# The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups

Authors: Jeong Soon Han; Hee Sik Kim; J. Neggers
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce the notion of hypergroupoids(HBinX,□), and show that (HBin(X),□) is a super-semigroup of the semigroup (Bin(X),□) via the identification x↔{x}. We prove that (HBin*(X),⊖,[∅]) is a BCK-algebra, and obtain several properties of (HBin*(X),□).


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## Body

## 1. Introduction

The notion of the semigroup(Bin(X),□) was introduced by Kim and Neggers [1]. Fayoumi [2] introduced the notion of the center ZBin(X) in the semigroup Bin(X) of all binary systems on a set X, and showed that if (X,•)∈ZBin(X), then x≠y implies {x,y}={x•y,y•x}. Moreover, she showed that a groupoid (X,•)∈ZBin(X) if and only if it is a locally zero groupoid. Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras [3, 4]. Neggers and Kim introduced the notion of d-algebras which is another useful generalization of BCK-algebras, and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs [5]. The present authors [6] defined several special varieties of d-algebras, such as strong d-algebras, (weakly) selective d-algebras, and pre-d-algebras, discussed the associative groupoid product (X;□)=(X;*)□(X;∘), where x□y=(x*y)∘(y*x). They showed that the squared algebra (X;□,0) of a pre-d-algebra (X;*,0) is a strong d-algebra if and only if (X;*,0) is strong.Zhan et al. [7] defined the T-fuzzy n-ary sub-hypergroups by using a norm T and obtained some related properties. Zhan, and Liu [8] introduced the notion of f-derivation of a BCI-algebras. They gave some characterizations of a p-semisimple BCI-algebras by using the idea of a regular f-derivation. Zhan et al. [9] defined the notion of hyperaction of a hypergroup as a generalization of the concept of action of a group. Recently, Davvaz and Leoreanu [10] published a beautiful book, Hyperring Theory and Applications, and provided useful information on the theory of the hypertheory.In this paper we introduce the notion of hypergroupoids(HBin(X),□), and show that (HBin(X),□) is a super-semigroup of the semigroup (Bin(X),□) via the identification x↔{x}. We prove that (HBin*(X),⊖,[∅]) is a BCK-algebra, and obtain several properties of (HBin*(X),□).


## 2. Preliminaries

Given a nonempty setX, we let Bin(X) the collection of all groupoids (X,*), where *:X×X→X is a map and where *(x,y) is written in the usual product form. Given elements (X,*) and (X,•) of Bin(X), define a product “□” on these groupoids as follows:
(2.1)(X,*)□(X,•)=(X,□),
where
(2.2)x□y=(x*y)•(y*x),
for any x,y∈X. Using the notion, H. S. Kim and J. Neggers showed the following theorem.Theorem 2.1 (see [1]).
(
Bin
(
X
)
,
□
) is a semigroup, that is, the operation “□” as defined in general is associative. Furthermore, the left zero semigroup is an identity for this operation.


## 3. Hypergroupoid Semigroups

Instead of a groupoid(X,*) on X, we may also consider a hypergroupoid(X,φ) on X, where φ:X×X→P*(X) is a hyperproduct with P*(X), the set of all non-empty subsets of X. We denote the set of all hypergroupoids (X,φ) on X by HBin(X), that is,
(3.1)HBin(X):={(X,φ)∣φ:ahypergroupoidon⁡X}.
The product “□” discussed in Bin(X) can be generalized in HBin(X) as follows: given (X,φ),(X,ψ)∈HBin(X), for any x,y∈X,
(3.2)xy:=(xφy)ψ(yφx).
If we identify x∈X with {x}∈P*(X), then we have an inclusion: X⊆P*(X) and thus for φ(x,y)=xφy∈P*(X), we have xφy⊆X and hence also xφy⊆P*(X) via this identification.IfA,B⊆X, then for the groupoid (X,*)∈Bin(X), we have
(3.3)A*B:={a*b∣a∈A,b∈B},
hence {a}*{b}={a*b} in a natural way. Similarly, given a hypergroupoid (X,φ)∈HBin(X), AφB is defined by AφB=∪{xφy|x∈A,y∈B}.Given hypergroupoids(X,φ),(X,ψ), we let (X,θ):=(X,φ)□(X,ψ). Then, for any x,y∈X, we have
(3.4)xθy=(xφy)ψ(yφx)=∪{aψb∣a∈xφy,b∈yφx}.Suppose that(X,*) and (X,•) are groupoids and that we determine the following:
(3.5)xθy=(x*y)•(y*x)=∪{a•b∣a∈{x*y},b∈{y*x}}={(x*y)•(y*x)}={x□y}=x□y,
via the identification x↔{x}. Hence (X,*)□(X,•) is the same as a product of groupoids or as a product of hypergroupoids.It can be shown that(Bin(X),□)→(HBin(X),□) is an injection (an into homomorphism) via the identification x↔{x} and the associated identification xθy={x□y}=x□y.Example 3.1.
LetX:=R2 and for any x,y∈X, let xφy denote the undirected line segment connecting x with y. Then xφx={x} and xφy=yφx. Let (X,θ):=(X,φ)□(X,φ). Then xθy=∪{aφba∈xφy,b∈yφx} for any x,y∈X. Since xφy=yφx, aφb⊆xθy for any a,b∈xφy. Since x,y∈xφy,  xφy⊆xθy. We claim that xθy⊆xφy. If α∈xθy, then α∈aφb for some a∈xφy and b∈yφx. Since xφy=yφx, α∈aφb for some a,b∈xφy, which shows that α∈xφy. This proves that (X,φ)=(X,θ)=(X,φ)□(X,φ), that is, (X,φ) is an idempotent hypergroupoid in (HBin(X),□).Theorem 3.2.
(
H
Bin
(
X
)
,
□
) is a supersemigroup of the semigroup (Bin(X),□) via the identification x↔{x}.Proof.
Suppose that(X,φ),(X,ψ) and (X,ω) are hypergroupoids and let (X,α):=(X,ψ)□(X,ω) and (X,β):=(X,φ)□(X,ψ). Then for any x,y∈X, we have xαy=(xψy)ω(yψx) and xβy=(xφy)ψ(yφx). Let (X,θ):=[(X,φ)□(X,ψ)]□(X,ω). Then (X,θ)=(X,β)□(X,ω) and hence we obtain the following
(3.6)xθy=(xβy)ω(yβx)=[(xφy)ψ(yφx)]ω[(yφx)ψ(xφy)].
If we let (X,μ):=(X,φ)□[(X,ψ)W(X,ω)], then (X,μ)=(X,φ)□(X,α) and hence xμy=(xφy)α(yφx) for any x,y∈X. Let p:=xφy,q:=yφx. Then
(3.7)xμy=pαq=(pψq)ω(qψp)=[(xφy)ψ(yφx)]ω[(yφx)ψ(xφy)].
This proves that (X,θ)=(X,μ), that is, (HBin(X),□) is a semigroup.Proposition 3.3.
The left-zero-semigroup(X,*), that is, x*y=x for any x,y∈X, is an identity of the semigroup (HBin(X),□).Proof.
Let(X,*) be a left-zero-semigroup. Then (X,*)∈Bin(X). By the identification x↔{x}, we have (X,*)∈(HBin(X),□). Given (X,ν)∈HBin(X), let (X,θ):=(X,*)□(X,ν). Then for any x,y∈X, we have
(3.8)xθy=(x*y)ν(y*x)={x}ν{y}=∪{aνb∣a∈{x},b∈{y}}=xνy,
that is, (X,θ)=(X,ν). This proves that (X,*) is a left identity on HBin(X).
Similarly, if we let(X,θ)=(X,ν)□(X,*), then for any x,y∈X,
(3.9)xθy=(xνy)*(yνx)={a*b∣a∈xνy,b∈yνx}={a∣a∈xνy}=xνy,
that is, (X,θ)=(X,ν). This proves that (X,*) is a right identity on HBin(X).Given an element(X,φ)∈HBin(X), xφy∈P*(X), that is, ∅≠xφy⊆X. We extend (X,φ) to (P*(X),φ^) as
(3.10)φ^:P*(X)×P*(X)⟶P*(P*(X))
by φ^(A,B):=Aφ^B, where Aφ^B=∪{aφb∣a∈A,b∈B}. In particular,
(3.11){x}φ^{y}=∪{aφb∣a∈{x},b∈{y}}=xφy.
This produces a mapping π:HBin(X)→BinP*(X). Let (X,θ):=(X,φ)□(X,ψ). Then xθy=∪{aψb∣a∈xφy,b∈yφx} for any x,y∈X. Since xφy,yφx∈P*(X), we have
(3.12)(xφy)ψ^(yφx)=∪{aψb∣a∈xφy,b∈yφx}=xθy.
Since xφy={x}φ^{y} via the identification x↔{x}, we obtain
(3.13)xθy=(xφy)ψ^(yφx)=({x}φ^{y})ψ^({y}φ^{x})=xθ^y,
where (P*(X),θ^)=(P*(X),φ^)□(P*(X),ψ^) in (BinP*(X),□). We claim that π is a homomorphism. In fact, π((X,φ)□(X,ψ))=π((X,θ))=(P*(X),θ^)=(P*(X),φ^)□(P*(X),ψ^)=π((X,φ))□π((X,ψ)).GivenHBin(X), we may order it according to the rule
(3.14)(X,φ)≤(X,ψ)⟺xφy⊆xψy,∀x,y∈X.
We define a mapping [∅]:X×X→P(X) by [∅](x,y):=∅ for all x,y∈X. If we let HBin*(X):=HBin(X)∪{(X,[∅])}, then (X,[∅]) is the minimal element of (HBin*(X),≤).Proposition 3.4.
Let(X,φ)∈HBin(X) and (X,*)∈Bin(X). If (X,φ)≤(X,*), then (X,φ)=(X,*).Proof.
If(X,φ)≤(X,*), then ∅≠xφy⊆{x*y} for any x,y∈X. It follows that xφy={x*y}=x*y, proving that (X,φ)=(X,*).Proposition 3.5.
Let(X,*),(X,•)∈Bin(X). If (X,*)≤(X,•), then (X,*)=(X,•), that is, Bin(X) is an antichain in (HBin*(X),≤).Proof.
If(X,*)≤(X,•), then {x*y}⊆{x•y} for any x,y∈X. It follows that x*y=x•y for any x,y∈X, proving that (X,*)=(X,•).


## 4.BCK-Algebras on HBin*(X)

In this section we discussBCK-algebras on HBin*(X) by introducing a binary operation as follows: given hypergroupoids (X,φ),(X,ψ)∈HBin*(X), we define a binary operation “⊖” by
(4.1)(X,φ)⊖(X,ψ):=(X,φ∖ψ),
where x(φ∖ψ)y:=xφy∖xψy for any x,y∈X.Theorem 4.1.
(
H
Bi
n
*
(
X
)
,
⊖
,
[
∅
]
) is a BCK-algebra.Proof.
For any(X,φ)∈HBin*(X), since x[∅]y∖xφy=∅ for any x,y∈X, we have (X,[∅])⊖(X,φ)=(X,[∅]).
Given(X,φ)∈HBin*(X), since xφy∖xφy=∅ for any x,y∈X, we have (X,φ)⊖(X,φ)=(X,[∅]).
Assume that(X,φ)⊖(X,ψ)=(X,[∅])=(X,ψ)⊖(X,φ). Then xφy∖xψy=∅,xψy∖xφy=∅ for any x,y∈X, which shows that xφy=xψy for any x,y∈X, that is, (X,φ)=(X,ψ).
Given(X,φ),(X,ψ)∈HBin*(X), since [xφy∖[xφy∖xψy]]∖xψy=∅ for any x,y∈X, we obtain [(X,φ)⊖[(X,φ)⊖(X,ψ)]]⊖(X,ψ)=(X,[∅]).
Given(X,φ),(X,ψ),(X,δ)∈HBin*(X), since [(xφy∖xψy)∖(xφy∖xδy)]∖(xδy∖xψy)=∅ for any x,y∈X, we obtain [((X,φ)⊖(X,ψ))⊖((X,φ)⊖(X,δ)]⊖[(X,δ)⊖(X,ψ)]=(X,[∅]). This proves the theorem.


## 5. Several Properties onHBin(X)

In this section, we discuss some properties onHBin(X).Proposition 5.1.
The product “□” is order-preserving, that is, if (X,φ)≤(X,ξ),(X,ψ)≤(X,ω), then (X,φ)□(X,ψ)≤(X,ξ)□(X,ω).Proof.
Let(X,φ)≤(X,ξ),(X,ψ)≤(X,ω) in HBin(X). If we let (X,θ):=(X,φ)□(X,ψ) and (X,ρ):=(X,ξ)□(X,ω), then for any x,y∈X,
(5.1)xθy=(xφy)ψ(yφx)⊆(xξy)ψ(yξx)⊆(xξy)ω(yξx)=xρy,
proving that (X,θ)≤(X,ρ).We define a mapping[X]:X×X→P(X) by [X](x,y):=X for all x,y∈X. Then (X,[X]) is the maximal element of (HBin*(X),≤). Given (X,φ)∈HBin(X), if we let (X,θ):=(X,[X])□(X,φ), then xθy=(x[X]y)φ(y[X]x)=XφX=∪{aφb∣a,b∈X} for any x,y∈X.Proposition 5.2.
If(X,φ)∈HBin(X), then (X,φ)□(X,[X])=(X,[X]).Proof.
Let(X,θ):=(X,φ)□(X,[X]). Then, for any x,y∈X, we have
(5.2)xθy=(xφy)[X](yφx)=∪{a[X]b∣a∈xφy,b∈yφx}=X,
proving that (X,θ)=(X,[X]).Given(X,φ)∈HBin*(X), we define a hypergroupoid (X,φC) by xφCy:=X∖xφy, for any x,y∈X. We call it the complementary hypergroupoid of (X,φC).For example, if(X,·,e) is a group, then x·Cy=X∖{x·y}, where x,y∈X. It follows that x·Ce=e·Cx=X∖{x} and x·Cx-1=x-1·Cx=X∖{e} for any x∈X.A hypergroupoid(X,φ) is said to be a complementaryd-algebra if there exists 0∈X such that (i) xφx=X∖{0}; (ii) 0φx=X∖{0}; (iii) xφy=yφx=X∖{x} implies x=y, for any x,y∈X.The following proposition can be easily seen.Proposition 5.3.
Given(X,φ)∈HBin*(X), (X,φ) is a d-algebra if and only if (X,φC) is a complementary d-algebra.Example 5.4.
LetX:=R be the set of all real numbers and f:X→X be a mapping. Define a map φf:X×X→P*(X) by φf(x,y):=[x-|f(y)|,x+|f(y)|]. Then (X,φf) be a hypergroupoid for which xφfy=[x-|f(y)|,x+|f(y)|] has a midpoint x where x,y∈X.
In particular, letf(x):=x2 for any x∈X and let (X,θ):=(X,φf)□(X,φf). Then xθy=(xφfy)φf(yφfx) = ∪{aφfb|a∈[x-|f(y)|,x+|f(y)|],b∈[y-|f(x)|,y+|f(x)|]} = ∪{[a-b2,a+b2]|a∈[x-y2,x+y2],b∈[y-x2,y+x2]} = [x-2y(y+x2)-x4,x+2y(y+x2)+x4], an interval of length y2+(y+x2)2≥0, where x=y=0 implies 0θ0=[0,0]={0}, corresponding to 0 in the identification.A hypergroupoid(X,φ) is said to be left inclusive if x∈xφy for any x,y∈X.Note that the only left inclusive hypergroupoid which is a groupoid is the left-zero-semigroup. In fact, let(X,*) be a left inclusive hypergroupoid which is a groupoid. Then x∈{x*y} for any x,y∈X. It follows that x=x*y for any x,y∈X, that is, (X,*) is a left-zero-semigroup.Proposition 5.5.
The left inclusive hypergroupoids onX relative to the product “□” on HBin(X) form a subsemigroup of (HBin(X),□).Proof.
Let(X,φ),(X,ψ) be left inclusive hypergroupoids and let (X,θ):=(X,φ)□(X,ψ). Then xθy=(xφy)ψ(yφx)=∪{aψb|a∈xφy,b∈yφx} for any x,y∈X. Since (X,φ) is left inclusive, x∈xφy,y∈yφx, and hence xψy⊆xθy for any x,y∈X. Moreover, (X,ψ) is left inclusive implies that x∈xψy, which proves that x∈xθy.Proposition 5.6.
Let(X,φ)≤(X,ψ) in HBin(X). If (X,φ) is left inclusive, then (X,ψ) is also left inclusive.Proof.
Let(X,φ)≤(X,ψ). Then xφy⊆xψy for any x,y∈X. Since (X,φ) is left inclusive, we have x∈xφy⊆xψy, proving the proposition.Proposition5.6 means that the collection of all left inclusive hypergroupoids is a filter in the poset (HBin(X),≤).A hypergroupoid(X,φ) is said to be left-self-avoiding if x∉xφy for any x,y∈X.Proposition 5.7.
The complementary hypergroupoid(X,φC) of a left inclusive hypergroupoid (X,φ) is left-self-avoiding.Proof.
Let(X,φC) be the complementary hypergroupoid of a left inclusive hypergroupoid (X,φ). Then xφCy=X∖xφy for any x,y∈X. Since (X,φ) is left inclusive, x∈xφy for any x,y∈X, and hence x∉xφCy, proving the proposition.Proposition 5.8.
The complementary hypergroupoid(X,φC) of a left-self-avoiding hypergroupoid (X,φ) is left inclusive.Proof.
Straightforward.Proposition 5.9.
Let(X,θ)=(X,φ)□(X,ψ) where (X,φ) is left inclusive and (X,θ) is left-self-avoiding. Then (X,ψ) is left-self-avoiding.Proof.
Let(X,θ) be a left-self-avoiding hypergroupoid. Then (X,θC) is left inclusive by Proposition 5.8. It follows that x∈xθCy=X∖∪{aψb∣a∈xφy,b∈yφx}. This means that x∉aψb for any a∈xφy and b∈yφx where x,y∈X. Since (X,φ) is left inclusive, x∈xφy,y∈yφx. Hence x∉xψy, proving that (X,ψ) is left-self-avoiding.


## 6. Conclusion

In this paper we have introduced the notion of hypergroupoids as a generalization of groupoids in a manner analogous to the introduction of the notion of hypergroups as a generalization of the notion of groups. Since the semigroup(Bin(X),□) can still benefit from more detailed investigation it follows that the same is even more true for (HBin(X),□). In the latter case one must rely on proper adaptations obtained from (Bin(X),□) and certainly on results obtained from studies on hypergroupoids available in the literature [7–10] as a general plan for the organization of the subject, with parts to be completed as time and opportunity permits.



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