝑉_𝑋(𝑎) or simply 𝑉(𝑎) to denote the set {𝑥∈𝑋∣𝑎∗𝑥=0} which is called the branch of 𝑋 with respect to 𝑎. It is easy to see that 𝑋=⋃_𝑎∈𝑃𝑉(𝑎), and for any distinct elements 𝑎,𝑏∈𝑃, we have(𝑃1)𝑉(𝑎)∩𝑉(𝑏)=∅;(𝑃2)if 𝑥∈𝑉(𝑎) and 𝑦∈𝑉(𝑏), then 𝑥∗𝑦∈𝑉(𝑎∗𝑏);(𝑃3)if 𝑥∈𝑉(𝑎), then {𝑦∈𝑋∣𝑥≤𝑦}∪{𝑦∈𝑋∣𝑦≤𝑥}⊆𝑉(𝑎);(𝑃4)for all 𝑥∈𝑋, 𝑥∗𝑎=0 implies 𝑥=𝑎.A 𝐵𝐶𝐼-algebra 𝑋 is called commutative if 𝑥≤𝑦 implies 𝑥=𝑦∗(𝑦∗𝑥), for any 𝑥,𝑦∈𝑋. Moreover, 𝑋 is called branchwise commutative if 𝑥∗(𝑥∗𝑦)=𝑦∗(𝑦∗𝑥), for all 𝑥,𝑦∈𝑉(𝑎) and all 𝑎∈𝑃. For more details, we refer to [5–11].

Lemma 2.2 (see [9]). A 𝐵𝐶𝐼-algebra X is commutative if and only if it is branchwise commutative.

Theorem 2.3 (see [10]). Let 𝑋={𝑥₀,𝑥₁,…,𝑥_𝑛} and (𝑋,∗,0) be a 𝐵𝐶𝐾-chain with 𝑥₀ as the zero element, whose 𝐵𝐶𝐼-ordering is supposed as follows: 𝑥₀<𝑥₁<⋯<𝑥_𝑛. Then, 𝑋 is commutative if and only if the relation ∗ on 𝑋 is given by 𝑥_𝑖∗𝑥_𝑗=𝑥_𝑖∗𝑗, where 𝑖∗𝑗=max{0,𝑖−𝑗}, for any 𝑖,𝑗∈{0,1,…,𝑛} (see [10, Theorem 2.3.3]).

A partial-order set (𝑃,≤) is said to be of finite length if the lenghts of all chains of 𝑃 are bounded. Let 𝑥,𝑦∈𝑃. Then, a chain of length 𝑛 between 𝑥 and 𝑦 is a chain 𝑎₀<𝑎₁<⋯<𝑎_𝑛−1<𝑎_𝑛 such that {𝑎₀,𝑎_𝑛}={𝑥,𝑦}. For any 𝑥,𝑦∈𝑃, 𝑙(𝑥,𝑦) is the greatest number in the lenghts of all chain between 𝑎 and 𝑏.

A 𝐵𝐶𝐼-algebra 𝑋 is said to be finite length if it is finite length as a partial-order set.

Note 1 (see [4]). Let 𝑋 be a 𝐵𝐶𝐼-algebra and 𝐴⊆𝑋. We will use the notations 𝑈_𝑋(𝐴) and 𝐿_𝑋(𝐴) or simply 𝑈(𝐴) and 𝐿(𝐴) to denote the sets {𝑥∈𝑋∣𝑎∗𝑥=0,∀𝑎∈𝐴} and {𝑥∈𝑋∣𝑥∗𝑎=0,∀𝑎∈𝐴}, respectively, that is, 𝑈(𝐴)={𝑥∈𝑋∣𝑎≤𝑥,∀𝑎∈𝐴} and 𝐿(𝐴)={𝑥∈𝑋∣𝑥≤𝑎,∀𝑎∈𝐴}.

Definition 2.4 (see [4]). A nonempty subset 𝐼 of 𝐵𝐶𝐼-algebra 𝑋 is called a quasi-ideal of 𝑋 if 𝑥∗𝑦=0 implies 𝑥∈𝐼, for any 𝑦∈𝐼 and 𝑥∈𝑋.

Proposition 2.5 (see [12]). Let (𝑋,∗₁,0) and (𝑌,∗₂,0) be two 𝐵𝐶𝐾-algebras such that 𝑋∩𝑌={0} and ∗ be the binary operation on 𝑋∪𝑌 as follows: for any 𝑎,𝑏∈𝑋∪𝑌; 𝑎∗𝑏=𝑎∗₁𝑏if𝑎,𝑏∈𝑋,𝑎∗₂𝑏if𝑎,𝑏∈𝑌,𝑎otherwise. (2.1) Then, (𝑋∪𝑌,∗,0) is a 𝐵𝐶𝐾-algebra. We denote (𝑋∪𝑌,∗,0) by 𝑋⊕𝑌.

Definition 2.6 (see [4]). For any 𝑥∈𝑋, the set {𝑦∈𝑋∣𝐿({𝑥,𝑦})={0}} is called the set of all zero divisors of 𝑥.

Theorem 2.7 (see [4]). For any element 𝑥 of 𝐵𝐶𝐾-algebra 𝑋, the set of all zero divisors of 𝑥 is a quasi-ideal of 𝑋 containing the zero element {0}.

Let 𝐺 be a graph, 𝐸(𝐺) be the set of all edges of 𝐺 and 𝑉(𝐺) be the set of all vertexes of 𝐺. For any 𝑆⊆𝐸(𝐺), the graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺)−𝑆 is denoted by 𝐺−𝑆. The edge which connect two vertices 𝑥,𝑦 is denoted by 𝑥𝑦. Note that, 𝑥𝑦 and 𝑦𝑥 are the same. For any 𝑇⊆𝑉(𝐺), the graph with vertex set 𝑉(𝐺)−𝑇 and edge set 𝐸(𝐺)−𝑇^′ is denoted by 𝐺−𝑇, where 𝑇^′={𝑥𝑦∈𝐸(𝐺)∣𝑥∈𝑇,𝑦∈𝐺}. A graph 𝐻 is called a subgraph of 𝐺 if 𝑉(𝐻)⊆𝑉(𝐺) and 𝐸(𝐻)⊆𝐸(𝐺). A graph 𝐺=(𝑉,𝐸) is connected, if any vertices 𝑥,𝑦 of 𝐺 linked by a path in 𝐺, otherwise the graph is disconnected. A tree is a connected graph with no cycles. The degree of a vertex 𝑣 in a graph 𝐺, denoted by deg(𝑣), is the number of edges of 𝐺 incident with 𝑣 and deg(𝐺)=∑_𝑥∈𝑉deg(𝑥)=2|𝐸|. We denote by 𝛿(𝐺) and Δ(𝐺) the minimum and maximum degrees of the vertices of 𝐺, respectively. If 𝐺 and 𝐻 are two graphs such that 𝑉(𝐺)∩𝑉(𝐻)=∅, then the graph with vertex set 𝑉(𝐺)∪𝑉(𝐻) and edge set 𝐸(𝐺)∪𝐸(𝐻) is called the disjoint union of 𝐺 and 𝐻 and we denote it by 𝐺∪𝐻. Any graph 𝐺=(𝑉,𝐸) may be expressed uniquely as a disjoint union of connected graphs. These graphs are called the connected components or simply components of 𝐺. The number of components of 𝐺 is denoted by 𝑐(𝐺). A graph 𝐺 is called empty graph if 𝐸(𝐺)=∅. Moreover, a graph 𝐾 is called complete graph if 𝑥𝑦∈𝐸(𝐺), for any distinct elements 𝑥,𝑦∈𝑉(𝐺). For more details we refer to [13, 14].

Definition 2.8. Let 𝐺 and 𝐻 be two graphs such that 𝑉(𝐺)∩𝑉(𝐻)=∅. We denote the graph 𝐺+𝐻, for the graph, whose vertex set and edge set are 𝑉(𝐺)∪𝑉(𝐻) and 𝐸(𝐺)∪𝐸(𝐻)∪{𝑥𝑦∣𝑥∈𝑉(𝐺),𝑦∈𝑉(𝐻)}, respectively.

Definition 2.9 (see [4]). By the associated graph of 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra 𝑋, denoted Γ(𝑋), we means the graph whose vertices are just the elements of 𝑋, and for distinct 𝑥,𝑦∈Γ(𝑋), there is an edge connecting 𝑥 and 𝑦 if and only if 𝐿({𝑥,𝑦})={0}.

Definition 2.10 (see [13, 15]). Let 𝐺 be a finite graph with |𝑉(𝐺)|=𝑛. Then, the adjacency matrix of 𝐺 is an 𝑛×𝑛 matrix 𝐴_𝐺=[𝑎_𝑖,𝑗] such that 𝑎_𝑖,𝑗 is the number of edges joining 𝑣_𝑖 and 𝑣_𝑗. Moreover, we denote the characteristic polynomial of the matrix 𝐴_𝐺, by 𝜒(𝐺,𝜆). That is 𝜒(𝐺,𝜆)=det(𝜆𝐼−𝐴_𝐺), where 𝐼 is a 𝑛×𝑛 identity matrix.

Proposition 2.11 (see [15, Proposition 2.3]). Let 𝐺 be a graph and 𝜒(𝐺,𝜆)=𝜆^𝑛+𝑐₁𝜆^𝑛−1+𝑐₂𝜆^𝑛−2+⋯+𝑐_𝑛. be the characteristic polynomial of the adjacent matrix 𝐺. Then, 𝑐₁=0, −𝑐₂ is the number of edges of 𝐺 and −𝑐₃ is twice the number of triangles in 𝐺.

3. Graph Based on BCI-Algebras

From now on, in this paper, (𝑋,∗,0) or simply 𝑋 is a 𝐵𝐶𝐼-algebra, 𝐵 is a 𝐵𝐶𝐾-part and 𝑃 is a 𝑝-semisimple part of 𝑋, unless otherwise state. For all 𝑥,𝑦∈𝑋, we use 𝑥≨𝑦 to denote 𝑥≤𝑦 and 𝑥≠𝑦.

Definition 3.1. Note that the set of all 0-divisors of 𝑥and the set of all zero divisors of 𝑥 are the same. Let 𝑥∈𝑋. Then, there exists 𝑎∈𝑃 such that 𝑥∈𝑉(𝑎). We will use the notation 𝑍_𝑥 to denote the set of all 𝑦∈𝑋 such that 𝐿({𝑥,𝑦})={𝑎}, that is, 𝑍_𝑥={𝑦∈𝑋∣𝐿({𝑥,𝑦})={𝑎}}, which is called the set of 𝑎-divisor of 𝑋.

Note that the set of all 0-divisors of 𝑥 and the set of all zero divisors of 𝑥 are the same.

Lemma 3.2. Let 𝑎,𝑏∈𝑃 and 𝑥,𝑦∈𝑋. Then(i)one has 𝐿=(𝑎),({𝑥,𝑎})𝑎if𝑥∈𝑉∅otherwise, (3.1)(ii)if 𝑎≠𝑏, 𝑥∈𝑉(𝑎) and 𝑦∈𝑉(𝑏), then 𝐿({𝑥,𝑦})=∅.

Proof. (i) Clearly, if 𝑥∈𝑉(𝑎), then 𝐿({𝑥,𝑎})={𝑎}. Let 𝑥∉𝑉(𝑎). Then, 𝑦∈𝐿({𝑥,𝑎}) implies 𝑦≤𝑥 and 𝑦≤𝑎. Since 𝑎∈𝑃, we have 𝑎=𝑦 and so 𝑎≤𝑥, which is impossible. Therefore, 𝐿({𝑥,𝑎})=∅.
(ii) Since 𝑎≠𝑏, we have 𝐿({𝑥,𝑦})⊆𝑉(𝑎)∩𝑉(𝑏)=∅.

Lemma 3.3. For any 𝑥,𝑦∈𝑋, if 𝑥∗𝑦=0, then 𝐿({𝑥})⊆𝐿({𝑦}) and 𝑍_𝑦⊆𝑍_𝑥.

Proof. Let 𝑥∗𝑦=0, for some 𝑥,𝑦∈𝑋. Then, 𝑥≤𝑦. Clearly, 𝐿{𝑥}⊆𝐿{𝑦}. If 𝑥∈𝑉(𝑎), where 𝑎∈𝑃, then 𝑎≤𝑥≤𝑦 and so by (𝑃3), 𝑦∈𝑉(𝑎), too. Now, let 𝑢∈𝑍_𝑦. Then, 𝐿({𝑢,𝑦})={𝑎} and so 𝑎∈𝐿({𝑢,𝑥})⊆𝐿({𝑢,𝑦})={𝑎}. Hence, 𝐿({𝑥,𝑢})=𝑎 and so 𝑢∈𝑍_𝑥. Therefore, 𝑍_𝑦⊆𝑍_𝑥.

Jun and Lee in [4] proved that if 𝑥 is an element of 𝐵𝐶𝐾-algebra 𝑋, then the set of all zero divisors of 𝑥 is a quasi-ideal of 𝑋. In Theorem 3.4, we will show that if 𝑥 is an element of 𝐵𝐶𝐼-algebra 𝑋, then 𝑍_𝑥 is a quasi-ideal of 𝑋.

Theorem 3.4. 𝑍_𝑥 is a quasi-ideal of 𝑋, for any 𝑥∈𝑋.

Proof. Let 𝑥∈𝑋 such that 𝑥∈𝑉(𝑎), for some 𝑎∈𝑃 and 𝑢,𝑣∈𝑋 such that 𝑢∗𝑣=0 and 𝑣∈𝑍_𝑥. Then, 𝑢≤𝑣 and 𝐿({𝑣,𝑥})={𝑎}. Hence, 𝑣∈𝑉(𝑎) and so by (𝑃3), 𝑢∈𝑉(𝑎). Since 𝑎∈𝐿({𝑢,𝑥})⊆𝐿({𝑣,𝑥})={𝑎}, then 𝐿({𝑢,𝑥})={𝑎}. Therefore, 𝑢∈𝑍_𝑥 and so 𝑍_𝑥 is a quasi-ideal of 𝑋.

Definition 3.5. Let 𝑌⊆𝑋, and 𝐺(𝑌) be a simple graph, whose vertices are just the elements of 𝑌 and for distinct 𝑥,𝑦∈𝑌, there is an edge connecting 𝑥 and 𝑦, denoted by 𝑥𝑦 if and only if 𝐿({𝑥,𝑦})={𝑎}, for some 𝑎∈𝑃. If 𝑌=𝑋, then 𝐺(𝑋) is called a 𝐵𝐶𝐼-graph of 𝑋.

Clearly, if 𝑋 is a 𝐵𝐶𝐾-algebra, then 𝐺(𝑋)=Γ(𝑋). But it is not true, in general.

Example 3.6. (i) Let 𝑋={0,1,𝑎,𝑏,𝑐}. Define the binary operation “*’’ on 𝑋 by the following table:

Then, (𝑋,∗,0) is a 𝐵𝐶𝐼-algebra, 𝑃={0,𝑎}, 𝑉(0)={0,1}, and 𝑉(𝑎)={𝑎,𝑏,𝑐}. Moreover, 𝐿({1,0})={0} and 𝐿({𝑎,𝑏})=𝐿({𝑎,𝑐})=𝐿({𝑏,𝑐})={𝑎} and so 𝐸(𝐺(𝑋))={10,𝑎𝑐,𝑏𝑐,𝑎𝑏}. Therefore, the graphs 𝐺(𝑋) and Γ(𝑋) which are given by Figure 1 are different.
(ii) Let (𝑋,∗,0) be the 𝐵𝐶𝐼-algebra in Example 3.6(i) and 𝑌={0,1,𝑏,𝑐}. Then, 𝐺(𝑌) is given by Figure 2.

Figure 1 

Figure 2 

Example 3.7 (see [4]). Let 𝑋={0,𝑎,𝑏,𝑐}. Define the binary operation “*’’ on 𝑋 by the following table:

Then, (𝑋,∗,0) is a 𝐵𝐶𝐾-algebra and 𝑃={0} and so 𝐺(𝑋)=Γ(𝑋). Moreover, 𝐿({0,𝑎})=𝐿({0,𝑏})=𝐿({0,𝑐})=𝐿({𝑎,𝑐})=𝐿({𝑏,𝑐})=𝐿({𝑎,𝑏})={0}. Therefore, the graph 𝐺(𝑋) is given by Figure 3.

Figure 3 

Proposition 3.8. (i)  𝐺(𝑉(𝑎)) is a connected graph, for any 𝑎∈𝑃,
(ii)  𝐺(𝑋)=⋃_𝑎∈𝑃𝐺(𝑉(𝑎)),
(iii)  𝐺(𝑋) is a graph with |𝑃| components.

Proof. (i) Let 𝑎∈𝑃 and 𝑥,𝑦∈𝑉(𝑎). Then, by Lemma 3.2(i) 𝑥𝑎,𝑦𝑎∈𝐸(𝐺(𝑋)) and so there is a path from 𝑥 to 𝑦 in 𝐺(𝑋).
(ii) Since 𝑋=⋃_𝑎∈𝑃𝑉(𝑎), then 𝑉(𝐺(𝑋))=𝑉(⋃_𝑎∈𝑃𝐺(𝑉(𝑎))). Clearly, 𝐸(⋃_𝑎∈𝑃𝐺(𝑉(𝑎)))⊆𝐸(𝐺(𝑋)). Now, let 𝑥𝑦∈𝐸(𝐺(𝑋)). Then, 𝐿({𝑥,𝑦})={𝑎}, for some 𝑎∈𝑃 and so 𝑥𝑦∈𝐺(𝑉(𝑎)). Hence, 𝐸(⋃_𝑎∈𝑃𝐺(𝑉(𝑎)))=𝐸(𝐺(𝑋)). Therefore, 𝐺(𝑋)=⋃_𝑎∈𝑃𝐺(𝑉(𝑎)).
(iii) We want to show that there is not any path between elements of 𝑉(𝑎) and 𝑉(𝑏), for all distinct elements 𝑎,𝑏∈𝑃. Let 𝑎,𝑏 be distinct elements of 𝑃, 𝑥∈𝑉(𝑎) and 𝑦∈𝑉(𝑏). If there is a path 𝑥,𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦, which link 𝑥 to 𝑦, then 𝑎=𝐿({𝑥,𝑥₁})≠∅ and so by Lemma 3.2(ii), 𝑥₁∈𝑉(𝑎). By a similar way, we have 𝑥₂,…,𝑥_𝑛,𝑦∈𝑉(𝑎), which is impossible. Hence, there is not any path between 𝑥 and 𝑦. Therefore, by (i), 𝐺(𝑋) is a graph with |𝑃| components.

Theorem 3.9. 𝑋 is a 𝑝-semisimple 𝐵𝐶𝐼-algebra if and only if 𝐺(𝑋) is an empty graph.

Proof. Clearly, if 𝑋 is a 𝑝-semisimple 𝐵𝐶𝐼-algebra, then by Lemma 3.2(ii), 𝐺(𝑋) is an empty graph. Conversely, let 𝐺(𝑋) be an empty graph and 𝑥∈𝑋. Since, 0∗(0∗𝑥)∈𝑃 and 𝑥∈𝑉(0∗(0∗𝑥)), then 𝐿({0∗(0∗𝑥),𝑥})={0∗(0∗𝑥)}. If 0∗(0∗𝑥)≨𝑥, then there exists an edge between 𝑥 and 0∗(0∗𝑥) in 𝐺(𝑋), which is impossible. Hence, 𝑥=0∗(0∗𝑥) and so 𝑋 is a 𝑝-semisimple 𝐵𝐶𝐼-algebra.

Definition 3.10. Let 𝑎∈𝑃. The element 𝑥∈𝑉(𝑎) is called an 𝑎-atom if 𝑥≠𝑎 and 𝑦∗𝑥=0 implies 𝑦=𝑎 or 𝑦=𝑥, for all 𝑦∈𝑋.

Note that if 𝑋 is a 𝐵𝐶𝐾-algebra then the concept of 0-atom and atom are the same.

Lemma 3.11. Let 𝑋 be a 𝐵𝐶𝐼-algebra of finite order. Then, for any 𝑥∈𝑋−𝑃, there is an 𝑎-atom 𝑏∈𝑋, such that 𝑏≤𝑥, for some 𝑎∈𝑃.

Proof. Let 𝑥∈𝑋−𝑃. Then, there is 𝑎∈𝑃 such that 𝑥∈𝑉(𝑎). Let 𝑆={𝐿(𝑥,𝑢)∣𝑢∗𝑥=0, 𝑢∈𝑉(𝑎)−{𝑎}}. Clearly, 0∈𝑆. Since 𝑋 is of finite length, then 𝑆 has the greatest element. Let 𝐿(𝑥,𝑢) be the greatest element of 𝑆. Then, we show that 𝑣 is an 𝑎-atom of 𝑋. Let 𝑤∗𝑣=0, for some 𝑤∈𝑋. Then, 𝑤≤𝑣 and so by (𝑃3), 𝑤∈𝑉(𝑎). If 𝑤≨𝑣, then 1+𝐿(𝑥,𝑢)≤𝐿(𝑥,𝑤), which is impossible. Therefore, 𝑤=𝑣 or 𝑤=𝑎 and so 𝑣 is an atom of 𝑋.

Theorem 3.12. Let 𝑋 be a finite leght 𝐵𝐶𝐼-algebra and 𝑎∈𝑃. Then(i)𝐺(𝑉(𝑎)) is a tree if and only if 𝑉(𝑎)={𝑎} or 𝑉(𝑎) has only one 𝑎-atom.(ii)𝐺(𝑋) is a tree if and only if 𝑋={0} or 𝑋 is a 𝐵𝐶𝐾-algebra with only one atom.

Proof. (i) Let 𝐺(𝑉(𝑎)) be a tree. If 𝑉(𝑎)={𝑎}, we do not have any thing to prove. Let 𝑉(𝑎)≠{𝑎} and 𝐴 be the set of all 𝑎-atoms of 𝑉(𝑎). Then, by Lemma 3.11, we have 1≤|𝐴|. Let 𝑥,𝑦∈𝐴, 𝑢≤𝑥 and 𝑢≤𝑦, for some 𝑢∈𝑋. Then, by (𝑃3), 𝑢∈𝑉(𝑎). Since 𝑥,𝑦 are 𝑎-atoms of 𝑉(𝑎), then 𝑢=𝑎 or 𝑥=𝑢=𝑦. Hence, 𝑥=𝑦 or 𝑥𝑦∈𝐸(𝐺(𝑋)). If 𝑥𝑦∈𝐸(𝐺(𝑋)), then by Lemma 3.2(i), we have 𝑥𝑦,𝑎𝑥,𝑎𝑦∈𝐸(𝐺(𝑋)). Hence, 𝐺(𝑋) has a cycle, which is impossible. Therefore, 𝑥=𝑦 and so |𝐴|=1. Conversely, let 𝑉(𝑎) has only one 𝑎-atom. By Proposition 3.8, 𝐺(𝑉(𝑎)) is a connected graph. If 𝑉(𝑎)={𝑎}, then clearly, 𝐺(𝑉(𝑎)) is a tree. Let 𝑉(𝑎)≠{𝑎}, 𝑥,𝑦∈𝑉(𝑎)−{𝑎} and 𝑢 be an 𝑎-atom of 𝑋. Then, by Lemma 3.11, 𝑢∈𝐿({𝑥,𝑦}) and so 𝐿({𝑥,𝑦})≠{𝑎}. Hence, 𝐸(𝐺(𝑋))={𝑥𝑎∣𝑥∈𝑉(𝑎)−{𝑎}} and so 𝐺(𝑉(𝑎)) does not have any cycle. Therefore, 𝐺(𝑉(𝑎)) is a tree.
(ii) Let 𝐺(𝑋) be a tree. Then, 𝐺(𝑋) is a connected graph and so by Proposition 3.8, |𝑃|=1. Hence, 𝑃={0} and so, 𝑋 is a 𝐵𝐶𝐾-algebra. Since 𝑋 is a 𝐵𝐶𝐾-algebra, we have 𝑋=𝑉(0) and so by (i), 𝑋={0} or 𝑋 is a 𝐵𝐶𝐾-algebra with only one atom. The converse is straight consequent of (i).

Example 3.13. Let 𝑋=ℝ⁺={𝑥∈ℝ∣0≤𝑥} and 𝑥∗𝑦=max{𝑥−𝑦,0}, for any 𝑥,𝑦∈𝑋. Then, (𝑋,∗,0) is a 𝐵𝐶𝐾-algebra. Clearly, 𝑋 does not have any atoms. Since 𝑋 is a 𝐵𝐶𝐾-algebra, then by Proposition 3.8, 𝐺(𝑋) is a connected graph. Moreover, 𝐿({𝑥,𝑦})≠{0}, for any 𝑥,𝑦∈𝑋−{0}. Hence, 𝐸(𝐺(𝑋))={𝑥0∣𝑥∈𝑋−{0}}. Therefore, 𝐺(𝑋) is a tree.

Example 3.14. Let “*’’ be the binary operation in Example 3.13, 𝐴={𝑎_𝑖∣𝑖∈ℝ⁺} and 𝐵={𝑏_𝑖∣𝑖∈ℝ⁺}, where 𝑎₀=0=𝑏₀ and 𝐴∩𝐵={0}. Let a binary operation ∗^′ on 𝑋 is defined as follows: for all 𝑖,𝑗∈ℝ⁺, 𝑎_𝑖∗^′𝑎_𝑗=𝑎_𝑖∗𝑗;𝑏_𝑖∗^′𝑏_𝑗=𝑏_𝑖∗𝑗;𝑎_𝑖∗^′𝑏_𝑗=𝑎_𝑖;𝑏_𝑖∗^′𝑎_𝑗=𝑏_𝑖. (3.2) By Example 3.13, (𝐴,∗₁,0) and (𝐵,∗₂,0) are two 𝐵𝐶𝐾-algebras, where 𝑎_𝑖∗₁𝑎_𝑗=𝑎_𝑖∗𝑗 and 𝑏_𝑖∗₂𝑏_𝑗=𝑏_𝑖∗𝑗, for any 𝑖,𝑗∈ℝ⁺. Hence, by Proposition 2.5, (𝑋,∗^′,0) is a 𝐵𝐶𝐾-algebra and so by Proposition 3.8, 𝐺(𝑋) is a connected graph. Let 𝑎_𝑖∈𝐴−{𝑎₀} and 𝑏_𝑖∈𝐵−{𝑎₀}. Since 𝑋 is a 𝐵𝐶𝐾-algebra, then 0∈𝐿({𝑎_𝑖,𝑏_𝑖}). Let 𝑢∈𝐿({𝑎_𝑖,𝑏_𝑖}), for some 𝑢∈𝑋. If 𝑢=𝑏_𝑗, for some 𝑗∈ℝ⁺, then 𝑢=𝑢∗^′𝑎_𝑖=0. If 𝑢=𝑎_𝑗, for some 𝑗∈ℝ⁺, then 𝑢=𝑢∗^′𝑏_𝑖=0. Hence, 𝐿({𝑎_𝑖,𝑏_𝑖})={0} and so 𝑎_𝑖𝑏_𝑖∈𝐸(𝐺(𝑋)). Since 0𝑎_𝑖,0𝑏_𝑖,𝑎_𝑖𝑏_𝑖∈𝐸(𝐺(𝑋)), then 𝐺(𝑋) has a cycle and so it is not a tree. But, 𝑋 does not have any atoms.

Proposition 3.15. Let 𝑋 be a 𝐵𝐶𝐼-algebra such that the set {|𝑉(𝑎)|∶𝑎∈𝑃} is bounded. Then, Δ(𝐺(𝑋))=max{|𝑉(𝑎)|−1∶𝑎∈𝑃}.

Proof. Since the set {|𝑉(𝑎)|∶𝑎∈𝑃} is bounded, then there is a 𝑢∈𝑃, such that |𝑉(𝑢)|−1=max{|𝑉(𝑎)|−1∶𝑎∈𝑃}. We show that deg(𝑥)≤deg(𝑢), for any 𝑥∈𝑋. Let 𝑥∈𝑋. Then, there is 𝑎∈𝑃 such that 𝑥∈𝑉(𝑎). Since 𝑉(𝑎)∩𝑉(𝑏)=∅, for any 𝑏∈𝑃−{𝑎}, then by Proposition 3.8(ii) and Lemma 3.2(i), deg(𝑥)≤|𝑉(𝑎)|−1≤|𝑉(𝑢)|−1=deg(𝑢). Therefore, Δ(𝐺(𝑋))=deg(𝑢)=|𝑉(𝑢)|−1=max{|𝑉(𝑎)|−1∶𝑎∈𝑃}.

Corollary 3.16. (i) If 𝑋 is a finite 𝐵𝐶𝐼-algebra, then Δ(𝐺(𝑋))=max{|𝑉(𝑎)|−1∶𝑎∈𝑃}.
(ii) If 𝑋 is a finite 𝐵𝐶𝐾-algebra, then Δ(𝐺(𝑋))=|𝑋|−1.

Proof. (i) If 𝑋 is a finite 𝐵𝐶𝐼-algebra, then {|𝑉(𝑎)|∶𝑎∈𝑃} is bounded. Therefore, by Proposition 3.15, Δ(𝐺(𝑋))=max{|𝑉(𝑎)|−1∶𝑎∈𝑃}.
(ii) If 𝑋 is a 𝐵𝐶𝐾-algebra, then 𝑃={0} and so {|𝑉(𝑎)|−1∶𝑎∈𝑃}={𝑉(0)−1}={|𝑋|−1}. Hence, by Proposition 3.15, Δ(𝐺(𝑋))=|𝑋|−1.

Definition 3.17. Let 𝐺 be a graph with 𝑛+1 vertices. Then 𝐺 is called an 𝑛-star graph if it has Figure 4.

Figure 4 

𝑛-star.

Theorem 3.18. Let 𝑋 be a 𝐵𝐶𝐾-algebra with 𝑚+1 elements and only one atom. Then 𝐺(𝑋) is an 𝑚-star graph. Moreover, if 𝐺 is an 𝑚-star graph, then there is a 𝐵𝐶𝐾-algebra 𝑋 such that 𝐺(𝑋)=𝐺.

Proof. Since, 𝑋 is a 𝐵𝐶𝐾-algebra, then 𝐺(𝑋) is a connected graph. Let 𝑎 be an atom of 𝑋. Then by Lemma 3.11, 𝑎≤𝑥, for all 𝑥∈𝑋−{0} and so 𝐸(𝐺(𝑋))={0𝑥∣𝑥∈𝑋−{0}}. Therefore, 𝐺(𝑋) is a 𝑚-star graph. Now, let 𝐺 be a 𝑚-star graph, for some 𝑚∈ℕ and 𝑋=𝑉(𝐺)={𝑥₀,𝑥₁,…,𝑥_𝑚} be a chain such that 𝑥_𝑖<𝑥_𝑖+1, for any 𝑖∈{0,1,…,𝑚−1}. Define the binary operation ∗^′ on 𝑋 by 𝑥_𝑖∗^′𝑥_𝑗=𝑥_𝑖∗𝑗, where 𝑖∗𝑗=max{0,𝑖−𝑗}, for any 𝑖,𝑗∈{0,1,…,𝑚}. By Theorem 2.3, 𝑋 is a commutative 𝐵𝐶𝐾-chain and so 𝐿({𝑥,𝑦})≠{0}, for any 𝑥,𝑦∈𝑋−{0}. Hence, by Lemma 3.2(i), 𝐸(𝐺(𝑋))={𝑥₀𝑥₁,𝑥₀𝑥₂,…,𝑥₀𝑥_𝑚} and so 𝐺(𝑋) is an 𝑚-star graph.

Theorem 3.19. 𝑋 is a commutative 𝐵𝐶𝐼-algebra and 𝑥∗𝑦=𝑥∗𝑎, for any 𝑎∈𝑃 and 𝑥,𝑦∈𝑉(𝑎), where 𝑥≠𝑦 if and only if 𝐺(𝑋) is a graph with complete components.

Proof. Let 𝐺(𝑋) be a graph with complete components, 𝑎∈𝑃 and 𝑥,𝑦∈𝑉(𝑎) such that 𝑥≠𝑦. Since 𝑥∗𝑦∈𝑉(0) by (𝐵𝐶𝐼6), we get 𝑥∗(𝑥∗𝑦)∈𝐿({𝑥,𝑦}). Now, since 𝑥,𝑦∈𝑉(𝑎) and 𝐺(𝑋) is a graph with complete components, then by Proposition 3.8, 𝑥𝑦∈𝐸(𝐺(𝑋)) and so 𝐿({𝑥,𝑦})={𝑎}. Hence, 𝑥∗(𝑥∗𝑦)=𝑎 and so by (𝐵𝐶𝐼6) and (𝐵𝐶𝐼4), we get 𝑥∗𝑎≤𝑥∗𝑦. On the other hand, since 𝑎≤𝑦, then by (𝐵𝐶𝐼7), 𝑥∗𝑦≤𝑥∗𝑎. Therefore, 𝑥∗𝑦=𝑥∗𝑎. Now, we show that 𝑋 is a commutative 𝐵𝐶𝐼-algebra. Let 𝑎∈𝑃 and 𝑥,𝑦∈𝑉(𝑎). Clearly, if 𝑥=𝑦, then 𝑥∗(𝑥∗𝑦)=𝑦∗(𝑦∗𝑥). If 𝑥≠𝑦, then 𝑥∗(𝑥∗𝑦)=𝑥∗(𝑥∗𝑎)≤𝑎. Since 𝑎∈𝑃, we get 𝑥∗(𝑥∗𝑎)=𝑎. By the similar way, we get 𝑦∗(𝑦∗𝑥)=𝑎 and so 𝑥∗(𝑥∗𝑦)=𝑎=𝑦∗(𝑦∗𝑥). Hence, 𝑋 is branchwise commutative. Therefore, by Lemma 2.2, 𝑋 is commutative 𝐵𝐶𝐼-algebra. Conversely, let 𝑋 be a commutative 𝐵𝐶𝐼-algebra such that 𝑥∗𝑦=𝑥∗𝑎, for any 𝑎∈𝑃 and distinct elements 𝑥,𝑦∈𝑉(𝑎). Let 𝑎∈𝑃 and 𝑥,𝑦∈𝑉(𝑎). If 𝑢∈𝐿({𝑥,𝑦}), then by (𝑃3), 𝑢∗𝑎=𝑢∗𝑥=0 and so 𝑢≤𝑎. Since 𝑢∈𝑃, we have 𝑎=𝑢. Hence, 𝐿({𝑥,𝑦})={𝑎} and so 𝑥𝑦∈𝐸(𝐺(𝑋)). Therefore, by Proposition 3.8, all components of 𝐺(𝑋) are complete graph.

Corollary 3.20. The graph 𝐺(𝑋) is a complete graph if and only if 𝑋 is a 𝐵𝐶𝐾-algebra and 𝑥∗𝑦=0if𝑦=𝑥,𝑥otherwise. (3.3)

Definition 3.21 (see [10]). Let (𝑋₁,∗₁,0) be a 𝐵𝐶𝐾-algebra and (𝑋₂,∗₂,0) be a 𝐵𝐶𝐼-algebra such that 𝑋₁∩𝑋₂={0}. Define the binary operation “*’’ on 𝑋₁∪𝑋₂ by 𝑥∗𝑦=𝑥∗₁𝑦if𝑥,𝑦∈𝑋₁;𝑥∗₂𝑦if𝑥,𝑦∈𝑋₂;0∗₂𝑦if𝑥∈𝑋₁,𝑦∈𝑋₂−{0};𝑥if𝑥∈𝑋₂,𝑦∈𝑋₁. (3.4) The (𝑋₁∪𝑋₂,∗,0) is a 𝐵𝐶𝐼-algebra, whose 𝐵𝐶𝐾-part 𝐵 contains 𝑋₁ and 𝑝-semisimple part 𝑃 is contained in 𝑋₂. This algebra is called the Li’s union of 𝑋₁ and 𝑋₂ and is denoted by 𝑋₁∪_𝐿𝑋₂.

Lemma 3.22. Let (𝑋₁,∗₁,0) be a 𝐵𝐶𝐾-algebra, (𝑋₂,∗₂,0) be a 𝐵𝐶𝐼-algebra such that 𝑋₁∩𝑋₂={0} and 𝑌 be the Li’s union of 𝑋₁ and 𝑋₂. Then 𝐿_𝑌=𝐿𝑥,𝑦^𝑋1𝑥,𝑦if𝑥,𝑦∈𝑋₁;𝐿_𝑋2𝑥,𝑦if𝑥,𝑦∈𝑋₂−𝑉_𝑋2(0);𝐿_𝑋2𝑥,𝑦∪𝑋₁if𝑥,𝑦∈𝑉_𝑋2(0)−{0};∅otherwise. (3.5)

Proof. If 𝑥,𝑦∈𝑋₁, then 𝑢∈𝐿_𝑋1𝑥,𝑦⟺𝑢∗₁𝑥=𝑢∗𝑥=0=𝑢∗𝑦=𝑢∗₁𝑦⟺𝑢∈𝐿_𝑌𝑥,𝑦. (3.6) Hence, 𝐿_𝑋1({𝑥,𝑦})=𝐿_𝑌({𝑥,𝑦}). Let 𝑥,𝑦∈𝑉_𝑋2(𝑎), for some 𝑎∈𝑃−{0}. Then by Definition 3.21, 𝑎∈𝑄−{0}, where 𝑄 is a 𝑝-semisimple part of 𝑌. Let 𝑢∈𝐿_𝑌({𝑥,𝑦}). Then by (𝑃3), 𝑢∈𝑉_𝑌(𝑎) and so 𝑢∈𝑋₂−{0}. Hence, 𝑢∗₂𝑥=𝑢∗𝑥=0=𝑢∗𝑦=𝑢∗₂𝑦 and so 𝑢∈𝐿_𝑋2({𝑥,𝑦}). Now, let 𝑢∈𝐿_𝑋2({𝑥,𝑦}). Then 𝑢∗𝑥=𝑢∗₂𝑥=0=𝑢∗₂𝑦=𝑢∗𝑦 and so 𝑢∈𝐿_𝑌({𝑥,𝑦}). Hence, 𝐿_𝑋2({𝑥,𝑦})=𝐿_𝑌({𝑥,𝑦}), for any 𝑎∈𝑃−{0} and 𝑥,𝑦∈𝑉(𝑎). If 𝑥,𝑦∈𝑉_𝑋2(0)−{0}, then 𝑢∗𝑥=0∗₂𝑥=0 and 𝑢∗𝑦=0∗₂𝑦=0, for any 𝑢∈𝑋₁. Clearly, by definition of “*’’, 𝐿_𝑋2({𝑥,𝑦})⊆𝐿_𝑌({𝑥,𝑦}) and so 𝐿_𝑋2({𝑥,𝑦})∪𝑋₁⊆𝐿_𝑌({𝑥,𝑦}). Let 𝑢∈𝐿_𝑌({𝑥,𝑦}). Then 𝑢∗𝑥=0. Since 𝑥∈𝑉_𝑋2(0)−{0} by definition of “*’’, 𝑢∈𝑋₁ or 𝑢∈𝐿_𝑋2({𝑥,𝑦}) and so 𝐿_𝑌({𝑥,𝑦})⊆𝐿_𝑋2({𝑥,𝑦})∪𝑋₁. The final part is straight consequent of Lemma 3.2.

Proposition 3.23. Let (𝑋₁,∗₁,0) be a 𝐵𝐶𝐾-algebra, (𝑋₂,∗₂,0) be a 𝐵𝐶𝐼-algebra such that 𝑋₁∩𝑋₂={0}, 𝑋₁≠{0} and 𝑆={𝑥𝑦∈𝐸(𝐺(𝑋₂))∣𝑥,𝑦∈𝑉_𝑋2(0)−{0}}. Then 𝐺(𝑋₁∪_𝐿𝑋₂)=(𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆.

Proof. Clearly, 𝑉(𝐺(𝑋₁∪_𝐿𝑋₂))=𝑋₁∪𝑋₂=𝑉((𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆). Let 𝑌=𝑋₁∪_𝐿𝑋₂, 𝑄 be a 𝑝-semisimple part of 𝑌 and 𝑥𝑦∈𝐸(𝐺(𝑌)). Then 𝑥≠𝑦 and 𝐿_𝑌{𝑥,𝑦}={𝑎}, for some 𝑎∈𝑄. If 𝑎≠0, then by Definition 3.21, 𝑥,𝑦∈𝑋₂ and so by Lemma 3.22, 𝐿_𝑋2({𝑥,𝑦})={𝑎}. Hence, 𝑥𝑦∈𝐸(𝐺(𝑋₂)−𝑆)⊆𝐸((𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆). Let 𝑎=0. Then 𝐿_𝑌({𝑥,𝑦})={0} and so 𝑥,𝑦∈𝑉_𝑌(0)=𝑉_𝑋1(0)∪𝑉_𝑋2(0). If 𝑥,𝑦∈𝑉_𝑋1(0), then by Lemma 3.22, 𝑥𝑦∈𝐸(𝐺(𝑋₁))⊆𝐸((𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆). If 𝑥=0 or 𝑦=0, then clearly, 𝑥𝑦∈𝐸(𝐺(𝑋₁)∪𝐺(𝑋₂)−𝑆). If 𝑥,𝑦∈𝑉_𝑋2(0)−{0}, then by Lemma 3.22, 𝐿_𝑋2({𝑥,𝑦})∪𝑋₁⊆𝐿_𝑌({𝑥,𝑦})={0} and so 𝑋=0, which is impossible. If 𝑥∈𝑉_𝑋1(0)−{0} and 𝑦∈𝑉_𝑋2(0)−{0}, then 𝑥∗𝑦=0∗𝑦=0 and so 𝑥∈𝐿_𝑌({𝑥,𝑦})={0}, which is impossible. By the similar way, we get that 𝑥∈𝑉_𝑋2(0)−{0} and 𝑦∈𝑉_𝑋1(0)−{0} is impossible. Hence, 𝐸(𝐺(𝑌))⊆𝐸(𝐺(𝑋₁)∪𝐺(𝑋₂)−𝑆). Now, let 𝑥𝑦∈𝐸((𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆). Then 𝑥≠𝑦 and 𝑥𝑦∈𝐸(𝐺(𝑋₁)) or 𝑥𝑦∈𝐸(𝐺(𝑋₂)−𝑆). By Lemma 3.22, 𝐸(𝐺(𝑋₁))⊆𝐸(𝐺(𝑌)). Let 𝑥𝑦∈𝐸(𝐺(𝑋₂)−𝑆). Then 𝐿_𝑋2({𝑥,𝑦})={𝑎}, for some 𝑎∈𝑃₂ and 𝑎∉𝑆, where 𝑃₂ is a 𝑝-semisimple part of 𝑋₂. If 𝑎≠0, then 𝑎∈𝑄−{0} and by Lemma 3.22, 𝐿_𝑌({𝑥,𝑦})={𝑎}. Hence, by Lemma 3.2, 𝑥𝑦∈𝐸(𝐺(𝑌)). If 𝑎=0, then 𝑥,𝑦∈𝑉_𝑋2(0) and so 𝑥𝑦∈𝐸(𝐺(𝑋₂)−𝑆) implies 𝑥=0 or 𝑦=0. Hence, 𝑥𝑦∈𝐸(𝐺(𝑌)). Therefore, 𝐸((𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆)⊆𝐸(𝐺(𝑌)) and so 𝐸((𝐺(𝑋₁)∪𝐺(𝑋₂))−𝑆)=𝐸(𝐺(𝑌)).

Corollary 3.24. Let (𝑋₁,∗₁,0) be a 𝐵𝐶𝐾-algebra, (𝑋₂,∗₂,0) be a 𝐵𝐶𝐼-algebra such that 𝑋₁∩𝑋₂=∅ and the graph 𝐺(𝑉_𝑋2(0)) be a tree. Then 𝐺(𝑋₁∪_𝐿𝑋₂)=𝐺(𝑋₁)∪𝐺(𝑋₂).

Proof. If 𝑋₁={0}, then 𝑋₂=𝑋₁∪_𝐿𝑋₂ and so 𝐺(𝑋₁∪_𝐿𝑋₂)=𝐺(𝑋₂)=𝐺(𝑋₁)∪𝐺(𝑋₂). Now, let 𝐺(𝑉_𝑋2(0)) be a tree. Then for any distinct elements 𝑥,𝑦∈𝑉_𝑋2(0)−{0}, we have 𝑥𝑦∉𝐸(𝐺(𝑉_𝑋2(0)). Hence, by Proposition 3.8(ii), 𝑥𝑦∉𝐸(𝐺(𝑋₂)) and so 𝑆=∅, where 𝑆={𝑥𝑦∈𝐸(𝐺(𝑋₂))∣𝑥,𝑦∈𝑉_𝑋2(0)−{0}}. Therefore, by Proposition 3.23, 𝐺(𝑋₁∪_𝐿𝑋₂)=𝐺(𝑋₁)∪𝐺(𝑋₂).

Example 3.25. (i) Let 𝑋₁={0,𝑎,𝑏,𝑐} and 𝑋₂={0,1,2,3}. Define the binary operations ∗₁ and ∗₂ on 𝑋₁ and 𝑋₂, respectively, by the following tables:

Then (𝑋₁,∗₁,0) and (𝑋₂,∗₂,0) are 𝐵𝐶𝐾-algebras (see [10]). Also, 𝐺(𝑋₁), 𝐺(𝑋₂) and 𝐺(𝑋₁)∪𝐺(𝑋₂) are given by Figure 5.
Let 𝑌=𝑋₁∪_𝐿𝑋₂. Then 𝐿_𝑌({𝑥,0})={0}, for any 𝑥∈𝑌. Hence, by Lemma 3.22, 𝑥∈𝐿_𝑌({𝑥,𝑦}) and 𝐿_𝑌({0,𝑧})={0}=𝐿_𝑌({𝑎,𝑏}), for any 𝑥∈{1,2,3}, 𝑧∈𝑋₁ and 𝑦∈{𝑎,𝑏,𝑐}. Moreover, 𝑋₁⊆𝐿_𝑌({𝑎,𝑏})∩𝐿_𝑌({𝑎,𝑐})∩𝐿_𝑌({𝑏,𝑐}). Hence, 𝐸(𝐺(𝑌))={0𝑥∣𝑥∈𝑌−{0}}∪{𝑎𝑏} and so 𝐺(𝑌) is given by Figure 6.
If 𝑆 is the set was defined in the Proposition 3.23, then 𝑆={13,23}. Therefore, 𝐺(𝑌)=𝐺(𝑋₁)∪(𝐺(𝑋₂)−𝑆).
(ii) Let 𝑋 be the 𝐵𝐶𝐼-algebra was defined in Example 3.6(i) and 𝑋₁ be the 𝐵𝐶𝐾-algebra in (i). Then the 𝐵𝐶𝐾-part of 𝑋₂ is a tree. Hence, by Corollary 3.24, 𝐺(𝑋₁∪_𝐿𝑋)=𝐺(𝑋₁)∪𝐺(𝑋).

Figure 5 

Figure 6 

Lemma 3.26. Let (𝑋,∗₁,0) and (𝑌,∗₂,0) be two 𝐵𝐶𝐾-algebras and 𝑥,𝑦∈𝑋⊕𝑌. Then 𝐿_𝑋⊕𝑌=𝐿𝑥,𝑦𝐿^𝑋𝑥,𝑦if𝑥,𝑦∈𝑋;_𝑌{0}𝑥,𝑦if𝑥,𝑦∈𝑌;otherwise. (3.7)

Proof. Let 𝑥,𝑦∈𝑋∪𝑌.(1)If 𝑥,𝑦∈𝑋, then for any 𝑢∈𝐿_𝑋{𝑥,𝑦}, we have 𝑢∗𝑥=0=𝑢∗𝑦. If 𝑢∈𝑌, then by Proposition 2.5, 𝑢=𝑢∗𝑥=0 and so 𝑢∈𝑋. Moreover, if 𝑢∈𝑋, then 𝑢∗₁𝑥=𝑢∗𝑥=0 and 𝑢∗₁𝑦=𝑢∗𝑦=0 and so 𝑢∈𝐿_𝑋{𝑥,𝑦}. Hence, 𝐿_𝑋⊕𝑌{𝑥,𝑦}⊆𝐿_𝑋{𝑥,𝑦}. Now, let 𝑢∈𝐿_𝑋{𝑥,𝑦}. Then 𝑢,𝑥,𝑦∈𝑋 and 𝑢∗₁𝑥=0=𝑢∗₁𝑦 and so 𝑢∗𝑥=𝑢∗₁𝑥=0=𝑢∗₁𝑦=𝑢∗𝑦. Hence, 𝐿_𝑋{𝑥,𝑦}⊆𝐿_𝑋⊕𝑌{𝑥,𝑦}. Therefore, 𝐿_𝑋⊕𝑌{𝑥,𝑦}=𝐿_𝑋{𝑥,𝑦}, for all 𝑥,𝑦∈𝑋. By the similar way, we can prove that 𝐿_𝑋⊕𝑌{𝑥,𝑦}=𝐿_𝑌{𝑥,𝑦}, for all 𝑥,𝑦∈𝑌.(2)If 𝑥∈𝑋 and 𝑦∈𝑌. Since 𝑋⊕𝑌 is a 𝐵𝐶𝐾-algebra, we have 0∈𝐿_𝑋⊕𝑌{𝑥,𝑦}. Let 𝑢∈𝐿_𝑋⊕𝑌{𝑥,𝑦}. Then 𝑢∗𝑥=0=𝑢∗𝑦. Since 𝑥∈𝑋 and 𝑦∈𝑌, then by definition of “*’’, we have 𝑢∗𝑥=𝑢 or 𝑢∗𝑦=𝑢 and so 𝑢=0. Therefore, 𝐿_𝑋⊕𝑌{𝑥,𝑦}={0}. By the similar way, if 𝑥∈𝑌 and 𝑦∈𝑋, then we can prove that 𝐿_𝑋⊕𝑌{𝑥,𝑦}={0}.

Theorem 3.27. Let (𝑋,∗₁,0) and (𝑌,∗₂,0) be two 𝐵𝐶𝐾-algebras. Then 𝐺(𝑋⊕𝑌)−{0}=(𝐺(𝑋)−{0})+(𝐺(𝑌)−{0}).

Proof. Clearly, 𝑉(𝐺(𝑋⊕𝑌)−{0})=𝑉((𝐺(𝑋)−{0})+(𝐺(𝑌)−{0})). Let 𝑥𝑦∈𝐸(𝐺(𝑋⊕𝑌)−{0}). Then 𝑥≠0, 𝑦≠0 and 𝐿_𝑋⊕𝑌{𝑥,𝑦}={0}. If 𝑥,𝑦∈𝑋 or 𝑥,𝑦∈𝑌, then by Lemma 3.26, 𝐿_𝑋{𝑥,𝑦}={0} or 𝐿_𝑌{𝑥,𝑦}={0} and so 𝑥𝑦∈𝐸(𝐺(𝑋)−{0}) or 𝑥𝑦∈𝐸(𝐺(𝑌)−{0}). Hence, by Definition 2.8, 𝑥𝑦∈𝐸((𝐺(𝑋)−{0})+(𝐺(𝑌)−{0})). If 𝑥∈𝑋 and 𝑦∈𝑌 or 𝑥∈𝑌 and 𝑦∈𝑋, then 𝑥∈𝑉((𝐺(𝑋)−{0})) and 𝑦∈𝑉(𝐺(𝑌)−{0}) and so 𝑥𝑦∈𝐸((𝐺(𝑋)−{0})+(𝐺(𝑌)−{0})). Now, let 𝑥𝑦∈𝐸((𝐺(𝑋)−{0})+(𝐺(𝑌)−{0})). Then 𝑥𝑦∈𝐸(𝐺(𝑋)−{0})∪𝐸(𝐺(𝑌)−{0}) or 𝑥∈𝑋−{0} and 𝑦∈𝑌−{0}. If 𝑥𝑦∈𝐸(𝐺(𝑋)−{0})∪𝐸(𝐺(𝑌)−{0}), then 𝑥≠0, 𝑦≠0 and 𝐿_𝑋{𝑥,𝑦}={0} or 𝐿_𝑌{𝑥,𝑦}={0}. Hence, by Lemma 3.26, 𝑥𝑦∈𝐸(𝐺(𝑋⊕𝑌)−{0}). If 𝑥∈𝑋−{0} and 𝑦∈𝑌−{0}, then by Lemma 3.26, 𝐿_𝑋⊕𝑌={0} and so 𝑥𝑦∈𝐸(𝐺(𝑋⊕𝑌)−{0}). Therefore, 𝐸(𝐺(𝑋⊕𝑌)−{0})=𝐸((𝐺(𝑋)−{0})+(𝐺(𝑌)−{0})).

Corollary 3.28. Let (𝑋,∗₁,0) and (𝑌,∗₂,0) be two finite 𝐵𝐶𝐾-algebras. Then |𝐸(𝑋⊕𝑌)|=(|𝑋|+|𝑌|−1)+(|𝑋|−1)⋅(|𝑌|−1).

Proof. Straightforward.

Example 3.29. Let 𝑋={𝑎₀,𝑎₁,𝑎₂,𝑎₃}, 𝑌={𝑏₀,𝑏₁,𝑏₂}, 𝑎₀=𝑏₀ and “*’’ be the operation was defined in Example 3.13. Then (𝑋,∗₁,𝑎₀) and (𝑌,∗₂,𝑏₀) are 𝐵𝐶𝐾-chain, where 𝑎_𝑖∗₁𝑎_𝑗=𝑎_𝑖∗𝑗 and 𝑏_𝑖∗₁𝑏_𝑗=𝑏_𝑖∗𝑗. Since they are 𝐵𝐶𝐾-chain, then by Lemma 3.2, 𝐺(𝑋) is 3-star graph and 𝐺(𝑌) is 2-star graph and so 𝐺(𝑋)−{0} and 𝐺(𝑌)−{0} have Figure 7.
Hence, by Definition 2.8, (𝐺(𝑋)−{0})+(𝐺(𝑌)−{0}) is given by Figure 8.
On the other hand, by Lemma 3.26, 𝐺(𝑋⊕𝑌) is given by Figure 9 and so 𝐺(𝑋⊕𝑌)−{0} is the graph on Figure 8.

Figure 7 

Figure 8 

(𝐺(𝑋)−{0})+(𝐺(𝑌)−{0}).

Figure 9 

𝐺(𝑋⊕𝑌)−{0}.

Let 𝐼 be an ideal of 𝑋. Define a binary relation 𝜃 on 𝑋 as follows: (𝑥,𝑦)∈𝜃 if and only if 𝑥∗𝑦,𝑦∗𝑥∈𝐼, for all 𝑥,𝑦∈𝑋. Then, 𝜃 is a congruence relation and it is called the equivalence relation induced by 𝐼. If 𝑋/𝐼={[𝑥]∣𝑥∈𝑋}, then (𝑋/𝐼,∗,[0]) is a 𝐵𝐶𝐼-algebra, where [𝑥]∗[𝑦]=[𝑥∗𝑦], for all 𝑥,𝑦∈𝑋 (see [10]). Moreover, let 𝐺=(𝑉,𝐸) be a graph and Π be a partition of 𝑋. The graph whose vertexes are the elements of Π and for distinct elements 𝑢,𝑣∈Π, there is an edge connecting 𝑢 and 𝑣 if and only if 𝑥𝑦∈𝐸, for some 𝑥∈𝑢 and 𝑦∈𝑣, is denoted by 𝐺/Π. Now, we want to verify the relation between the 𝐺(𝑋/𝐼) and 𝐺(𝑋)/Π, where 𝐼 is an ideal of 𝑋, 𝜃 is a congruence relation induced by 𝐼 and Π is a partition induced by 𝜃.

Theorem 3.30. Let 𝐼 be an ideal of 𝑋 and Π be the partition of 𝑋 induced by 𝐼.(i)If 𝑋 is a 𝐵𝐶𝐾-algebra, then 𝐺(𝑋)/Π is a subgraph of 𝐺(𝑋/𝐼).(ii)If 𝑋/𝐼 is a commutative 𝐵𝐶𝐼-algebra, then 𝐺(𝑋)/Π is a subgraph of 𝐺(𝑋/𝐼).

Proof. (i) Clearly, 𝑉(𝐺(𝑋/𝐼))={[𝑥]∣𝑥∈𝑋}=𝑉(𝐺(𝑋)/Π). Let [𝑥][𝑦]∈𝐸(𝐺(𝑋)/Π). Then there are 𝑢∈[𝑥] and 𝑣∈[𝑦], such that 𝑢𝑣∈𝐸(𝐺(𝑋)). Hence, 𝐿_𝑋{𝑢,𝑣}={0}. Let [𝑧]∈𝐿_𝑋/𝐼{[𝑥],[𝑦]}. Since [𝑢]=[𝑥] and [𝑣]=[𝑦], then [𝑧∗𝑢]=[𝑧]∗[𝑢]=[0]=[𝑧]∗[𝑣]=[𝑧∗𝑣] and so 𝑧∗𝑢∈𝐼 and 𝑧∗𝑣∈𝐼. Hence, 𝑧∗𝑢=𝑎 and 𝑧∗𝑣=𝑏, for some 𝑎,𝑏∈𝐼. Since 𝑋 is a 𝐵𝐶𝐾-algebra, then by (𝐵𝐶𝐼6), (𝑧∗𝑎)∗𝑏∈𝐿_𝑋{𝑢,𝑣}={0}. Since 𝐼 is an ideal of 𝑋 and 𝑎,𝑏∈𝐼, then 𝑧∈𝐼 and so 𝑧∗0∈𝐼 and 0∗𝑧=0∈𝐼. Hence, [𝑧]=[0] and so 𝐿_𝑋/𝐼{[𝑥],[𝑦]}={[0]}. Therefore, [𝑥][𝑦]∈𝐸(𝐺(𝑋/𝐼)) and so 𝐺(𝑋)/Π is a subgraph of 𝐺(𝑋/𝐼).
(ii) Clearly, 𝑉(𝐺(𝑋/𝐼))={[𝑥]∣𝑥∈𝑋}=𝑉(𝐺(𝑋)/Π). Let [𝑥][𝑦]∈𝐸(𝐺(𝑋)/Π). Then there are 𝑢∈[𝑥] and 𝑣∈[𝑦], such that 𝑢𝑣∈𝐸(𝐺(𝑋)) and so 𝐿_𝑋{𝑢,𝑣}={𝑎}, for some 𝑎∈𝑃. Since 𝑢,𝑣∈𝑉(𝑎), then by (𝑃2), 𝑣∗𝑢,𝑢∗𝑣∈𝑉(0) and so by (𝐵𝐶𝐼6), 𝑣∗(𝑣∗𝑢),𝑢∗(𝑢∗𝑣)∈𝐿_𝑋{𝑢,𝑣}={𝑎}. Hence, 𝑣∗(𝑣∗𝑢)=𝑎=𝑢∗(𝑢∗𝑣) and so [𝑣]∗([𝑣]∗[𝑢])=[𝑎]=[𝑢]∗([𝑢]∗[𝑣]). Let [𝑧]∈𝐿_𝑋/𝐼{[𝑥],[𝑦]}. Then by (𝑃3), [𝑧]∈𝑉_𝑋/𝐼([𝑎]). Since 𝑋/𝐼 is a commutative 𝐵𝐶𝐼-algebra, then by Lemma 2.2, [𝑣]∗([𝑣]∗[𝑧])=[𝑧]∗([𝑧]∗[𝑣])=[𝑧] and so by (𝐵𝐶𝐼7), we have [𝑧]=[𝑣]∗([𝑣]∗[𝑧])≤[𝑣]∗([𝑣]∗[𝑢])=[𝑎]. Clearly, 𝑎∈𝑄, where 𝑄 is a 𝑝-semisimple part of 𝑋/𝐼. Hence, [𝑎]=[𝑧] and so 𝐿_𝑋/𝐼{[𝑥],[𝑦]}={[𝑎]}. Therefore, [𝑥][𝑦]∈𝐸(𝐺(𝑋/𝐼)) and so 𝐺(𝑋)/Π is a subgraph of 𝐺(𝑋/𝐼).

Example 3.31. Let 𝑌={0,1,2,𝑎,𝑏} and 𝑋={0,1,2,3,4}. Define the binary operations “∗₁’’ and “∗₂’’ on 𝑌 and 𝑋, respectively by the following tables:

(i)It is easy to see that (𝑌,∗₁,0) is a 𝐵𝐶𝐼-algebra and 𝐽={0,1} is an ideal of 𝑌 (see [10, Example 2.5.5]). Moreover, [0]={0,1,𝑎,𝑏} and [2]={2}. Hence, 𝑌/𝐽={[0],[2]} is a commutative 𝐵𝐶𝐼-algebra and so 𝐸(𝐺(𝑌/𝐼))={[0][2]}. Moreover, Since 𝐿_𝑌{2,0}={0}, then [0][2]∈𝐸(𝐺(𝑋)/Φ, where Φ is the partition of 𝑌 induced by 𝐽. Therefore, 𝐺(𝑋/𝐽)=𝐺(𝑋)/Φ.(ii)We can prove that (𝑋,∗₂,0) is a 𝐵𝐶𝐾-algebra and 𝐼={0,1,2} is an ideal of 𝑋 (see [10]). Moreover, 𝑋/𝐼={[0],[3],[4]}, where [0]={0,1,2}, [3]={3} and [4]={4}. It is easy to check that 𝐸(𝐺(𝑋/𝐼))={[0][3],[0][4],[3][4]} and 𝐸(𝐺(𝑋)/Π)={[0][3],[0][4]}.

4. Characteristic Polynomials of Graphs of BCI-Algebras

In this section, we verify the characteristic polynomials of graph of 𝐵𝐶𝐼-algebras. Then we fined the relation between Characteristic polynomial of 𝐺(𝑋) and Characteristic polynomial of graphs of branches of 𝑋, for any 𝐵𝐶𝐼-algebra 𝑋.

Theorem 4.1. Let 𝑋 be a finite 𝐵𝐶𝐼-algebra. Then 𝜒(𝐺(𝑋),𝜆)=∏_𝑎∈𝑃𝜒(𝐺(𝑉(𝑎)),𝜆).

Proof. Let 𝑚∈ℕ, 𝑃={𝑎₁,…,𝑎_𝑚}, 𝑉(𝑎_𝑡)={𝑥_1,𝑡,…,𝑥_{𝑟𝑡,𝑡}} and 𝐴_𝑡=[𝑏_𝑖,𝑗] be the adjacency matrix of 𝐺(𝑉(𝑎_𝑖)), for all 𝑡∈{1,2,…,𝑚}. Then 𝑋={𝑥_1,1,𝑥_2,1,…,𝑥_𝑟1,1,𝑥_1,2,𝑥_2,2,…,𝑥_𝑟2,2,…,𝑥_1,𝑚,𝑥_2,𝑚,…,𝑥_{𝑟𝑚,𝑚}}. Since 𝑉(𝑎_𝑖)∩𝑉(𝑎_𝑗)=∅, for all distinct 𝑖,𝑗∈{1,2,…,𝑚}, then by Proposition 3.8(ii), the adjacent matrix of 𝐺(𝑋) is of the form 𝐴₁0⋯00𝐴₂⋯0⋮⋮⋱⋮00⋯𝐴_𝑚, (4.1) where 𝐴_𝑗 is a 𝑚_𝑗×𝑚_𝑗 matrix, for all 𝑗∈{1,2,…,𝑚}. By the properties of determinant, we have 𝜒(𝐺(𝑋),𝜆)=det(𝜆𝐼−𝐴)=det(𝜆𝐼₁−𝐴₁)×det(𝜆𝐼₂−𝐴₂)×⋯×det(𝜆𝐼_𝑚−𝐴_𝑚)=∏_𝑎∈𝑃𝜒(𝐺(𝑉(𝑎)),𝜆), where 𝐼_𝑗 is a 𝑚_𝑗×𝑚_𝑗 identity matrix, for all 𝑗∈{1,2,…,𝑚}.

Corollary 4.2. Let 𝑋 be a finite 𝐵𝐶𝐼-algebra and 𝑡 is the number of elements 𝑎∈𝑃 such that |𝑉(𝑎)|=1.(i)𝜒(𝐺(𝑋),𝜆)=𝜆^𝑡×𝑓(𝜆), for some polynomial 𝑓(𝜆).(ii)𝑋 is a 𝑝-semisimple 𝐵𝐶𝐼-algebra if and only if 𝜒(𝐺(𝑋),𝜆)=𝜆^𝑛, for some 𝑛∈ℕ.

Proof. (i) Let |𝑋|=𝑛 and {𝑎₁,…,𝑎_𝑡} be the set of all elements of 𝑃 such that |𝑉(𝑎_𝑖)|=1, for all 𝑖∈{1,2,…,𝑡}. Then by using the proof of Theorem 4.1, the adjacent matrix of 𝐺(𝑋) is of the form 𝐵000_𝑛×𝑛, (4.2) where 𝐵 is an (𝑛−𝑡×𝑛−𝑡) matrix. Hence, by properties of determinant, we have 𝜒(𝐺(𝑋),𝜆)=det(𝜆𝐼_𝑡)×det(𝜆𝐼_𝑛−𝑡−𝐵)=𝜆^𝑡×det(𝜆𝐼_𝑛−𝑡−𝐵). Let 𝑓(𝜆)=det(𝜆𝐼_𝑛−𝑡−𝐵), then 𝜒(𝐺(𝑋),𝜆)=𝜆^𝑡×𝑓(𝜆).
(ii) Since 𝑋 is a 𝑝-semisimple 𝐵𝐶𝐼-algebra, then |𝑉(𝑎)|=1, for all 𝑎∈𝑋. Therefore, by (i), 𝜒(𝐺(𝑋),𝜆)=𝜆^𝑛, where |𝑋|=𝑛. Conversely, let 𝜒(𝐺(𝑋),𝜆)=𝜆^𝑛, for some 𝑛∈ℕ. Then by Proposition 2.11, 𝐺(𝑋) is an empty graph. Therefore, by Theorem 3.9, 𝑋 is a 𝑝-semisimple algebra.

Theorem 4.3. Let 𝑋 be a finite 𝐵𝐶𝐼-algebra and 𝑎∈𝑃. If 𝜒(𝐺(𝑉(𝑎)),𝜆)=𝜆^𝑛+𝑐₁𝜆^𝑛−1+𝑐₂𝜆^𝑛−2+⋯+𝑐_𝑛−1𝜆+𝑐_𝑛, for some 𝑛∈ℕ, then(i)𝑐₁=0 and 2𝑐₂=−∑_𝑥∈𝑋|𝑍_𝑥|+|𝑃|,(ii)let 𝐴 be the set of all {𝑥,𝑦,𝑧}⊆𝑋 such that 𝑥,𝑦,𝑧 are distinct elements of 𝑋 and 𝑢∗𝑣=𝑢∗𝑎, for all 𝑢,𝑣∈{𝑥,𝑦,𝑧} and some 𝑎∈𝑃. If 𝑋 is a commutative 𝐵𝐶𝐼-algebra, then 𝑐₃=−2|𝐴|.

Proof. (i) By Proposition 2.11, 𝑐₁=0. If 𝑥∈𝑋−𝑃, then by Lemma 3.2, deg(𝑥)=𝑦∈𝑋−{𝑥}∣𝐿𝑥,𝑦=𝑎,forsome𝑎∈𝑃=𝑍_𝑥. (4.3) Now, if 𝑥∈𝑃, then by Lemma 3.2, deg(𝑥)=|{𝑦∈𝑋−{𝑥}|𝐿{𝑥,𝑦}=𝑎,forsome𝑎∈𝑃}|=|𝑍_𝑥|−1. Hence, by Proposition 2.11, we have 2𝑐₂=−2|𝐸(𝐺(𝑋))|=−deg(𝐺)=−_{𝑥∈𝑋−𝑃}𝑍_𝑥+_𝑥∈𝑃𝑍_𝑥−1=−_𝑥∈𝑋𝑍_𝑥+|𝑃|. (4.4)
 (ii) By Proposition 2.11, 𝑐₃=−2𝑡, where 𝑡 is the number of triangles of 𝐺(𝑋). Since by Proposition 3.8, 𝐺(𝑋)=⋃_𝑎∈𝑃𝐺(𝑉(𝑎)) and 𝑉(𝑎)∩𝑉(𝑏)=∅, for any distinct elements 𝑎,𝑏∈𝑃, then 𝑡=∑_𝑎∈𝑃𝑡_𝑎, where 𝑡_𝑎 is the number of triangles of 𝐺(𝑉(𝑎)), for all 𝑎∈𝑃. Now, let 𝑎∈𝑃 and 𝑥,𝑦,𝑧 be three vertexes of a triangles of 𝐺(𝑉(𝑎)). Then 𝑥,𝑦,𝑧 are distinct elements of 𝑋 and 𝐿{𝑥,𝑦}=𝐿{𝑥,𝑧}=𝐿{𝑦,𝑧}={𝑎}. Since 𝑥∗(𝑥∗𝑦)∈𝐿{𝑥,𝑦}, then 𝑥∗(𝑥∗𝑦)=𝑎 and so by (𝐵𝐶𝐼4) and (𝐵𝐶𝐼6), 𝑥∗𝑎≤𝑥∗𝑦. Moreover, 𝑎≤𝑦 and (𝐵𝐶𝐼7), imply 𝑥∗𝑦≤𝑥∗𝑎 and so 𝑥∗𝑦=𝑥∗𝑎. By the similar way, we can prove that 𝑥∗𝑧=𝑥∗𝑎, 𝑦∗𝑧=𝑦∗𝑎=𝑦∗𝑥 and 𝑧∗𝑥=𝑧∗𝑎=𝑧∗𝑦. Hence, {𝑥,𝑦,𝑧}∈𝐴. Conversely, let {𝑥,𝑦,𝑧}∈𝐴. Then there exists 𝑎∈𝑃 such that 𝑢∗𝑣=𝑢∗𝑎, for any 𝑢,𝑣∈{𝑥,𝑦,𝑧}. Let 𝑢,𝑣∈{𝑥,𝑦,𝑧} and 𝑤∈𝐿{𝑢,𝑣}. Then by (𝑃2), 𝑤∈𝑉(𝑎). Since 𝑋 is commutative, then by Lemma 2.2, (𝐵𝐶𝐼6) and (𝐵𝐶𝐼7), 𝑤=𝑤∗0=𝑤∗(𝑤∗𝑢)=𝑢∗(𝑢∗𝑤)≤𝑢∗(𝑢∗𝑣)=𝑢∗(𝑢∗𝑎)≤𝑎. Hence, 𝑤=𝑎 (since 𝑎∈𝑃), and so 𝐿{𝑢,𝑣}={𝑎}. Therefore, 𝑥𝑦,𝑥𝑧,𝑦𝑧∈𝐸(𝐺(𝑉(𝑎))) and so 𝑥,𝑦,𝑧 are three vertexes of a triangle of 𝐺(𝑉(𝑎)), for some 𝑎∈𝑃. Hence, 𝑡=∑_𝑎∈𝑃𝑡_𝑎=|𝐴|. This complete the proof.