**2. Preliminaries**


 

 

**Definition 1.** *A hypergroupoid* (*L*, )) *is a nonvoid set L with a hyperoperation* ), *which is a map* ) : *L* × *L* → *P*∗(*L*), *where P*∗(*L*) *implies the family of all nonvoid subsets of L [9]. Denote c* ) *d as the hyperproduct of c and d for every c*, *d* ∈ *L*. *A hypergroupoid* (*L*, )) *is described as a semihypergroup if L has associative property, i.e.,* (*c* ) *d*) ) *e* = *c* ) (*d* ) *e*) *for all c*, *d*,*e* ∈ *L*. *A hypergroup is a semihypergroup along with reproductivity axiom, that is e* ) *L* = *L* ) *e* = *L for all e* ∈ *L*. *A hypergroupoid* (*L*, )) *is called quasihypergroup if the reproductivity property holds. The hypergroup is commutative if e* ) *f* = *f* ) *e for all e*, *f* ∈ *L*. *A nonvoid subset M of a hypergroup L is a subhypergroup of L if z* ) *M* = *M* ) *z* = *M for every z* ∈ *M*.

*Assume E and F are nonvoid subsets of L*, *hence E* ) *F* = *<sup>e</sup>*∈*E*, *<sup>f</sup>*∈*<sup>F</sup> e* ) *f* . *Moreover, l* ∈ *L and E* ⊆ *L*, *we have l* ) *E* = *<sup>e</sup>*∈*<sup>E</sup> l* ) *e*. *If associativity holds, then we denote the hyperproduct of elements x*1,..., *xn of L by* ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *xi* := *x*<sup>1</sup> ) *x*<sup>2</sup> ) ... ) *xn.*

*Suppose that* (*L*, )) *and* (*L* , ) ) *are two hypergroups. A map ψ* : *L* −→ *L is determined as a* homomorphism *if ψ*(*k* ) *l*) = *ψ*(*k*) ) *ψ*(*l*) *for all k*, *l* ∈ *L*. *Furthermore, ψ is named an* isomorphism *if it is one to one and onto homomorphism written by L* ∼= *L* .

The following Definition 2, Proposition 1, Theorem 1, Proposition 2, and Theorem 2 are taken from [31].

**Definition 2.** *Assume that L is a nonvoid set and σ is a binary relation on L*. *Consider the following hypercomposition "*◦*" on L as:*

$$x \circ y = \{ z \in L : (x, z) \in \sigma, \ (z, y) \in \sigma \}\tag{1}$$

(*L*, ◦) *is a hypergroupoid provided there exists z* ∈ *L so that* (*x*, *z*) ∈ *σ and* (*z*, *y*) ∈ *σ for every couple of elements x*, *y* ∈ *L*.

*Denote the hypercompositional structure in Equation* (1) *by Lσ*. *The reproductivity property in L<sup>σ</sup> is satisfied if and only if* (*x*, *y*) ∈ *σ for all x*, *y* ∈ *Lσ*.

#### **Proposition 1.**


**Theorem 1.** *Let σ be a binary relation on the nonvoid set L*. *Then, the hypercomposition x* ◦ *y satisfies the reproductivity or associativity only when L<sup>σ</sup> is total (i.e., x* ◦ *y* = *Lσ).*

Each relation *σ* on finite set *L* = {*a*1, *a*2, ... , *an*} can be represented through a Boolean matrix *M<sup>σ</sup>* with *n* × *n* elements. The Boolean matrix *M<sup>σ</sup>* = (*mij*) is defined as follows:

$$m\_{i\bar{j}} = \begin{cases} 1, & \text{if } (a\_i, a\_{\bar{j}}) \in \sigma \\ 0, & \text{otherwise} \end{cases}$$

In Boolean algebra, we have

$$\begin{aligned} 0 + 1 &= 1 + 0 = 1 + 1 = 1, \; 0 + 0 = 0 \\ 0.0 = 0.1 &= 1.0 = 0, \; 1.1 = 1 \end{aligned}$$

*L<sup>σ</sup>* is hypergroupoid if and only if *M*<sup>2</sup> *<sup>σ</sup>* = *S*, where *S* = (*sij*) with *sij* = 1 for all *i*, *j*.
