An algorithm for generating permutation algebras using soft spaces

ABSTRACT Soft set theory has recently gained significance for finding rational and logical solutions to various real-life problems, which involve uncertainty, impreciseness and vagueness. In this paper, we introduced an algorithm to find permutation algebras using soft topological space. Moreover, this class of permutation algebras is called even (odd) permutation algebras if its permutation is even (odd). Furthermore, new concepts in permutation algebras are investigated such as splittable permutation algebra and ambivalent permutation algebra. Furthermore, several examples are given to illustrate the concepts introduced in this paper.


Introduction
The concept of soft sets is a novel notion was introduced by Molodtsov [1]. Next, Shabir and Naz [2] introduced the notion of soft topological spaces. Some notions of this concept with its applications fundamental concepts of fuzzy soft topology and Intuitionistic fuzzy soft topology are studied by many mathematicians see the following references [3][4][5][6][7][8][9][10][11][12][13][14][15]. BCK-algebra as a class of abstract algebras is introduced by Imai and Iseki [16,17]. Next, the concept of d-algebras, which is another useful generalization of BCK-algebras, is introduced (see refs [18][19][20]). Also, the notion of d*-algebras is investigated [21]. After then, the concepts of ρ-algebras and r/r-ideals are introduced and studied [22]. In 2009, the concept of soft d-algebras is introduced (see Jun et al. [23]). Also, some extensions using power set are introduced such as soft ρ-algebra and soft edge ρ-algebra of the power set [24], soft BCL-algebras of the power set [25] and soft BCH-algebra of the power set [26]. Next, the notion of G-fuzzy d-algebra is given [27]. In this paper, we introduced an algorithm to find permutation algebras using soft topological space. Moreover, this class of permutation algebras is called even (odd) permutation algebras if its permutation is even (odd). Furthermore, new concepts in permutation algebras are investigated such as splittable permutation algebra and ambivalent permutation algebra. Furthermore, several examples are given to illustrate the concepts and introduced in this paper.

Preliminaries
In this section, we recall basic definitions and results that are needed later. Definition 2.1: We call the partition a = a(b) = (a 1 (b), a 2 (b), . . . , a c(b) (b)) the cycle type of b [28].
Definition 2.2: Let a be a partition of n. We define C a , S n to be the set of all elements with cycle type a [28].
where b is a permutation in alternating group A n . A(b) conjugacy class of b in A n is defined by [29] where C a+ are two classes of equal order in alternating group A n such that C a = C a+ < C a− and H n ={C a of S n | n . 1, with all parts a k of a different and odd}.

Proposition 2.4:
The conjugacy classes C a+ of A n are ambivalent if 4|(a i − 1) for each part a i of a [30].
Remark 2.6: Suppose that l b i and l b j are b-sets in V, where |l i | = s and |l j | = y. Then, the known definitions will be written as follows.
Definition 2.7: We call l b i and l b j are disjoint b-sets in V, if and only if s k=1 b i k = y k=1 b j k and there exists 1 ≤ d ≤ s, for each 1 ≤ r ≤ y such that b i d = b j r [31]. .
Definition 2.11: For any collection of not disjoint b-sets Definition 2.12: Let b be permutation in a symmetric group S n , and b composite of pairwise disjoint cycles n is a collection of b-set of the family {l i } c(b) i=1 union V and empty set [31]. Definition 2.15: Let U be an initial universe set and let E be a set of parameters [1]. A pair (F, A) is called a soft set (over X) where A # E and F is a multivalued function F:A P(X). In other words, the soft set is a parameterized family of subsets of the set X. Every set F(e), e [ E, from this family may be considered as the set of e-elements of the soft set (F, A), or as the set of e-approximate elements of the soft set. Clearly, a soft set is not a set. For two soft sets (F, A) and (G, B) over the common universe X, we say that (F, A) is a soft subset of (G, B) if A # B and for all e [ A, F(e) and G(e) are identical approximations. We write (F, A)#(G, B). (F, A) is said to be a soft superset of (G, B), if (G, B) is a soft subset of (F, A). Two soft sets (F, A) and (G, B) over a common universe X are said to be soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A). A soft set (F, A) over X is called a null soft set, denoted by F =(f,f), if for each F(e) = f, ∀e [ A. Similarly, it is called universal soft set, denoted by (X, E), if for each F(e) = X, ∀e [ A. The collection of soft sets (F, A) over a universe X and the parameter set A is a family of soft sets denoted by SS(X A ).  Definition 2.18: Let t be the collection of soft sets over X [2]. Then t is called a soft topology on X if t satisfies the following axioms: (i) F and (X, E) belong to t.
(ii) The union of any number of soft sets in t belongs to t. (iii) The intersection of any two soft sets in t belongs to t.
The triplet (X, E, t) is called a soft topological space over X. The members of t are called soft open sets in X and complements of them are called soft closed sets in X. Furthermore, (X, E, t) is said to be a soft indis- is said to be a soft discrete space over X, if t is the collection of all soft sets which can be defined over X. Some results on permutations 2.19: [29] ( Remark 2.20: In this work, for any set D = {d 1 , d 2 , . . . , d k } of k distinct objects and for any cycle , we will use the same symbol (||) to refer to the cardinality of set X and to refer to the length of the cycle B. Hence |D| = k and |B| = m.

Definition 2.21:
A d-algebra is a non-empty set X with a constant 0 and a binary operation* satisfying the following axioms [18]: x * y = 0 and y * x = 0 imply that y = x for all x, y in X.
Definition 2.24: Let (X, * , 0) be a d-algebra [21]. Then X is called a d * -algebra if it satisfies the identity (x * y) * x = 0, for all x, y [ X.
Here we used normal union (<), normal intersection (>) and empty set (f) then we have (X, Here we used normal intersection (>) between pairwise sets to find the set B. For each ) G is called permutation subspace induced by soft topology G where t g l b m is a family of all g l b -sets of disjoint cycles decomposition of g l b together with V ′ and the empty set.
3. An algorithm to generate permutation algebras from soft spaces In this section, we will introduce an algorithm to generate permutation algebra by analysis soft topological space and this class of permutation algebra is called even (odd) permutation algebra if its permutation is even (odd). Moreover, some basic properties of permutation spaces are studied.
Steps of the work 3.1: Let (X, E, G) be a soft topological space, where X = {s 1 , s 2 , . . . , s k }, E = {e 1 , e 2 , . . . , e n } and G = {F, (X, E), N be a map from X into natural numbers N defined by d i (s j ) = j + (i − 1)k, for all s j [ X and (1 ≤ i ≤ n) where k = |X|. Then s = n i=1 s i is called permutation in a symmetric group S nk where for all 1 ≤ i ≤ n, s i = P Hence, by Definition 2.5, we consider that: (V, t s h ) G is permutation topological space induced by soft topology G, where V = {1, 2, . . . , h}, h = nk and t s h is a collection of s-set of the family {s i } n i=1 together with V and the empty set. Now, we need to follow these steps: n which is holed as follows: n , #, f) G is a permutation d-algebra induced by soft topologyG.  Permutation subalgebras induced by soft topology G 3.3: Let (V, t b n ) G be a permutation space induced by soft topology G and l b , V. By Definition 2.7, we consider that (V ′ , t g l b m ) G is called permutation subspace induced by soft topology G where t g l b m is a family of all g l b -sets of disjoint cycles decomposition of g l b together with V ′ and the empty set. Also, (t g l b m , #, f) G is called permu- Remarks 3.4: If there are two permutation subspaces (V ′ , t g l b m ) G and (V ′′ , t g p b k ) G of (V, t b n ) G , and (t g l b m , #, f) G is permutation d/d * /BCK/r-subalgebra of (t b n , #, f) G induced by soft topology G, then (t g p b k , #, f) G need not be permutation d/d * /BCK/r-subalgebra of (t b n , #, f) G too. In addition, we assume that the "best house" in the opinion of his friend, say Mr Moreover, for all 1 ≤ i ≤ n, Then (t s 12 , #, f) G is permutation d-algebra, since # satisfies conditions for d-algebra. Also, (t g p s 10 , #, f) G is permutation d-subalgebra of (t s 12 , #, f) G . But (t g l s 4 , #, f) G is not permutation d-subalgebra of (t s 12 , #, f) G since t g l s 4 å t s 12 .    t b 9 be a binary operation defined in the following table:

Some notions of permutation algebras
Remark 4.5: It is clearly that every even (odd) permutation BCK/d * /r-algebra is even (odd) permutation d-algebra, but the converse need not be true, see Example 3.5 (t s 12 , #, f) G is an even permutation d-algebra but is not even permutation r-algebra.  The maps induced by soft topologies 4.8: Suppose that (X, E, t 1 ), (X, E, t 2 ) and (X, E, t 3 ) are three soft topological spaces over the common universe X the parameter set E with their permutations b, m and d in a symmetric group S n , and let d: c and this contradiction with our hypothesis. Hence Proof: Let (t b nk , #, f) G be a permutation d/BCK/d * /r-algebra induced by soft topological space (X = {x j } k j=1 , E = {e r } n r=1 , G) and for any pair c, for any ( 1 ≤ i ≤ n) and (1 ≤ j ≤ k). Then by Theorem 4.10), we consider that (V, t b nk ) G is a permutation space induced by soft topology G and its permutation b in a symmetric group S nk satisfies and hence nk − c(b) is even. This implies that b is even permutation in a symmetric group S nk . Then (t b nk , #, f) G is an even permutation d/BCK/d * /r-algebra. n , #, f) G be a permutation d/BCK/d * /r-algebra induced by soft topological space (X, E, G). Then (t b n , #, f) G is odd permutation d/BCK/d * /r-algebra, if (X, E, G) is a soft indiscrete topological space and 2|n.
Lemma 4.13: Let (t b n , #, f) G be a permutation r-algebra and H # t b n . Then H is permutation d-subalgebra induced by soft topology G, if H is permutation r-subalgebra induced by soft topology G.
Proof: Suppose that (t b n , #, f) G is permutation r-algebra and H is permutation r-subalgebra induced by soft topology G. Then we consider that (t b n , #, f) G is permutation d-algebra induced by soft topology G.     nk , #, f) G is an ambivalent permutation d/BCK/d * /r-algebra.

Methodology flowchart
The flow of this work is explained by using the chart in Figure 1 to generate permutation algebras from soft topological spaces.