Entropy Treatment of Evolution Algebras.

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.

1. Introduction

The theory of non-associative algebras is an important branch of abstract algebra. Such kinds of algebras include baric, evolution, Bernstein, train, and stochastic algebras. These types of objects were tied up with the abstract description of biological systems [1,2,3,4].

Let be an algebra over a field , where is called an evolution algebra if it admits a basis such that

The matrix is called the structure matrix of relative to B. A basis B satisfying (2) is called the natural basis of We say that is a non-negative evolution algebra if and the structure matrix entries are non-negative.

These kinds of algebras were first considered in [5,6,7] and have been exhaustively studied over the recent years (see [8,9,10,11,12,13,14,15,16] and references therein for a review of some of the main results achieved on this topic [17]). These algebras are related to a wide variety of mathematical subjects, including Markov chains and dynamical systems [18,19]. The relationship between evolution algebras and homogeneous discrete-time Markov chains was settled in [6]. We recall that Markov evolution algebra is a non-negative evolution algebra whose structure matrix A has row sums equal to 1.

Tian [6] proposed one of the most fruitful further topics of research: the development of the theory of continuous evolution algebras and their connection to continuous-time Markov processes. He outlined continuous evolution algebras to be evolution algebras using multiplication, with respect to a natural basis , such that for some functions .

Recently, Markov evolution algebras have been strongly connected with group theory, Markov processes, the theory of knots, dynamic systems, and graph theory [11,20,21,22,23,24]. In [25], Markov evolution algebras, whose stricture matrices obey semi-group property, were investigated. This type of study is related to the chain of evolution algebras [26].

On the other hand, recently, in [27], we introduced a new class of evolution algebras called S-evolution algebras. These algebras are not nilpotent and naturally extended Lotka—Volterra evolution algebras [18]. It is stressed that directed weighted graphs associated with S-evolution algebras have meaning, whereas those connected with Lotka—Volterra algebras do not.

Due to [28], the intersection of information theory and algebraic topology is fertile ground. For example, in [29], it was established that the Shannon entropy defines a derivation of the operad of topological simplices. On the other hand, it is important to construct invariants for evolution algebras which can detect their isomorphism. It turns out that such an invariant can be defined via the Shannon entropy for S-evolution algebras. In the present paper, we demonstrate how this entropy allows for the treatment of the isomorphism of S-evolution algebras. To be precise, we demonstrate that, if S-evolution algebras (symmetric) have different entropies, they are not isomorphic. This result enables the construction of many examples of non-isomorphic evolution algebras. As a result of the primary finding, we propose a non-isomorphic family of Markov evolution algebras. This result sheds new light on the Markov evolution algebras and their isomorphism problems.

Let us briefly describe the structure of this paper. Section 2 contains preliminary definitions of evolution algebra. In Section 3, we define the entropy of the structural matrix of the Markov evolution algebra, and we demonstrate that any isomorphic evolution algebra would produce the same Markov evolution algebra. Furthermore, we derive Markov evolution algebra through the Hadamard product of the structural matrix A of evolution algebra. We show that the entropy of such a matrix will be constant if whereas the entropy will be decreasing if This result allows us to construct a lot of non-isomorphic chains of Markov evolution algebras (see [26]). Finally, in Section 4, the relative entropy is defined, and we prove that such a function is a measure of the ‘distance’, even though it is not a metric space, since the symmetric axiom, in general, is not satisfied. In the case of symmetric evolution algebra, we show that this property is satisfied only in the class of isomorphic algebras.

2. Preliminaries

In this section, we recall the definitions of evolution algebra and some definitions which are needed throughout the paper. Let be a real non-negative evolution algebra with structure matrix and natural basis B. If and for any , then is called Markov evolution algebra. The name is due to the fact that there is an interesting one-to-one correspondence between and a discrete time Markov chain with the stated space and transition probabilities given by , i.e., for : for any .

For the sake of completeness, we wish to state that a discrete-time Markov chain can be thought of as a sequence of random variables defined in the same probability space, taking values from the same set , and such that the Markovian property is satisfied, i.e., for any set of values , and any , it holds

Notice that, in the correspondence between the evolution algebra and the Markov chain , what we have is each state of identified with a generator of B.

([27]). A matrix

We notice that, if is a , then there is a family of injective functions , with such that for all . Hence, each S-matrix is uniquely defined by off diagonal upper triangular matrix and a family of functions . This allows us to construct lots of examples of S-matrices.

Given an upper triangular matrix , one can construct several examples of S-matrices as follows:

symmetric matrices, i.e., ;

skew-symmetric matrices, i.e., ;

.

An evolution algebra

We note that evolution algebras corresponding to skew-symmetric matrices are called Lotka–Volterra evolution algebras. Such kinds of algebras have been investigated in [

One can see that the conical form of the table of multiplication of evolution algebra with respect to natural basis is given by

We note that, if , then the first part of (4) is zero, if , then the second part is zero.

The motivation behind introducing S-evolution algebra is that such algebras have certain applications in the study of electrical circuits, finding the shortest routes and constructing a model for analysis and solution of other problems [

A linear map is called the homomorphism of evolution algebras if for any Moreover, if ψ is bijective, then it is called an isomorphism. In this case, the last relationship is denoted by

Let

A graph graph attached to the evolution algebra relative to the natural basis B.

The triple weighted graph attached to the evolution algebra relative to the natural basis B.

Recall that if every two vertices of a graph are connected by an edge, then such a graph is called complete.

Using the graph , in [27], we have established the isomorphism of S-evolution algebras.

([27]). Let

3. -Evolution Algebras and Corresponding Markov Evolution Algebras

In what follows, we always assume that is a non-negative, symmetric S-evolution algebra with structure matrix and natural basis B. Using the matrix A, one can define a stochastic matrix as follows: where . Sometimes, is denoted by .

An evolution algebra with the natural basis B and structural matrix is a Markov evolution algebra corresponding to which is denoted by .

Our task now is to examine the isomorphism between and

Let

We notice that and are evolution algebras. So, the isomorphism between these two algebras can be checked by Theorem 1. Hence, the proof is straightforward. □

Consider a discrete random variable with possible values and probability mass function The entropy can be explicitly written as: where it is assumed that .

Now, given a non-negative symmetric evolution algebra with structure matrix , we define its entropy as follows: where is defined by (5).

We notice that the considered entropy has a relationship with the Jamiolkowski entropy of a stochastic matrix [

The Jamiolkowski entropy of

Hence, the entropy given by (

The obtained formula

On the other hand, if one defines the entropy of a stochastic matrix in the sense of [

This will allow us to further investigate the algebraic structure of

Let

Let . Due to the isomorphism between theses two algebras, we have

The corresponding Markov evolution algebras have the following matrices of structural constants: respectively. Let be an arbitrary entry of the matrix then

We may assume that (since the matrices are S-matrices). Hence,

From (12), one has

Hence,

Due to the arbitrariness of we obtain . This completes the proof. □

Assume that all conditions of Theorem 3 are satisfied. Then

We stress that the converse of Corollary 1 need not be true. Indeed, let

Using the condition from

The advantage of Theorem 3 is that, for any two non-negative

The natural question that arises is: if we have arbitrary isomorphic evolution algebras, are their entropies equal? The following example gives a negative answer.

Let

Clearly,

Now, we may calculate the entropies for both, which are

Given the evolution algebra the Hadamard product is defined as

Let us denote by By we denote the evolution algebra whose structural matrix is .

Assume that all conditions of Theorem 3 are satisfied and

If

Let and be the structure matrices of and , respectively, then , if and only if

Assume that then for any Next, let Then if and only if □

From above lemma, we emphasize the following points:

If

If

Now, let us consider , which is defined by

Let us denote

Then one has

Let

The entropy of

Let us calculate the entropy of (13). Due to the symmetry of , it is enough to find the value for the first row, the rest will process in the same manner. Hence, the value of at i will be denoted by , where

The last equality can be rewritten as

Now, let then, from (14) we have

Thus, Equation (16) can be rewritten as follows:

Therefore,

The last expression leads to the required assertion. This completes the proof. □

Let us denote with the set of all maximum entries of the row In what follows, we assume that ,

Let

(i). From Theorem 4, we infer that where and So, the first derivative of is given by Next, computing the second derivative of we have

Clearly, from the last equation, we find , then As , then Hence, is decreasing. This completes the proof of (i).

Now consider (ii). For the sake of simplicity of calculations, we may assume that where represent the row of where

Simple calculations yields that

Since then

Therefore, This completes proof (ii).

From the (ii), the maximum value of occurs at Putting in the expression we get But This implies that the maximum value of On the other hand, from (i) and (ii), we obtain that the minimum value of Hence, which yields (iii). □

If

From Theorem 5, we emphasize the following points:

In Theorem 5 (i), if

In Theorem 5 (i), if

From Theorem 5 and Corollary 1, if the

All non-negative

Let us consider the following examples:

Let

Then,

Then, the corresponding Markov evolution algebras have the following structure matrices:

One can see that

Then,

The following example is related to (4) of remark (18).

Let

Then,

Then, the corresponding Markov evolution algebras have the following structure matrices:

One can see that

Then,

Since

4. Relative Entropy

Suppose that we have two sets of discrete events, and , with the corresponding probability distributions, and . The relative entropy between these two distributions is defined by

This function is a measure of the ‘distance’ between and , even though it is not metric space, since the symmetric axiom in general is not satisfied

Let , be non-negative symmetric evolution algebras with matrix of structural matrices and . Let and be the corresponding stochastic matrices (see (5)). We define the relative entropy of A and B as follows:

Let

One can see that if we write as the row vectors and .

Now, we are going to compute For a fixed i, one has

If then

Substituting the last expression into (19), we obtain

The last expression can be rewritten as follows:

Since we have and Hence, from (21), we obtain

Due to the arbitrariness of we arrive at for any i. This completes the proof. □

5. Conclusions

In this paper, we introduced an entropy of Markov evolution algebras, and treated the isomorphism of the corresponding S-evolution algebras. It turns out that the considered entropy is a semi-invariant of non-negative symmetric evolution algebras. This work opens new insight to the isomorphism problem through the entropy theory. Moreover, we have pointed out that entropy can be investigated by means of quantum channels. Furthermore, a family of Markov evolution algebras is defined through the Hadamard product of the structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied through entropy. The isomorphism of any algebra is considered a crucial task. So, it is necessary to find a shortcut way that is effective and accurate to study such a problem. This paper treats this problem by using the entropy value in the class of evolution algebras. However, this property is not valid for general evolution algebras, as we have shown in Example 1. Therefore, for other types of algebras, it is better to find other kinds of entropies.