A new approach to the entropy of a transitive BE-algebra with countable partitions

Abstract

The concept of entropy and information gain of BE-algebras in scientific disciplines such as information theory, data science, supply chain and machine learning assists us to calculate the uncertanity of the scientific processes of phenomena. In this respect the notion of filter entropy for a transitive BE-algebra is introduced and its properties are investigated. The notion of a dynamical system on a transitive BE-algebra is introduced. The concept of the entropy for a transitive BE-algebra dynamical system is developed and, its characteristics are considered. The notion of equivalent transitive BE-algebra dynamical systems is defined, and it is proved the fact that two equivalent BE-algebra dynamical systems have the same entropy. Theorems to help calculate the entropy are given. Specifically, a new version of Kolmogorov– Sinai Theorem has been proved. The study introduces the concept of information gain of a transitive BE-algebra with respect to its filters and investigates its properties. This study proposes the use of filter entropy to approximate the level of risk introduced by a BE-algebra dynamical system. This aim is reached by defining the information gain with respect to the filters of a BE-algebra. This methodology is well developed for use in engineering, especially in industrial networks. This paper proposes a novel approach to assess the quantity of uncertainty, and the impact of information gain of a BE-algebra dynamical system.

Keywords

Generator transitive BE-algebra dynamical system entropy information gain

1 Introduction

Entropy is a useful tool in machine learning to know various concepts such as feature selection, building decision trees, and fitting classification models, etc. Entropy can be cosidered as an assessment tool of supply chain information [1–4]. The term entropy was first used in 1865 by the German physicist, Rudolf Clausius, to denote the thermodynamic function that he introduced in 1854.

In 1948, Glude Shanon,the American mathematician, presented the notion of entropy in information theory. In 1958, the Russian mathematician,Kolmogorov, introduced the concept of measure - theoretic entropy in ergodic theory. Gradually, the notion of entropy was used for the algebraic structures. In 1975, Weiss considered the definition of entropy for endomorphisms of abelian groups [14].

In 1979, Peters gave a different definition of entropy for automorphisms of a discrete abelian group ‘G’. After proving the basic properties, similar to those proved by Weiss, he generalized Weiss’s main result to countable abelian groups, relating the entropy of an automorphism of ‘G’ to the measure-theoretic Kolmogorov– Sinai entropy of the adjoint automorphism of the dual of ‘G’.

In 2010, Dikranjan and Giordano Bruno [12, 13] extended the definition of the algebraic entropy, ‘h’, given by Peters for automorphisms to endomorphism of arbitrary abelian groups, and considered its attributes. Specifically, they proved ‘Bridge Theorem’ that connected the algebraic and the topological entropies.

Di Nola et al. [15] discussed the entropy of dynamical systems on effect algebras, and Rie $\overset{ˇ}{c}$ an [27] introduced the concept of the entropy for an MV-algebra dynamical system, and investigated its main features. In 2006, Ebrahimi [16] represented a different approach to the measure-theoretic notion of entropy using countable partitions, and studied its fundamental properties.

On the other hand, In 2007, Kim and Kim [22] introduced the notion of BE-algebras which was a generalization of the concept of BCK-algebras. In 2008, Ahn and So [5] introduced the notion of ideals in BE-algebras, and then they stated and proved several characterizations on ideals. In 2009 Walendziak [30] introduced the concept of commutative BE-algebras. The theory of filters of BE-algebras was introduced by B.L. Meng [24], and In 2009 Ahn and So [6] investigated some properties of filters of BE-algebras.

Subsequently, the algebraic entropy of the special linear character automorphisms of free groups was discussed by Brown [11]. Two years later, the notion of the algebraic entropy for the endomorphisms of modules were investigated by L.Salce [28].

In 2010, M.Ebrahimi and N.Mohammadi [18] investigated the entropy of countable partitions of F-structures using the generators of an m-preserving transformation of a discrete dynamical system. In 2015, Mehrpooya, Ebrahimi and Davvaz [23] introduced the entropy of semi-independent hyper MV-algebra dynamical systems and investigated its fundamental properties. Ebrahimi and Izadara [17] presented the notion of the entropy of dynamical systems on BCI-algebras. They introduced the concept of ideal entropy of BCI-algebras and investigated its application in the bilinear codes.

In 2022 Ghasemi and Jamalzadeh [19] introduced the notion of hypernormed entropy on a topological hypernormed hypergroup.

Mutual information is a quantity that measures relationship between two random variables that are sampled simultaneously i.e, it measures how much information is communicated, in one random variable about another. For random variables X, Y whose joint distribution is P (X, Y), the Mutual information of two random variables X, Y, is defined by: $I (X; Y) = \sum_{x \in X} \sum_{y \in Y} p (x, y) log \frac{p (x, y)}{p (x) p (y)} .$ (1) Mutual Information is one of the most popular techniques to measure the correlation between a feature X and class Y. Mutual Information is based on entropy which attempts to quantify the amount of a random variable X. Filter methods such as filter entropy calculate the entropy using Shanon’s information theory where the labeled information is utilized and is used to calculate the mutual information of the feature class pair. As mentioned above the mutual information between two random variables X and Y can be stated as: $I (X, Y) = H (X) - H (X | Y),$ (2) here H (X) is the entropy for X and H (X|Y) is the conditional entropy for X given Y. In order to acquire more information, one can see [8, 10, 20, 21, 26].

On the other hand information gain calculate the reduction in entropy. It is used in the construction of decision trees from a training by evaluating the information gain for each variables, and the variable maximizes the information gain suggests a minimum entropy group or groups of samples. In the other words, the information gain can be calculated as: $IG (s, a) = H (s) - H (s | a),$ (3) where IG (s, a) is the information gain of the data set s and for the random variable a. H (s) is the entropy for the data set for any change and H (s|a) is the conditional entropy for the data set given the variable a.

Hence there is a similar way to calculate the mutual information and information gain i.e one can say that they are equivalent. I (X, Y) = H (X) - H (X|Y) and IG (s, a) = H (s) - H (s|a).

In this paper, the notion of the entropy on BE-algebras is introduced, and its essential characteristics are studied. In this respect, in section 3, the concept of a BE-algebra dynamical system is introduced and the notion of the entropy of a BE-algebra dynamical system is defined. The basic properties are studied and their isomorphism problem is presented. In this section, the concept of generator of a BE-algebra dynamical system is defined and theorems that help us to calculate the entropy are given.

In particular, a new version of Kolmogorov-Sinai Theorem is established and proved. In the last of this section the information gain of a transitive BE-algebra is introduced and its properties are investigated.

Fig. 1

Illustration of the idea.

2 Preliminaries

In this section, the notion of transitive BE-algebras and some of their important properties are illustrated. The concept of congruence relation on a transitive BE-algebra is given, and the equivalence classes, as its partitions, are introduced.

An algebra (X, ∗ , 1) of type (2, 0) is said to be a BE-algebra [25], if it satisfies the following conditions:

x * x = 1,

x * 1 =1,

1 * x = x,

x * (y * z) = y * (x * z), for all x, y, z ∈ X.

Example 1. Let X = {1, a, b, c, d, 0} and consider the following table. Then (X, ∗ , 1) is a BE-algebra.

*	1	a	b	c	d	0
1	1	a	b	c	d	0
a	1	1	a	c	c	d
b	1	1	1	c	c	c
c	1	a	b	1	a	b
d	1	1	a	1	1	a
0	1	1	1	1	1	1

A BE-algebra (X,∗,1) is said to be self-distributive if for all x, y, z ∈ X : $x * (y * z) = (x * y) * (x * z) {and (x * y) * z = (x * z) * (y * z) .$ It’s introduced a relation ≤ on a BE-algebra X by x ≤ y if and only if x * y = 1, [25].

A non-empty subset Y of a BE-algebra X containing 1, is called a subalgebra of X if x * y ∈ Y, whenever x, y ∈ Y. A BE-algebra X, is said to be commutative if (x * y) * y = (y * x) * x, for all x, y ∈ X.

Corollary 2.1. [25] If X is a commutative BE-algebra, then the relation ≤ is a partially order on X.

The pair (X, ≤) is called an ordered BE-algebra.

A BE-algebra (X, * , 1) is said to be transitive if y * z ≤ (x * y) * (x * z), for all x, y, z ∈ X.

Example 2. Let X = {1, a, b, c}. Then (X, * , 1) is a transitive BE-algebra, when * is defined as follow:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Proposition 2.2. [25]

Every BE-algebra is a partially ordered BE-algebra.

Every commutative BE-algebra is transitive.

Every self distributive BE-algebra is transitive.

Example 3. Let X = [0, 1] and $x * y = {\begin{matrix} 1, & if x \leq y \\ y, & if y < x \end{matrix}$ Then (X, * , 1) is a self distributive and then a transitive BE-algebra.

Lemma 2.3. [25] A subalgebra Y of a commutative and distributive BE-algebra X is a commutative and distributive BE-algebra.

Let X be a BE-algebra, a nonempty subset F of X is said to be a filter of X [25], if it satisfies the following conditions:

1 ∈ F,

x ∈ F and x * y ∈ F imply that y ∈ F.

It is obvious that {1} is a filter of a BE-algebra X, say trivial filter. The set of all filters of a BE-algebra X is denoted by $F (X) .$ It’s clear that $⋂_{F \in F (X)} F$ is a filter of X.

Example 4. Let X = {1, a, b, c, d} and consider the following table. Then (X, * , 1) is a BE-algebra and F = {1, b, c} is a filter of X.

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	b	a
c	1	a	1	1	a
d	1	1	1	b	1

A filter F of a BE-algebra (X, * , 1) is said to be a closed filter if it is closed under the multiplication * on X.

Let (X, * , 1) be a BE-algebra. A binary relation θ on X is called a congruence [25], if it satisfies the following properties:

(x, x) ∈ θ,

(x, y) ∈ θ then (y, x) ∈ θ,

(x, y) ∈ θ and (y, z) ∈ θ then (x, z) ∈ θ,

θ is compatible with the operation *. That is, if (x, y) ∈ θ and (z, w) ∈ θ then (x * z, y * w) ∈ θ, for all x, y, z, w ∈ X.

It is easy to see that an equivalence relation θ is a congruence relation on X if and only if (a, b) ∈ θ implies (c * a, c * b) ∈ θ as well as (a * c, b * c) ∈ θ, for all a, b, c ∈ X.

Example 5. Let X = [0, 1] and $x * y = {\begin{matrix} 1, & if x \leq y \\ y, & if y < x \end{matrix}$ Let $F = [\frac{1}{3}, 1]$ and define θ_F = {(x, y) ∈ X × X : x * y ∈ F andy * x ∈ F}. One can see that F is a filter of X, and θ_F = {(x, y) ∈ X × X : x = yorx, y ∈ F} is a congruence relation.

For any congruence θ on X and for any x ∈ X, the corresponding congruence class [x] _θ = θ_x is defined as: ${y \in X : (x, y) \in θ} .$ [1] _θ = θ₁ is called the kernel of the congruence θ. It’s clear that [1] _θ = θ₁ is a subalgebra of BE-algebra X. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes for the relation θ.

Let (X, * , 1) be a BE-algebra and θ be a congruence relation on X. The collection of all congruence classes of X is denoted by $\frac{X}{θ}$ and is given as follow: $\frac{X}{θ} = {θ_{x} : x \in X} .$

Proposition 2.4. [25] Let θ be a congruence on a BE-algebra (X, * , 1). Then the quotient algebra ( $\frac{X}{θ}$ , ⊗, [1] _θ) is a BE-algebra, where the operator ⊗ on X is defined by: $[a]_{θ} \otimes [b]_{θ} = [a * b]_{θ} {or θ_{a} \otimes θ_{b} = θ_{a * b} .$

Proposition 2.5. [25] Let (X, * , 1) be a commutative BE-algebra. Then for any congruence θ on X,

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	b	a
c	1	a	1	1	a
d	1	1	1	b	1

Proposition 2.6. [25] Let θ₁, θ₂ be two congruences on a commutative BE-algebra X. Then θ₁ ⊆ θ₂ if and only if [1] _θ₁ ⊆ [1] _θ₂.

Proposition 2.7. [25] Let (X, * , 1) be a BE-algebra and {θ_α} _α∈Δ be a family of congruences on X. Put θ = ⋂ _α∈Δθ_α, then the following conditions hold:

θ is a congruence on X,

[1] _θ = ⋂ _α∈Δ [1] _{θ_α}.

Let (X, * , 1) and (Y, ⊙ , e) be two BE-algebras. A mapping f : (X, * , 1) ⟶ (Y, ⊙ , e) is called a homomorphism of X into Y if it satisfies the following property:

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	b	a
c	1	a	1	1	a
d	1	1	1	b	1

A homomorphism of X into itself is called an endomorphism in X.

Example 6. Let X = {1, a, b, c}. Define a binary operation * on X as follows, and let f : (X, * , 1) ⟶ (X, * , 1) be defined as f (1) =1, f (a) = a, f (b) = c, f (c) = b .

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Proposition 2.8. [25] Let (X, * , 1) and (Y, ⊙ , e) be two BE-algebras and f : (X, * , 1) ⟶ (Y, ⊙ , e) be a homomorphism of BE-algebras. Then f^-1 (F) is a filter of X whenever F is a filter of Y.

Proposition 2.9. [25] Let (X, * , 1) and (Y, ⊙ , e) be two BE-algebras and f : (X, * , 1) ⟶ (Y, ⊙ , e) be a homomorphism of BE-algebras. For any x, y ∈ X define the binary operation θ on X by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Then θ is a congruence on X such that [1] _θ = ker f .

Proposition 2.10. [25] Let (X, * , 1) and (Y, ⊙ , e) be two BE-algebras and f : (X, * , 1) ⟶ (Y, ⊙ , e) be a homomorphism of BE-algebras. Then the quotient algebra $\frac{X}{θ}$ is isomorphic to the image of f, i.e $\frac{X}{θ} ≃ f (X) .$

Let (X, * , 1) be a BE-algebra and F be a filter of X. Define a binary operation θ_F on X by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Proposition 2.11. [25] Let F be a filter of a transitive BE-algebra X. Then θ_F is a unique congruence relation on X with the kernel [1] _{θ_F} = F.

The congruence θ_F is called the filter congruence on X by F. For the congruence θ_F on X by F one can usually denote x ≃ y (F) for x ≃ y (θ_F) and F_x for [x] _{θ_F} and $\frac{X}{F}$ for $\frac{X}{θ}$ .

Proposition 2.12. [25] Let F be a filter of a transitive BE-algebra X. Then F₁ = [1] _{θ_F} is the smallest filter containing F.

Proposition 2.13. [25] Let F be a filter of a transitive BE-algebra X and $\frac{X}{F} = {F_{x} : x \in X}$ . Then $(\frac{X}{F}, \otimes, F_{1})$ is a BE-algebra with the greatest element F₁.

Corollary 2.14. Let (X, * , 1) be a transitive BE-algebra and F be a filter of X. Then F = F₁ = [1] _{θ_F} is closed.

Proof. Suppose that x, y ∈ F since x ∈ F, x * 1 ∈ F and 1 * x ∈ F. Similarly y * 1 ∈ F and 1 * y ∈ F. θ_F is a congruence then 1 * (x * y) ∈ F and hence x * y ∈ F.□

Corollary 2.15. Let F and F′ be two filters of a transitive BE-algebra X. Then $\frac{X}{F} = \frac{X}{F^{'}}$ if and only if F = F′.

Proof. Suppose that $\frac{X}{F} = \frac{X}{F^{'}}$ and x ∈ F. Then there exists x′ ∈ X such that $F_{x} = F_{x^{'}}^{'}$ . 1 ∈ F_x then $1 \in F_{x^{'}}^{'}$ and hence 1 * x′, x′ * 1 ∈ F′. Thus x′ ∈ F′, $x \in F_{x} = F_{x^{'}}^{'}$ ⇒ x′ * x ∈ F′, x′ ∈ F′ and then x ∈ F′. This means that F ⊆ F′ and in the same manner F′ ⊆ F.□

Proposition 2.16. Let (X, * , 1) and (Y, ⊙ , e) be two BE-algebras. Let g : X ⟶ Y be an epimorphism of BE-algebras. Then the followings are satisfied:

If F is a filter of X, then g (F) is a filter of Y,

If F is a closed filter of X, then g (F) is a closed filter of Y.

Proof. (1) Since g (1) = g (1 *1) = g (1) ⊙ g (1), it follows that g (1) = e ∈ g (F). Now, let y₁ ∈ g (F), y₁ ⊙ y₂ ∈ g (F). Then there exist x₁, z ∈ F such that y₁ = g (x₁) and y₁ ⊙ y₂ = g (z). Since g is an epimorphism and y₂ ∈ Y, there is x₂ ∈ X that y₂ = g (x₂). Put u = (z * (x₁ * x₂)) * x₂ and consider $\begin{matrix} z * (x_{1} * u) & = (z * (x_{1} * (z * (x_{1} * x_{2})) * x_{2})) \\ = (z * (z * (x_{1} * x_{2})) * (x_{1} * x_{2})) \\ = (z * (x_{1} * x_{2})) * (z * (x_{1} * x_{2})) \\ = 1 \in F . \end{matrix}$ Then u ∈ F and $\begin{matrix} g (u) & = g (z * (x_{1} * x_{2}) * x_{2}) \\ = g (z * (x_{1} * x_{2})) ⊙ g (x_{2}) \\ = (g (z) ⊙ g (x_{1} * x_{2})) ⊙ g (x_{2}) \\ = ((y_{1} ⊙ y_{2}) ⊙ (y_{1} ⊙ y_{2})) ⊙ y_{2} \\ = e ⊙ y_{2} \\ = y_{2} \in F^{'} . \end{matrix}$

(2) Let y₁, y₂ ∈ F′, since g is onto, there are x₁, x₂ ∈ F such that y₁ = g (x₁) , y₂ = g (x₂) and y₁ * y₂ = g (x₁) * g (x₂) = g (x₁ * x₂) ∈ g (F) = F′ . This completes the proof. □

Lemma 2.17. If g : (X, * , 1) ⟶ (Y, ⊙ , e) is a homomorphism of BE-algebras then Img is a subalgebra of (Y, ⊙ , e) and hence is a BE-algebra.

Proof. It is straightforward. □

Lemma 2.18. Let g : (X, * , 1) ⟶ (Y, ⊙ , e) be a BE-algebra homomorphism and F be a filter of X such that ker g ⊆ F. Then g^-1 (F′) = F where F′ = g (F).

Proof. It is enough to show that g^-1 (F′) ⊆ F. Let x ∈ g^-1 (F′), then g (x) ∈ F′ = g (F) and hence there exists y ∈ F such that g (x) = g (y). Since kerg ⊆ F and F is a filter of X, x ∈ F. □

Lemma 2.19. Let g : (X, * , 1) ⟶ (Y, ⊙ , e) be a BE-algebra homomorphism and F, F′ be subsets of X and Y, respectively. Then the following are satisfied:

If F is a subalgebra of X, then g (F) is a subalgebra of Y,

If F′ is a subalgebra of Y, then g^-1 (F′) is a subalgebra of X.

Proof.

Suppose that F is a subalgebra of X, then 1 ∈ F and g (1) = e ∈ g (F). Now if z, w ∈ g (F) then z ⊙ w = g (x) ⊙ g (y) = g (x * y), for some x, y ∈ F. Since F is a subalgebra, z ⊙ w = g (x * y) ∈ g (F).

The proof is similar to the previous part.

□

3 Filter entropy of BE-algebras

In this section the notion of the entropy of a BE-algebra that is called the filter entropy, is introduced. Some definitions, propositions and lemmas are given to calculate some entropies of BE-algebras. A new version of Kolmogorov-Sinai Theorem is investigated and the concept of Information gain related to filters is defined.

If F is a filter of a transitive BE-algebra (X, * , 1) the collection $\frac{X}{F} = {F_{x} : x \in X}$ is a partition of X. From now on, every partition is considered countable.

Definition 3.1. Let E and F be two filters of a transitive BE-algebra (X, * , 1). In a way that $\frac{X}{E}, \frac{X}{F}$ are two partitions of X. $\frac{X}{F}$ is said to be a refinement of $\frac{X}{E}$ if each element of $\frac{X}{E}$ is a union of elements of $\frac{X}{F}$ and is denoted by $\frac{X}{E} < \frac{X}{F}$ .

Definition 3.2. Let E and F be two filters of a transitive BE-algebra (X, * , 1). If $\frac{X}{E} = {E_{x_{i}} : x_{i} \in X, i \in ℕ}$ , $\frac{X}{F} = {F_{y_{j}} : y_{j} \in X, j \in ℕ}$ are countable partitions of X. Then the join of partitions is defined by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

It is clear that $\frac{X}{E} \lor \frac{X}{F}$ is a partition of X. One may remember that for a transitive BE-algebra (X, * , 1) and a filter F of X the structure $(\frac{X}{F}, \otimes, F_{1})$ is a BE-algebra that is called a quotient algebra induced by F.

Now in the following proposition it is shown that the join of two quotient algebras is a quotient algebra.

Proposition 3.3. Let (X, * , 1) be a transitive BE-algebra and (Y, * , 1) be a subalgebra of (X, * , 1). Suppose that E and F are filters of Y and $\frac{Y}{E}, \frac{Y}{F}$ are quotient algebras of Y induced by E and F, respectively. Then $\frac{Y}{E} \lor \frac{Y}{F}$ is a quotient algebra induced by E ∩ F.

Proof. One knows that $\frac{Y}{E} = {E_{x_{i}} : x_{i} \in Y, i \in ℕ}$ and $\frac{Y}{F} = {F_{y_{j}} : y_{j} \in Y, j \in ℕ}$ are partitions of Y. Also, $\frac{Y}{E} \lor \frac{Y}{F} = {E_{x_{i}} \cap F_{y_{j}} : x_{i}, y_{j} \in Y and i, j \in ℕ}$ is a partition of Y. Define the relation ∼ on Y by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

One can easily see that ∼ is an equivalence relation on Y. Let x, y ∈ Yandx ∈ E_{x
_i}, y ∈ F_{y
_j}. Since x ∼ y, it follows that there is one and only one $A \in \frac{Y}{E}$ and $B \in \frac{Y}{F}$ such that x, y ∈ A ∩ B . Then x * y ∈ E ∩ F and y * x ∈ E ∩ F and this implies that $\frac{Y}{E} \lor \frac{Y}{F}$ is a quotient algebra induced by E ∩ F and hence $\frac{Y}{E} \lor \frac{Y}{F} = \frac{Y}{E \cap F}$ . □

Corollary 3.4. Let ${F_{i}}_{i = 1}^{n}$ be a family of countable partitions of a transitive BE-algebra X. Then the following holds:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Proposition 3.5. Let Y be a subalgebra of a transitive BE-algebra X, and let E and F be two filters of X with F ⊂ E. Then $\frac{Y}{E} < \frac{Y}{F}$ .

Proof. Suppose that $\frac{Y}{E} = {A_{i} : i \in ℕ}$ and $\frac{Y}{F} = {C_{j} : j \in ℕ}$ and let $a, b \in \frac{Y}{E}$ such that a, b ∈ A_i andA_i = [a_i] _E for some a_i ∈ Y and for fixed arbitrary i ≥ 1 .

a * a_i ∈ E and a_i * a ∈ E, b * a_i ∈ E and a_i * b ∈ E. By transitivity a_i * b ≤ (a * a_i) * (a * b) and then (a * a_i) * (a * b) ∈ E . Hence a * b ∈ E, in the same manner b * a ∈ E and since F ⊂ E, a * b ∈ F and b * a ∈ F . Then $[a]_{F} = [b]_{F} \in \frac{Y}{F}$ and there is $j \in ℕ$ with [a] _F = [b] _F = C_j . Therefore A_i ⊆ C_j and $\frac{Y}{E} < \frac{Y}{F} .$ □

Corollary 3.6. Let X be a transitive BE-algebra and Y be a subalgebra of X. Then

If F = 〈1〉 is the filter induced by 1 ∈ X and E is an arbitrary filter of Y, then $\frac{Y}{E} < \frac{Y}{F}$ .

If E and F are filters of Y, then $\frac{Y}{E} < \frac{Y}{E \cap F}$ and $\frac{Y}{F} < \frac{Y}{E \cap F}$ .

If E ∪ F is a filter of Y, then $\frac{Y}{E \cup F} < \frac{Y}{E}$ and $\frac{Y}{E \cup F} < \frac{Y}{F}$ .

Corollary 3.7. Let Y be a subalgebra of a transitive BE-algebra X and F be a filter of Y. If F is closed, then $F \in \frac{Y}{F}$ and $\frac{F}{F} = F$ . Conversely, if $\frac{F}{F} = F$ , then F is a closed filter.

Let X be a non-empty set and * be a binary operation on X. The pair (X, *) is said to be a semi BE-algebra if

X is closed under *,

x * (y * z) = y * (x * z), for all x, y, z ∈ X .

Let X be transitive BE-algebra and let

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Define the binary operation ★ on β (X) by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

(β (X) , ★) is a semi BE-algebra. The function m : β (X) ⟶ [0, 1] is called a state if

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Definition 3.8. By a dynamical system on β (X) one can consider a couple (β (X) , T, m) with T : β (X) ⟶ β (X) and m : β (X) ⟶ [0, 1] satisfying the following conditions:

T (X) = X and T ∣ _X is a BE-algebra homomorphism,

T is a homomorphism of semi BE-algebras, i.e.,

m (T^-1E_x) = m (E_x),

T^-1 (E_x ★ F_y) = (T^-1E_x) ★ (T^-1F_y) , for all x, y ∈ X.

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Note that if X is a transitive BE-algebra and E, F are filters of X, it is easy to see that

If $\frac{X}{F} = {F_{x_{i}} : x_{i} \in X, i \in ℕ}$ , then $T^{- n} (\frac{X}{F}) = {T^{- n} (F_{x_{i}}) : x_{i} \in X, i \in ℕ}$ is a partition of X, for $n \in ℕ \cup {0}$ .

$T^{- n} (\frac{X}{E} \lor \frac{X}{F}) = T^{- n} (\frac{X}{E}) \lor T^{- n} (\frac{X}{F})$ , for $n \in ℕ \cup {0}$ .

If $\frac{X}{E} < \frac{X}{F}$ then $T^{- n} (\frac{X}{E}) < T^{- n} (\frac{X}{F})$ , for $n \in ℕ \cup {0}$ .

Definition 3.9. Let Y be a subalgebra of a transitive BE-algebra (X, * , 1) and F be a filter of Y. If $\frac{Y}{F} = {F_{y_{j}} : y_{j} \in Y, i \in ℕ}$ is a quotient algebra of Y induced by F. The entropy of $\frac{Y}{F}$ is defined by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Definition 3.10. Let (X, * , 1) be a transitive BE-algebra and E, F be filters of X. Then E and F are said to be independent if the quotient algebra $\frac{X}{E}$ and $\frac{X}{F}$ are independent in the meaning of:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Lemma 3.11. Let (X, * , 1) be a transitive BE-algebra and F = 〈1〉 be the filter of X induced by {1}, then the followings are satisfied:

If X is commutative then $\frac{X}{F}$ = {{x} : x ∈ X} is the discrete partition of X,

If E is an arbitrary filter of X, then H_E (X) ≥0,

H_X (X) =0,

If $\frac{X}{F} = {F_{x_{i}} : x_{i} \in X, i \in ℕ}$ and $m (F_{x_{i}}) = \frac{1}{k}$ , then H_F (X) = log k,

If (β (X) , T, m) is a semi BE-algebra then H_T
^-1
F (T^-1X) = H_F (X) .

Proposition 3.12. Let Y be a subalgebra of a transitive BE-algebra X and E, F be two filters of X. Then the followings are satisfied:

If E ⊆ F then H_F (Y) ≤ H_E (Y),

$H_{E} (Y) + H_{F} (Y) \leq 2 H (\frac{Y}{E} \lor \frac{Y}{F}) = 2 H_{E \cap F} (Y)$ .

Proof. (1) Let $\frac{Y}{E} = {E_{y_{i}} : y_{i} \in Y, i \in ℕ}$ and $\frac{Y}{F} = {F_{y_{j}} : y_{j} \in Y, j \in ℕ}$ . Since E ⊆ F, it follows that $\frac{Y}{F} < \frac{Y}{E}$ and then for each y_i ∈ E there exists y_{i
_j} ∈ F such that E_{y
_i} ⊆ F_{y
_{i
_j}}. Hence $m (E_{y_{i}}) \leq m (F_{y_{i_{j}}}) \leq sup_{y_{j} \in E} m (F_{y_{j}})$ . Now one can show:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

(2) It is clear. □

Definition 3.13. Let Y be a subalgebra of a transitive BE-algebra X, and E and F be two filters of X. The conditional entropy of $\frac{Y}{E}$ given $\frac{Y}{F}$ is defined by:

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

If m (F_{y
_j}) =0, it will not be considered.

Note that

H_F∣Y (Y) = H_F (Y) + log m (Y) ,

H_F∣Y (Y) ≤ H_F (Y) ,

H_F∣X (X) = H_F (X).

Lemma 3.14. Let F be a filter of a commutative BE-algebra X. Then the followings are satisfied:

H_F∣〈1〉 (X) =0,

H_〈1〉∣F (X) = H_〈1〉 (X) - H_F (X) .

Proof. (1) We have $\begin{matrix} H_{F ∣ 〈 1 〉} (X) & = - log sup {\frac{m (F_{x_{i}} \cap 〈 1 〉_{y_{j}})}{m (〈 1 〉_{y_{j}})} : x_{i}, y_{j} \in X, i, j \in ℕ} \\ = - log sup {\frac{m (F_{x_{i}} \cap {y_{j}})}{m ({y_{j}})} : x_{i}, y_{j} \in X, i, j \in ℕ} . \end{matrix}$ (2) We have $\begin{matrix} H_{〈 1 〉 ∣ F} (X) & = - log sup {\frac{m ({y_{j}} \cap F_{x_{i}})}{m (F_{x_{i}})} : x_{i}, y_{j} \in X, i, j \in ℕ} \\ = - log sup {\frac{m ({y_{j}})}{m (F_{x_{i}})} : x_{i}, y_{j} \in X, i, j \in ℕ} \\ = - log sup {m (y_{j}) : y_{j} \in X, j \in ℕ} - (- log sup {m (F_{x_{i}}) : x_{i} \in X, i \in ℕ}) \\ = H_{〈 1 〉} (X) - H_{F} (X) . \end{matrix}$ □

Proposition 3.15. Let (X, * , 1) be a transitive BE- algebra and E, F, G be filters of X and (β (X) , T, m) be a dynamical system. If $\frac{X}{E}$ , $\frac{X}{F}$ , $\frac{X}{G}$ are quotient algebras of X induced by E, F, G, respectively, then the followings are satisfied:

If F and E ∨ G are independent, then H_E∨F∣G (X) = H_E∣G (X) + H_F∣E∨G,

H_E∨F (X) = H_E (X) + H_F∣E (X) ,

If T preserves m and is invertible, then H_{T^-1E∣T^-1F} (T^-1 (X)) = H_E∣F (X),

If E ⊆ F then H_F∣G (X) ≤ H_E∣G (X),

If E ⊆ F, then H_E (X) ≤ H_E∣F (X) + H_F (X).

Proof. (1) Since F and E ∨ G are independent, it follows that $\begin{matrix} H_{E \lor F ∣ G} (X) & = - log sup {\frac{m ((E_{x_{i}} \cap F_{y_{j}}) \cap G_{z_{k}})}{m (G_{Z_{k}})} : x_{i}, y_{j}, z_{k} \in X, i, j, k \in ℕ} \\ = - log sup {\frac{m (F_{y_{j}} \cap (E_{x_{i}} \cap G_{z_{k}}))}{m (G_{z_{k}})} : x_{i}, y_{j}, z_{k} \in X, i, j, k \in ℕ} \\ = - log sup {\frac{m (F_{y_{j}}) m (E_{x_{i}} \cap G_{z_{k}})}{m (G_{z_{k}})} : x_{i}, y_{j}, z_{k} \in X, i, j, k \in ℕ} \\ = - log sup {m (F_{y_{j}}) : y_{j} \in X, j \in ℕ} - log sup {\frac{m (E_{x_{i}} \cap G_{z_{k}})}{m (G_{z_{k}})} : \\ x_{i}, z_{k} \in X, i, k \in ℕ} \\ = H_{F ∣ E \lor G} (X) + H_{E ∣ G} (X) . \end{matrix}$

(2) Let G = X .

(3) Since T preserves m, the proof is straightforward.

(4) One has E ⊆ F and then $\frac{X}{F} < \frac{X}{E}$ , let $\frac{X}{F} = {F_{x_{i}} : x_{i} \in X, i \in ℕ}$ and $\frac{X}{E} = {E_{x_{j}} : x_{j} \in X, j \in ℕ}$ . For each x_i ∈ X and for each $i \in ℕ$ , E_{x
_i} ⊆ F_{x
_i} that implies m (E_{x
_i} ∩ G_{z
_k}) ≤ m (F_{x
_i} ∩ G_{z
_k}) ∀z_k ∈ X. Then, we have $sup {m (E_{x_{i}} \cap G_{z_{k}}) : x_{i}, z_{k} \in X, i, k \in ℕ} \leq$ $sup {m (F_{x_{i}} \cap G_{z_{k}}) : x_{i}, z_{k} \in X, i, k \in ℕ},$ and so $\begin{matrix} - log sup {\frac{m (F_{x_{i}} \cap G_{z_{k}})}{m (G_{z_{k}})} : x_{i}, z_{k} \in X, i, k \in ℕ} \\ \leq - log sup {\frac{m (E_{x_{i}} \cap G_{z_{k}})}{m (G_{z_{k}})} : x_{i}, z_{k} \in X, i, k \in ℕ} . \end{matrix}$ Therefore, we get H_F∣G (X) ≤ H_E∣G (X) .

(5) It is similar to (4). □

Definition 3.16. Let (X, * , 1) be a transitive BE-algebra and E, F be filters of X. If (β (X) , T, m) is a dynamical system define: $\frac{X}{E} \overset{\circ}{\subseteq} \frac{X}{F} \Leftrightarrow \forall E_{x_{i}} \in \frac{X}{E}$ there exists $F_{y_{j}} \in \frac{X}{F}$ such that m (E_{x
_i} ▵ F_{y
_j}) =0, where E_{x
_i} ▵ F_{y
_j} = (E_{x
_i} - F_{y
_j}) ∪ (F_{y
_j} - E_{x
_i}), and

*	1	a	b	c
1	1	a	b	c
a	1	1	a	a
b	1	1	1	a
c	1	1	a	1

Proposition 3.17. Let (X, * , 1) be a transitive BE-algebra and E, F be two filters of X. If $\frac{X}{E} \overset{\circ}{=} \frac{X}{F}$ then H_E (X) = H_F (X) .

Proof. Since $\frac{X}{F} \overset{\circ}{=} \frac{X}{E}$ , it follows that for each $F_{y_{j}} \in \frac{X}{F}$ there exists $E_{x_{i}} \in \frac{X}{E}$ such that m (E_{x
_i} ▵ F_{y
_j}) =0. Let $\frac{X}{E} = {E_{x_{i}} : x_{i} \in X, i \in ℕ}$ and $\frac{X}{F} = {F_{y_{j}} : y_{j} \in X, j \in ℕ}$ . One can see that m (E_{x
_i} - F_{y
_j}) = m (F_{y
_j} - E_{x
_i}), on the other hand m (F_{y
_j}) = m (F_{y
_j} - E_{x
_j}) + m (F_{y
_j} ∩ E_{x
_i}) and m (E_{x
_i}) = m (E_{x
_i} - F_{y
_j}) + m (E_{x
_i} ∩ F_{y
_j}). Hence for each $j \in ℕ$ , there is $i \in ℕ$ such that m (F_{y
_j}) = m (E_{x
_i}) . Then $m (F_{y_{j}}) \leq sup {m (E_{x_{i}}) : E_{x_{i}} \in X, i \in ℕ}$ ∀y_j ∈ X, $\forall j \in ℕ .$ Therefore $sup {m (F_{y_{j}}) : y_{j} \in X, j \in ℕ} \leq sup {m (E_{x_{i}}) : E_{x_{i}} \in X, i \in ℕ}$ , and H_E (X) ≤ H_F (X). In the same manner one has H_F (X) ≤ H_E (X). □

Sometimes $E \overset{\circ}{=} F$ , is used instead of $\frac{X}{E} \overset{\circ}{=} \frac{E}{F}$ . Using the following example one can show that the converse of Proposition 3.17 is not generally true.

Example 7. Let X = {1, a, b, c, d} and consider * as bellow:

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	c	c
c	1	1	b	1	b
d	1	1	1	1	1

(X, * , 1) is a transitive BE-algebra. Now let E = {1, a, c} and F = {1, a, b}. Then $H_{E} (X) = H_{F} (X) = log \frac{5}{3}$ , but $E \overset{\circ}{\neq} F$ because $E \overset{\circ}{⊈} F$ . Consider m : β (X) ⟶ [0, 1] defined by $m (E_{x}) = \frac{∣ E_{x} ∣}{5} .$ Let E_c = {1, a, c}, $m (E_{c} ▵ F_{1}) = m (E_{c} ▵ F_{a}) = m (E_{c} ▵ F_{b}) = \frac{2}{5},$ and $m (E_{c} ▵ F_{c}) = \frac{3}{5}$ , m (E_c ▵ F_d) =1. But $H_{E} (X) = H_{F} (X) = log \frac{5}{3}$ .

Proposition 3.18. Let (X, * , 1) be a transitive BE-algebra and E, F be two filters of X. If $E \overset{\circ}{=} F$ , then H_E∣K (X) = H_F∣K (X) for each arbitrary filter K of X.

Proof. One has $E \overset{\circ}{\subseteq} F$ and for each x_i ∈ X there is y_j ∈ X such that m (E_{x
_i}) = m (F_{y
_j}). Now, assume that K is an arbitrary filter of X. Then one can write $\begin{matrix} \frac{m (E_{x_{i}} \cap K_{z_{l}})}{m (K_{z_{l}})} = \frac{m (F_{y_{j}} \cap K_{z_{l}})}{m (K_{z_{l}})}, for all z_{l} \in X and l \in ℕ, \\ \frac{m (E_{x_{i}} \cap K_{z_{l}})}{m (K_{z_{l}})} \leq sup {\frac{m (F_{y_{j}} \cap K_{z_{l}})}{m (K_{z_{l}})} : y_{j} \in X, j \in ℕ}, \\ for all x_{i} \in X and i \in ℕ . \end{matrix}$ Then, it is given $sup {\frac{m (E_{x_{i}} \cap K_{z_{l}})}{m (K_{z_{l}})} : x_{i}, z_{l} \in X, i, l \in ℕ} \leq$ $sup {\frac{m (F_{y_{j}} \cap K_{z_{l}})}{m (K_{z_{l}})} : y_{j}, z_{l} \in X, j, l \in ℕ} .$ Therefore we have H_F∣K (X) ≤ H_E∣K (X). Similarly, one can show that H_E∣K (X) ≤ H_F∣K (X). □

Using the following example, it is shown that the converse of the Proposition 3.18 is not generally true.

Example 8. Let X = {1, a, b, c, d} and * be as in Proposition 3.17. Let E = {1, a, c} and F = {1, a, b}. The filters of X are 〈1〉, {1, a},{1, b},{1, a, c},{1, a, b} . Then H_E∣K (X) = H_F∣K (X) for each filter K of X, but $E \overset{\circ}{\neq} F$ .

Proposition 3.19. Let (X, * , 1) be a transitive BE-algebra and E, F be two filters of X. If $E \overset{\circ}{=} F$ then H_K∣E (X) = H_K∣F (X), for each filter K of X.

Proof. See the proof of Proposition 3.18. □

The converse of Proposition 3.19 is not true. Example 8 is enough proof.

Lemma 3.20. If E, F are independent filters of a transitive BE-algebra (X, * , 1), then H_E∣F (X) = H_E (X) .

Note that the converse of Lemma 3.20 is not generally true.

Definition 3.21. Let (X, * , 1) be a transitive BE-algebra and (β (X) , T, m) be a dynamical system. If F is a filter of X, the entropy of T with respect to F is defined by:

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	c	c
c	1	1	b	1	b
d	1	1	1	1	1

Proposition 3.22. [31] The h_F (T, X) is well defined. It means that the limit exists.

Corollary 3.23. Let (X, * , 1) be a transitive BE-algebra and (β (X) , T, m) be a dynamical system. If E, F are filters of X, then the followings are satisfied:

h_F (T, X) ≥0, h_X (T, X) =0,

h_E∨F (T, X) = h_E∩F (T, X) .

Proof. It is straightforward. □

Definition 3.24. Let (X, * , 1) be a transitive BE-algebra and (β (X) , T, m) be a dynamical system. The entropy of the dynamical system (β (X) , T, m) is denoted by h_T (X) and

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	c	c
c	1	1	b	1	b
d	1	1	1	1	1

Definition 3.25. Let (β (X) , T, m) and (β (Y) , S, μ) be two BE-algebra dynamical systems. The above dynamical systems are said to be isomorphic if there exist closed filters E and F of X and Y, respectively, together with a BE-algebra isomorphism φ : E ⟶ F such that

T (E) ⊆ E, S (F) ⊆ F,

S ∘ φ = φ ∘ T,

m ∘ (φ^-1) _∣F = μ.

Note that when φ : E ⟶ F is a BE-algebra isomorphism then φ : β (E) ⟶ β (F) is a semi BE-algebra isomorphism.

Proposition 3.26. If (β (X) , T, m) and (β (Y) , S, μ) are isomorphic, then h (T) = h (S).

Proof. As one has seen $h_{E} (T, X) = h (T, \frac{X}{E}) = lim_{n \to \infty} \frac{1}{n} H (⋁_{i = 0}^{n - 1} T^{- i} (\frac{X}{E})) .$ At first, one would claim that for any integer k ≥ 0, $h_{E} (T, X) = h_{⋁_{i = 0}^{k} (T^{- i} (\frac{X}{E}))} (T, X)$ .

Proof of the claim: $\begin{matrix} h_{⋁_{i = 0}^{k} (T^{- i} (\frac{X}{E}))} (T, X) & = lim_{n \to \infty} \frac{1}{n} H (⋁_{j = 0}^{n - 1} T^{- j} (⋁_{i = 0}^{k} (T^{- i} \frac{X}{E}))) \end{matrix}$ $\begin{matrix} = lim_{n \to \infty} \frac{k + n}{n} \frac{1}{k + n} H (⋁_{t = 0}^{n + k - 1} (T^{- t} \frac{X}{E})) \\ = lim_{n \to \infty} \frac{1}{k + n} H (⋁_{t = 0}^{k + n - 1} (T^{- t} \frac{X}{E})) \\ = h (T, \frac{X}{E}) = h_{E} (T, X) . \end{matrix}$

Now since dynamical systems T and S are isomorphic, it follows that there exist closed filters E, F of X, Y, respectively, and a BE-algebra isomorphism φ : E ⟶ F such that T (E) ⊆ E, (SF) ⊆ F and φoT = Soφ, furthermore $m ((φ^{- 1} F_{y_{j}}^{'}) = μ (F_{y_{j}}^{'})$ , where F′ is a filter of Y. Then $\begin{matrix} H (φ^{- 1} (\frac{Y}{F^{'}})) & = - log sup {m (φ^{- 1} F_{y_{j}}^{'}) : y_{j} \in Y, j \in ℕ} \\ = - log sup {μ (F_{y_{j}}^{'}) : y_{j} \in Y, j \in ℕ} \\ = H (\frac{Y}{F^{'}}) . \end{matrix}$ Therefore, one can conclude that $\begin{matrix} H (⋁_{i = 0}^{n - 1} T^{- i} (φ^{- 1} \frac{Y}{F^{'}})) = H (φ^{- 1} (⋁_{i = 0}^{n - 1} S^{- i} \frac{Y}{F^{'}})) & = H (⋁_{i = 0}^{n - 1} S^{- i} \frac{Y}{F^{'}}) . \end{matrix}$ $lim_{n \to \infty} \frac{1}{n} H (⋁_{i = 0}^{n - 1} T^{- i} (φ^{- 1} \frac{Y}{F^{'}})) = lim_{n \to \infty} \frac{1}{n} H (⋁_{i = 0}^{n - 1} S^{- i} \frac{Y}{F^{'}}) .$ This yields that $h (T, φ^{- 1} \frac{Y}{F^{'}}) = h (S, \frac{Y}{F^{'}})$ , and so h_φ^-1(F′) (T, X) = h_F′ (S, Y), where F′ is an arbitrary filter of Y. Thus, it is obtained

*	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	c	c
c	1	1	b	1	b
d	1	1	1	1	1

Consequently, one may see h (S) ≤ h (T).

Similarly, one can show that h (T) ≤ h (S). □

Proposition 3.27. Let (β (X) , T, m) be a BE-algebra dynamical system. Then for any integer k > 0, h (T^k) = kh (T).

Proof. If E is a filter of X that $\frac{X}{E}$ is countable, then $\begin{matrix} h_{E} (T^{k}, X) & = h (T^{k}, \frac{X}{E}) \\ = h (T^{k}, \frac{X}{⋁_{i = 0}^{k - 1} (T^{- i} \frac{X}{E})}) \\ = lim_{n \to \infty} \frac{1}{n} H (⋁_{j = 0}^{n - 1} T^{- jk} (⋁_{i = 0}^{k - 1} T^{- i} \frac{X}{E})) \end{matrix}$ $\begin{matrix} = lim_{n \to \infty} \frac{1}{n} H (⋁_{j = 0}^{n - 1} ⋁_{i = 0}^{k - 1} (T^{- kj - i} \frac{X}{E})) \\ = lim_{n \to \infty} \frac{1}{n} H (⋁_{i = 0}^{nk - i - 1} (T^{- i} \frac{X}{E})) \\ = lim_{n \to \infty} \frac{nk}{n} \frac{1}{nk} H (⋁_{i = 0}^{nk - 1} T^{- i} \frac{X}{E}) \\ = {kh}_{E} (T, X), \end{matrix}$ and $\begin{matrix} h (T^{k}) & = sup {h_{E} (T^{k}, X) : E is a filter of X, \frac{X}{E} is countable} \\ = k sup {h_{E} (T, X) : E is a filter of X, \frac{X}{E} {is countable \end{matrix}} .$ Then, we have h (T^k) = h_{T
^k} (X) = kh_T (X) = kh (T). □

The well - known Kolmogorov-Sinai Theorem on generators is the main tool used to calculate the entropy of dynamical systems. Now the entropy of BE-algebra dynamical system is concluded which is the goal in this paper.

Definition 3.28. Let E be a filter of a transitive BE-algebra X. Then E is said to be a generator of a dynamical system (β (X) , T, m), if for any filter F of X there exists an integer k > 0 such that $\frac{X}{F} < ⋁_{i = 0}^{k} (T^{- i} \frac{X}{E})$ .

Proposition 3.29. Let E be a generator of a dynamical system (β (X) , T, m). Then h (T) = h_T (X) = h_E (T, X).

Proof. Let E be a generator of a dynamical system (β (X) , T, m), so for arbitrary filter F of X, there exists an integer k > 0 such that $\frac{X}{F} < ⋁_{i = 0}^{k} (T^{- i} \frac{X}{E})$ . Then one can get $\begin{matrix} h_{E} (T, X) & \leq h_{⋁_{i = 0}^{k} (T^{- i} \frac{X}{E})} (T, X) \\ = h (T, \frac{X}{⋁_{i = 0}^{k} (T^{- i} \frac{X}{E})}) \\ = h (T, \frac{X}{E}) = h_{E} (T, X) . \end{matrix}$ Therefore, it is obtained $\begin{matrix} h (T) & = h_{T} (X) = sup {h_{F} (T, X) : F is a filter of X, \frac{X}{F} is countable} \\ \leq h_{E} (T, X) \leq h (T) . \end{matrix}$ This completes the proof. □

Definition 3.30. Let E and F be two filters of a transitive BE-algebra X. By $\underset{(E, F)}{I (X)}$ , we call the information gain of E ∨ F related to E and F and is calculated as follows: $\underset{(E, F)}{I (X)} = ∣ \underset{E}{H (X)} - \underset{E | F}{H (X)} ∣ .$ (4)

Corollary 3.31. Let E and F be two independent filters of a transitive BE-algebra, then $\underset{(E, F)}{I (X)} = 0$ .

Proof. By Lemma 3.20, since E and F are independent, $\underset{E | F}{H (X)} = \underset{E}{H} (X)$ , $\underset{(E, F)}{I (X)} = 0$ . □

Proposition 3.32. Let E, F, G be filters of a transitive BE-algebra X. If E ⊆ F, then either $\underset{(E, G)}{I (X)} \leq ∣ \underset{E}{H (X)} - \underset{F}{H (X)} ∣ + \underset{(F, G)}{I (X)}$ or $\underset{(E, G)}{I (X)} \geq ∣ \underset{E}{H (X)} - \underset{F}{H (X)} - \underset{(F, G)}{I (X)} ∣$ .

Proposition 3.33. Let E, F be two filters of a transitive BE-algebra X such that $F \overset{\circ}{=} E$ . Then $\underset{(E, F)}{I (X)} = \underset{(F, E)}{I (X)}$ .

Proof. By prop. 3.17, since $F \overset{\circ}{=} E$ , $\underset{E}{H (X)} = \underset{F}{H (X)}$ , $\underset{E | K}{H} = \underset{F | K}{H}$ for each filter K of X. Then

$\begin{matrix} \underset{(E, F)}{I (X)} = ∣ \underset{E}{H (X)} - \underset{E | F}{H (X)} ∣ = ∣ \underset{E}{H (X)} ∣ = \underset{E}{H (X)} \\ = \underset{F}{H (X)} = ∣ \underset{F}{H (X)} ∣ = ∣ \underset{F}{H (X)} - \underset{F | E}{H (X)} ∣ = \underset{(F, E)}{I (X)} . \end{matrix}$ □

Proposition 3.34. Let E, F and G be filters of a transitive BE-algebra X, whenever F and E ∨ G be independent. Then $\underset{(E \lor F, G)}{I (X)} \leq \underset{(E, G)}{I (X)} + ∣ \underset{F | E}{H} - \underset{F | E \lor G}{H} ∣$ .

Proof.

$\begin{matrix} \underset{(E \lor F, G)}{I (X)} = ∣ \underset{E \lor F}{H (X)} - \underset{E \lor F | G}{H (X)} ∣ \leq ∣ \underset{E}{H (X)} - \underset{E | G}{H (X)} ∣ \\ + ∣ \underset{F | E}{H (X)} - \underset{F | E \lor G}{H (X)} ∣ = \underset{(E, G)}{I (X)} + ∣ \underset{F | E}{H (X)} - \underset{F | E \lor G}{H (X)} ∣ . \end{matrix}$ □

Example 9. Consider example 3, X = [0, 1] and $x * y = {\begin{matrix} 1, & x \leq y; \\ y, & y < x \end{matrix}$

(X, ∗ , 1) is a transitive BE-algebra.

$E = [\frac{1}{2}, 1]$ and $F = [\frac{1}{3}, 1]$ are filters of [0, 1]. If x ∗ y ∈ E, x ∈ E, then $\frac{1}{2} \leq x * y \leq 1$ , $\frac{1}{2} \leq x \leq 1$ , in the case x ≤ y, x ∗ y = 1 ∈ E then $\frac{1}{2} \leq y \leq 1 \Rightarrow y \in E$ . In the case y < x, $x * y = y \in E \Rightarrow \frac{1}{2} \leq y \leq 1$ . Similarly one can check that F so is.

$E_{x} = \frac{x}{E} = {y \in [0, 1] : x * y \in E, y * x \in E}$ , x ∈ [0, 1]. Let $y \in E_{x} = \frac{x}{E}$ .

If x ≤ y ⇒ x ∗ y = 1, $y * x = x \in [\frac{1}{2}, 1] \Rightarrow x \in E, y \in E$ .

If $y \leq x \Rightarrow x * y = y, y * x = 1 \Rightarrow y \in E \Rightarrow x \in [\frac{1}{2}, 1]$ . Then $E_{x} = \frac{x}{E} \subseteq E$ , for $x \geq \frac{1}{2}$ .

Conversely if $y \in E = [\frac{1}{2}, 1]$ , let x ∈ [0, 1] there are two cases:

if x ≤ y ⇒ x ∗ y = 1 ∈ E, y ∗ x = x ∈ E.

if y < x ⇒ x ∗ y = y ∈ E, y ∗ x = 1 ∈ E.

Then $y \in \frac{x}{E}$ and $x \geq \frac{1}{2}$ , i.e, E_x = E, for each $x \in [\frac{1}{2}, 1]$ and E_x = {x} for each $x \in [0, \frac{1}{2})$ .

For the same way one can shows that $F_{y} = F, \forall y \in [\frac{1}{3}, 1]$ and $F_{y} = {y}, \forall y \in [0, \frac{1}{3})$ . $H_{E} (X) = - log sup_{x \in [0, 1]} m (E_{x}) = - log sup_{x \in [0, 1]} m (x) = - log \frac{1}{2} = log 2 .$ (5)

$\begin{matrix} \underset{E | F}{H (X)} = - log sup_{x \in [0, 1]} \frac{m (E_{x} \cap F_{y})}{m (F_{y})} = - log \\ sup_{x \in [\frac{1}{2}, 1]} \frac{m (E_{x} \cap F_{y})}{m (F_{y})} = - log \frac{\frac{1}{2}}{\frac{1}{3}} = log 3 - log 2 . \end{matrix}$ $\underset{(E, F)}{I (X)} = ∣ \underset{E}{H (X)} - \underset{E | F}{H (X)} ∣ = 2 log 2 - log 3 = log \frac{4}{3} .$ (6) (m is the Lebesgue measure on [0, 1]).

4 Conclusion

In this paper, the concept of a BE-algebra dynamical system was put forward using the notion of state on this structure. The entropy of a BE-algebra dynamical system was introduced and the fundamental properties were studied. It was proved that isomorphic dynamical systems on a BE-algebra have the same entropy. By some examples it was shown that some of classic theorems in ergodic theory are not satisfied for BE-algebra structures. A new version of Kolmogorov - Sinai Theorem was given. The concept of information gain related to filter entropy was defined and its properties were proved.

Limitations: Unavailablity of some research resources including books and full text papers are our limitations.

Recommendation: One can define the entropy function on a hyper BE-algebra and study its characteristics for future research. In addition, it would be helpful if one consider the entropy and information gain simultaneously for calculating the impact of supply chain information disruption.

Footnotes

Compliance with ethical standards

Funding

The authors declare that no funds, grants or other supports were received during the preparation of this manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.

Informed consent

The authors have the informed consent and they are very indebted to the referees for valuable suggestions for improving the readability of the manuscript.

Authors contributions

All authors contributed to the study conception and design. The first draft of the manuscript was written by [Mohamad Ebrahimi] and all authors commented on previous versions of the manuscript.

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