Applications of (Neutro/Anti)sophications to Semihypergroups

In this paper, we extend the notion of semi-hypergroups (resp. hypergroups) to neutro-semihypergroups (resp. neutrohypergroups). We investigate the property of anti-semihypergroups (resp. anti-hypergroups). We also give a new alternative of neutro-hyperoperations (resp. anti-hyperoperations), neutro-hyperoperation-sophications (resp. anti-hypersophications). Moreover, we show that these new concepts are different from classical concepts by several examples.


Introduction
A hypergroup, as a generalization of the notion of a group, was introduced by F. Marty [1] in 1934. e first book in hypergroup theory was published by Corsini [2]. Nowadays, hypergroups have found applications to many subjects of pure and applied mathematics, for example, in geometry, topology, cryptography and coding theory, graphs and hypergraphs, probability theory, binary relations, theory of fuzzy and rough sets and automata theory, physics, and also in biological inheritance [3][4][5][6][7]. e first book in semihypergroup theory was published by Davvaz in 2016 (see [8]). In recent years, several other valuable books in hyperstructures have been written by Davvaz et al. [6,9,10].
M. Al-Tahan et al. introduced the Corsini hypergroup and studied its properties as a special hypergroup that was defined by Corsini. ey investigated a necessary and sufficient condition for the productional hypergroup to be a Corsini hypergroup, and they characterized all Corsini hypergroups of orders 2 and 3 up to isomorphism [3]. Semihypergroup, hypergroup, and fuzzy hypergroup of order 2 are enumerated in [7,11,12]. S. Hoskova-Mayerova et al. used the fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups, defined the different operations on fuzzy multi-hypergroups, and extended the fuzzy hypergroups to fuzzy multihypergroups [13].
In 2019 and 2020, within the field of neutrosophy, Smarandache [14][15][16] generalized the classical algebraic structures to neutroalgebraic structures (or neutroalgebras) (whose operations and axioms are partially true, partially indeterminate, and partially false) as extensions of partial algebra and to antialgebraic structures (or antialgebras) (whose operations and axioms are totally false). Furthermore, he extended any classical structure, no matter what field of knowledge, to a neutrostructure and an antistructure. ese are new fields of research within neutrosophy. Smarandache in [16] revisited the notions of neutroalgebras and antialgebras, where he studied partial algebras, universal algebras, effect algebras, and Boole's partial algebras and showed that neutroalgebras are the generalization of partial algebras. Also, with respect to the classical hypergraph (that contains hyperedges), Smarandache added the supervertices (a group of vertices put together to form a supervertex), in order to form a super-hypergraph. en, he extended the superhypergraph to n-super-hypergraph, by extending the power set P(V) to P n (V) that is the n-power set of the set V (the nsuper-hypergraph, through its n -super-hypergraph-vertices and n -superhypergraph-edges that belong to P n (V), can be the best (so far) to model our complex and sophisticated reality). Furthermore, he extended the classical hyperalgebra to n-ary hyperalgebra and its alternatives n -ary neutrohyperalgebra and n -ary anti-hyperalgebra [17]. e notion of neutrogroup was defined and studied by Agboola in [18].
Recently, M. Al-Tahan et al. studied neutro-ordered algebra and some related terms such as neutro-ordered subalgebra and neutro-ordered homomorphism in [19].
In this paper, the concept of neutro-semihypergroup and anti-semihypergroup is formally presented. And, new alternatives are introduced, such as neutro-hyperoperations (resp. anti-hyperoperations), neutro-hyperaxioms, and antihyperaxioms. We show that these definitions are different from classical definitions by presenting several examples. Also, we enumerate neutro-hypergroup and anti-hypergroup of order 2 (see Table 1) and obtain some known results (see Table 2).

Preliminaries
In this section, we recall some basic notions and results regarding hyperstructures.

On Neutro-hypergroups and Antihypergroups
F. Smarandache generalized the classical algebraic structures to the neutroalgebraic structures and antialgebraic structures. Neutro-sophication of an item C (that may be a concept, a space, an idea, a hyperoperation, an axiom, a theorem, a theory, an algebra, etc.) means to split C into three parts (two parts opposite to each other, and another part which is the neutral/indeterminacy between the opposites), as pertinent to neutrosophy ((〈A〉, 〈neutA〉, 〈antiA〉), or with other notation (T, I, F)), meaning cases where C is partially true (T), partially indeterminate (I), and partially false (F), while antisophication of C means to totally deny C (meaning that C is made false on its whole domain) (see [14,15,17,20]). Neutrosophication of an axiom on a given set X means to split the set X into three regions such that, on one region, the axiom is true (we say the degree of truth T of the axiom), on another region, the axiom is indeterminate (we say the degree of indeterminacy I of the axiom), and on the third region, the axiom is false (we say the degree of falsehood F of the axiom), such that the union of the regions covers the whole set, while the regions may or may not be disjoint, where (T, I, F) is different from (1, 0, 0) and from (0, 0, 1).
Antisophication of an axiom on a given set X means to have the axiom false on the whole set X (we say total degree of falsehood F of the axiom) or (0, 0, 1).  Neutrosophication of a hyperoperation defined on a given set X means to split the set X into three regions such that, on one region, the hyperoperation is well-defined (or inner-defined) (we say the degree of truth T of the hyperoperation), on another region, the hyperoperation is indeterminate (we say the degree of indeterminacy I of the hyperoperation), and on the third region, the hyperoperation is outer-defined (we say the degree of falsehood F of the hyperoperation), such that the union of the regions covers the whole set, while the regions may or may not be disjoint, where (T, I, F) is different from (1, 0, 0) and from (0, 0, 1).
Antisophication of a hyperoperation on a given set X means to have the hyperoperation outer-defined on the whole set X (we say total degree of falsehood F of the axiom) or (0, 0, 1).
In this section, we will define the neutro-hypergroups and anti-hypergroups.

Definition 6.
A neutro-hyperoperation is a map ∘: H × H ⟶ P(U), where U is a universe of discourse that contains H that satisfies the below neutrosophication process.
e neutrosophication (degree of truth, degree of indeterminacy, and degree of falsehood) of the hypergroup axiom of associativity is the following neutroassociativity (NA): and H are not all three equal, or some of them are indeterminate). Also, we define the neutrocommutativity (NC) on (H, ∘ ) as follows: Now, we define a neutro-hyperalgebraic system S � 〈H, F, A〉, where H is a set or neutrosophic set, F is a set of the hyperoperations, and A is the set of hyperaxioms, such that there exists at least one neutro-hyperoperation or at least one neutro-hyperaxiom and no anti-hyperoperation and no anti-hyperaxiom.
e anti-hypersophication (totally false) of the hypergroup is as follows: H)(H ∘ a, a ∘ H, and H are not equal) (antireproduction axiom) Also, we define the anticommutativity (AC) on (H, ∘ ) as follows: (AC) (∀a, b ∈ H with a ≠ b) (a ∘ b ≠ b ∘ a).
Definition 8. A neutro-semihypergroup is an alternative of semi-hypergroup that has at least (NH) or (NA), which does not have (AA). H � a, b, H � a, b,

Journal of Mathematics 3
Definition 10. A neutrocommutative hypergroup is a hypergroup that satisfies (NC). H � a, b, c, d,   en, (H, ∘ 5 , e) is a group and so is a natural hypergroup. Also, it is a neutrocommutative hypergroup, since

Definition 11.
A neutrohypergroup is an alternative of hypergroup that has at least (NH) or (NA) or (NR), which does not have (AA) and (AR).
Define the hyperoperation ∘ 6 on H with Cayley's table.
Definition 12. An anti-semihypergroup is an alternative of semi-hypergroup that has at least (AH) or (AA).
Define the hyperoperation ∘ 9 on H with Cayley's table.