1. Introduction

The structure of groups is used in algebra and their orders for finite groups are more important is used in many parts of mathematics, as well as in quantum chemistry and physics. For example Lagrange's theorem [16] about the orders of finite groups are studied to find the number of the solutions of equations in finite groups see ([9]-[11], [13]). algebra, class of algebra of logic, was introduced by Imai and Iseki [4]. In 1999, the concept of algebras, another generalization of the concept of algebras, was introduced by Neggers and Kim [14]. They studied some properties of this class of algebras. Since then many researchers have extensively studied these algebras (see [1]-[3], [5], [18]). In [6], Yonghong Liu introduced a new class of abstract algebra (BCL-algebra) and then he introduced a wide class of abstract algebras (algebra) ([7]). After that some fundamental properties of topological algebras are obtained ([8]).

In 2014, the concept of permutation topological space where is a permutation in symmetric group , was introduced by Shuker [12]. The aim of this paper is to introduce new class of algebra it's called algebra. Also, the relations between algebra and some algebras like algebra, algebra and algebra are studied. Further, the concept of subalgebra is introduced and showed that in algebra Lagrange's theorem is not true in general. So, there is no law determines the relation between cardinality of algebra and cardinalities of their subalgebras. In this work, the notations of ideal, and ideal in algebra are introduced and investigated their relations with importing types in algebra like ideal, subalgebra and ideal. Further, the multiplication permutation map is given and then a permutation topological algebra is defined and explained. In another words, permutation topological algebra has the algebraic structure of a algebra and the permutation topological structure of a topological space and they are linked by the requirement that multiplication permutation is continuous function. Moreover, several examples are given to illustrate the concepts introduced in this paper.

2. Preliminaries

In this section we recall the basic definition and information which are needed in our work.

Definition 2.1: [11]

A partition is a sequence of nonnegative integers with and . The length and the size of are defined as and We set for . An element of is called a partition of

Remark 2.2:

We only write the non zero components of a partition. Choose any and write it as . With disjoint cycles of length and is the number of disjoint cycle factors including the 1-cycle of . Since disjoint cycles commute, we can assume that . Therefore is a partition of and each is called part of (see [9]).

Definition 2.3: [10]

We call the partition the cycle type of .

Definition 2.4: [17]

Suppose first that Then the support of , is the set where . So we say and are disjoint cycles iff

Definition 2.5: [12]

Suppose is permutation in symmetric group on the set and the cycle type of is , then composite of pairwise disjoint cycles where For any cycle in we define set as and is called set of cycle . So the sets of are defined by

Remark 2.6: [12]

For any cycle in we put , Further, suppose that and are sets in where and We will give some definitions needed in this work.

Definition 2.7: [12]

We call and are disjoint sets in , if and only if and there exists , for each such that

Definition 2.8: [12]

We call and are equal sets in , if and only if for each there exists such that

Definition 2.9: [12]

We call is contained in and denoted by , if and only if .

Definition 2.10: [12]

We define the operations and on sets in as followers:

and

Remarks 2.11: [12]

1. The intersection of and is

2. The union of and is

3. The complement of is

4. The intersection and union of and are and , respectively.

5. The intersection and union of and are and , respectively.

Definition 2.12: [12]

Let be permutation in symmetric group , and composite of pairwise disjoint cycles , where , then is a permutation topological space where and is a collection of sets of the family union and empty set.

Definition 2.13: [12]

If is set in the space then is called closed set in the space and is smallest closed set containing or equal and any set is called closed set iff

Definition 2.14: [12]

The set is called the interior of the set in the permutation space

Remarks 2.15: [12]

1. We call belong to set iff for some

2. The condition means that Therefore, is an interior point of set if and only if there is an open set containing and such that

3. If and are disjoint sets in , then neither nor

Remark 2.16: [12] Any map between two permutation topological spaces is called permutation map.

Definition 2.17: [12]

Let and be three permutations in symmetric group and let be a function, where for each set , the image of under is called set and defined by the rule . In another direction, let be set, the inverse image of under is called set and defined by the rule . The usual properties relating images and inverse images of subsets of complements, unions, and intersections also hold for permutation sets.

Definition 2.18: [12]

Given permutation topological spaces and , a function is permutation continuous if the inverse image under of any open set in is an open set in (i.e whenever

Lemma 2.19: [12]

The identity permutation in symmetric group is a permutation continuous on a permutation space

Lemma 2.20: [12]

A composition of permutation continuous functions is permutation continuous.

Definition 2.21: [14] A algebra is a non-empty set with a constant 0 and a binary operation* satisfying the following axioms:

i)-

ii)-

iii)- and imply that for all x, y in X.

Remark 2.22: [14] Let be algebra. Then is called finite algebra if is a finite set.

Definition 2.23: [15] A algebra is called algebra if satisfying the following additional axioms:

(1).

(2). for all

Definition 2.24: [15] Let be a algebra and Then I is called a subalgebra of algebra if whenever and

Definition 2.25: [15] Let be a algebra and Then I is called a ideal of algebra if

(1). and

(2). and for all

Definition 2.26: [15] Let be a algebra and Then I is called a ideal of algebra if

(1).

(2). and for all

Definition 2.27: [15] Let be a algebra. Then is called a algebra if it satisfies the identity for all

Remark 2.28: [15] In algebra any ideal is ideal and subalgebra.

Theorem 2.29: [16] (Lagrange's theorem) Let be a finite group and a subgroup of Then divides

Definition 2.30: [14]: Let be a algebra and . Define . Then is said to be edge if , for all

3. Characterizations of ρ-algebra

Definition 3.1 A algebra is a non-empty set with a constant and a binary operation* satisfying the following axioms:

i)-

ii)-

iii)- imply that ,

iv)- For all imply that

Remark 3.2: It is clear every algebra is algebra, but the converse is not true in general.

Example 3.3: Let and let the binary operation * be defined as follows:

Table (1)

It is clear that is a algebra, but not algebra, since there are two elements and

Definition 3.4 Let be a algebra and . is called a subalgebra of if whenever and .

Theorem 3.5 Let be a algebra and . Then is subalgebra of , if is subalgebra of

Proof: Suppose that is algebra and is subalgebra of . Then we consider that is algebra. Also, satisfies whenever and Hence is subalgebra of algebra .

Remark 3.6 From theorem (3.5) we consider that every subalgebra is subalgebra. However, the converse is not true in general.

Example 3.7 Let be a algebra with the following table:

Table (2)

Then is subalgebra of . Further, is not algebra and hence is not subalgebra of

Definition: 3.8

For any positive integer Let be a finite set and Define binary operation on as follows:

Further, this type of algebra is denoted by

Proposition: 3.9 Let be any positive integer. Then,

1)- Each element in has inverse under the binary operation with right identity.

2)- The mathematical system is neither commutative system nor associative system.

3)- The mathematical system with a constant is algebra.

4)- If then the number of subalgebra orsubalgebra of is

Proof:

(1) It is clear for any there exists right identity element Moreover, for any there exists inverse element where

(2) Let we have . Then the mathematical system is not a commutative.

Now, we need to show that the binary operation is not associative, let Where and hence is not associative system.

(3) Since , for all Then for each we consider that is a constant element and hence the following are hold:

i)-

ii)-

iii)- and imply that for all Thus is algebra.

iv)- For all imply that Then is algebra.

4) Let , then for all we consider that . Also, for any we have ( by definition 3.4). Then is a subalgebra and subalgebra of and hence the number of subalgebra or subalgebra of is

Notations on Algebra Using Type : 3.10

We will show that Lagrange's Theory is also incorrect for finite algebra by a counterexample. Let , where is prime number. Then is algebra and it is clear that for each we consider that is subalgebra of finite algebra In another side Hence does not divide That means in algebra Lagrange's theorem is not true in general. Moreover, for any where we consider that Therefore algebra need not be algebra. Further, for any and , we consider that Hence algebra need not be algebra. Also, is not edge, if . Since for any we consider that and but neither nor . Moreover, is edge, if . Since and hence for any we have

Definition 3.11: Let be a algebra and Then K is called a ideal of algebra if (1). imply

(2). and imply , for all

Example 3.12: It is clear, and are ideal for any algebra X. Moreover, if is a algebra. Then every ideal of is a algebra with the same binary operation on and the constant

Remark 3.13: By condition (1) in definition 3.11, we consider that every ideal is subalgebra and hence subalgebra.

Theorem 3.14: In algebra every ideal is ideal.

Proof: Suppose that is a ideal in Now, we need to prove that:

(1). imply

(2). and imply , for all

Since is a ideal, then condition (2) is hold. Moreover, for any we have and (since ). This implies that (by condition (2) in definition 2.25). Also, since is algebra then we have is ideal.

Remark 3.15: In algebra above theory is not true in general.

Example 3.16: Let be a algebra with the following table:

Table (3)

Then is a ideal of . Further, is not algebra and hence is not -ideal of

Theorem 3.17: If is a ideal of algebra , then is a ideal of algebra .

Proof: Suppose that is a ideal in algebra . Then is non-empty subset of and is algebra. Thus, we need only to prove that:

(1).

(2). and imply , for all

Since is a ideal, then condition (2) is hold. Also, there is at least (since ). This implies that (by condition (1) in definition 3.11), but and hence . Then is a ideal of algebra .

Definition 3.18: Let be a algebra and be a subset of . Then is called ideal of algebra if

(1).

(2). and for all

Example 3.19: Let be a algebra with the following table:

Table (4)

Then is ideal of . Further, is not ideal of , since and , but

Remark 3.20 It is easy to show that every ideal is subalgebra. However, the converse is not true and the following example showing that

Example 3.21: Let be a algebra with the following table:

Table (5)

Then is subalgebra of . However, is not algebra and hence is not ideal of .

Remark 3.22: By the above results we have the following diagram:

Figure 1. Diagram showing relationships among some types of algebras

Definition: 3.23 (Multiplication Permutation Map)

Let and be two permutations in symmetric group . Then and are two permutation maps from onto . Further, is a product map of permutation maps where In another side, the map is a permutation in as this form

Now, let be a binary operation on and be a map defined by Then the permutation map from permutation space into () for any permutation in symmetric group is called multiplication permutation map. Further, it is called multiplication permutation continuous iff the inverse image under of any open set in is an open set in (i.e whenever ).

Example: 3.24 Suppose that and are permutations in symmetric group with and let be a binary operation on where We consider that the multiplication permutation map , where is a multiplication permutation continuous map.

Definition 3.25 For any permutation in symmetric group let be a permutation topological space and be a algebra. If is a continuous permutation mapping from permutation space into , where is product topology of , then we say that is a permutation topological algebra.

Example: 3.26 Let be a permutation in symmetric group . Then is permutation topological space, where and Also, let be a algebra with the following table:

Table (6)

It is clear that is an indiscrete permutation space. Thus is multiplication permutation continuous map, . Then is a permutation topological algebra.

Remark 3.27: Finally, our new notion (see, Definition 3.25) is given and hence this notion of permutation topological algebra can be considered a special case of topological algebra using member in finite group.

4. Conclusions

We have initiated a study of algebras and explained their relations with algebras. Moreover, the multiplication permutation map is given and then a permutation topological algebra is defined and explained. In future work, we will study the relation between algebras and algebras. Moreover, we will investigate some new types of permutation spaces using members in subgroups of symmetric groups instead symmetric groups like Mathieu group Alternating group Quaternion group and others. Further, we will consider new constructer in topological algebra is called sub permutation topological algebra, since each one of these groups on n letters is a subgroup of symmetric group