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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 717698, 8 pages
http://dx.doi.org/10.1155/2012/717698
Research Article

The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups

1Department of Applied Mathematics, Hanyang University, Ansan 426-791, Republic of Korea
2Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
3Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA

Received 8 March 2012; Accepted 14 July 2012

Academic Editor: Sazzad H.Β Chowdhury

Copyright Β© 2012 Jeong Soon Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the notion of hypergroupoids (𝐻Bin(𝑋),β–‘), and show that (𝐻Bin(𝑋),β–‘) is a super-semigroup of the semigroup (Bin(𝑋),β–‘) via the identification π‘₯↔{π‘₯}. We prove that (𝐻Binβˆ—(𝑋),βŠ–,[βˆ…]) is a 𝐡𝐢𝐾-algebra, and obtain several properties of (𝐻Binβˆ—(𝑋),β–‘).

1. Introduction

The notion of the semigroup (Bin(𝑋),β–‘) 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,β€’)βˆˆπ‘Bin(𝑋), then π‘₯≠𝑦 implies {π‘₯,𝑦}={π‘₯‒𝑦,𝑦‒π‘₯}. Moreover, she showed that a groupoid (𝑋,β€’)βˆˆπ‘Bin(𝑋) if and only if it is a locally zero groupoid. Y. Imai and K. IsΓ©ki introduced two classes of abstract algebras: 𝐡𝐢𝐾-algebras and 𝐡𝐢𝐼-algebras [3, 4]. Neggers and Kim introduced the notion of 𝑑-algebras which is another useful generalization of 𝐡𝐢𝐾-algebras, and then investigated several relations between 𝑑-algebras and 𝐡𝐢𝐾-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;β–‘)=(𝑋;βˆ—)β–‘(𝑋;∘), where π‘₯░𝑦=(π‘₯βˆ—π‘¦)∘(π‘¦βˆ—π‘₯). They showed that the squared algebra (𝑋;β–‘,0) of a pre-𝑑-algebra (𝑋;βˆ—,0) is a strong 𝑑-algebra if and only if (𝑋;βˆ—,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 𝐡𝐢𝐼-algebras. They gave some characterizations of a 𝑝-semisimple 𝐡𝐢𝐼-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),β–‘), and show that (𝐻Bin(𝑋),β–‘) is a super-semigroup of the semigroup (Bin(𝑋),β–‘) via the identification π‘₯↔{π‘₯}. We prove that (𝐻Binβˆ—(𝑋),βŠ–,[βˆ…]) is a 𝐡𝐢𝐾-algebra, and obtain several properties of (𝐻Binβˆ—(𝑋),β–‘).

2. Preliminaries

Given a nonempty set 𝑋, we let Bin(𝑋) the collection of all groupoids (𝑋,βˆ—), where βˆ—βˆΆπ‘‹Γ—π‘‹β†’π‘‹ is a map and where βˆ—(π‘₯,𝑦) is written in the usual product form. Given elements (𝑋,βˆ—) and (𝑋,β€’) of Bin(𝑋), define a product β€œβ–‘β€ on these groupoids as follows: (𝑋,βˆ—)β–‘(𝑋,β€’)=(𝑋,β–‘),(2.1) where π‘₯░𝑦=(π‘₯βˆ—π‘¦)β€’(π‘¦βˆ—π‘₯),(2.2) for any π‘₯,π‘¦βˆˆπ‘‹. Using the notion, H. S. Kim and J. Neggers showed the following theorem.

Theorem 2.1 (see [1]). (Bin(𝑋),β–‘) is a semigroup, that is, the operation β€œβ–‘β€ 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,βˆ—) on 𝑋, we may also consider a hypergroupoid (𝑋,πœ‘) on 𝑋, where πœ‘βˆΆπ‘‹Γ—π‘‹β†’π‘ƒβˆ—(𝑋) is a hyperproduct with π‘ƒβˆ—(𝑋), the set of all non-empty subsets of 𝑋. We denote the set of all hypergroupoids (𝑋,πœ‘) on 𝑋 by 𝐻Bin(𝑋), that is, 𝐻Bin(𝑋)∢={(𝑋,πœ‘)βˆ£πœ‘βˆΆahypergroupoidon𝑋}.(3.1) The product β€œβ–‘β€ discussed in Bin(𝑋) can be generalized in 𝐻Bin(𝑋) as follows: given (𝑋,πœ‘),(𝑋,πœ“)∈𝐻Bin(𝑋), for any π‘₯,π‘¦βˆˆπ‘‹, π‘₯π‘¦βˆΆ=(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯).(3.2) If we identify π‘₯βˆˆπ‘‹ with {π‘₯}βˆˆπ‘ƒβˆ—(𝑋), then we have an inclusion: π‘‹βŠ†π‘ƒβˆ—(𝑋) and thus for πœ‘(π‘₯,𝑦)=π‘₯πœ‘π‘¦βˆˆπ‘ƒβˆ—(𝑋), we have π‘₯πœ‘π‘¦βŠ†π‘‹ and hence also π‘₯πœ‘π‘¦βŠ†π‘ƒβˆ—(𝑋) via this identification.

If 𝐴,π΅βŠ†π‘‹, then for the groupoid (𝑋,βˆ—)∈Bin(𝑋), we have π΄βˆ—π΅βˆΆ={π‘Žβˆ—π‘βˆ£π‘Žβˆˆπ΄,π‘βˆˆπ΅},(3.3) hence {π‘Ž}βˆ—{𝑏}={π‘Žβˆ—π‘} in a natural way. Similarly, given a hypergroupoid (𝑋,πœ‘)∈𝐻Bin(𝑋), π΄πœ‘π΅ is defined by π΄πœ‘π΅=βˆͺ{π‘₯πœ‘π‘¦|π‘₯∈𝐴,π‘¦βˆˆπ΅}.

Given hypergroupoids (𝑋,πœ‘),(𝑋,πœ“), we let (𝑋,πœƒ)∢=(𝑋,πœ‘)β–‘(𝑋,πœ“). Then, for any π‘₯,π‘¦βˆˆπ‘‹, we have π‘₯πœƒπ‘¦=(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯)=βˆͺ{π‘Žπœ“π‘βˆ£π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯}.(3.4)

Suppose that (𝑋,βˆ—) and (𝑋,β€’) are groupoids and that we determine the following: π‘₯πœƒπ‘¦=(π‘₯βˆ—π‘¦)β€’(π‘¦βˆ—π‘₯)=βˆͺ{π‘Žβ€’π‘βˆ£π‘Žβˆˆ{π‘₯βˆ—π‘¦},π‘βˆˆ{π‘¦βˆ—π‘₯}}={(π‘₯βˆ—π‘¦)β€’(π‘¦βˆ—π‘₯)}={π‘₯░𝑦}=π‘₯░𝑦,(3.5) via the identification π‘₯↔{π‘₯}. Hence (𝑋,βˆ—)β–‘(𝑋,β€’) is the same as a product of groupoids or as a product of hypergroupoids.

It can be shown that (Bin(𝑋),β–‘)β†’(𝐻Bin(𝑋),β–‘) is an injection (an into homomorphism) via the identification π‘₯↔{π‘₯} and the associated identification π‘₯πœƒπ‘¦={π‘₯░𝑦}=π‘₯░𝑦.

Example 3.1. Let π‘‹βˆΆ=𝐑2 and for any π‘₯,π‘¦βˆˆπ‘‹, let π‘₯πœ‘π‘¦ denote the undirected line segment connecting π‘₯ with 𝑦. Then π‘₯πœ‘π‘₯={π‘₯} and π‘₯πœ‘π‘¦=π‘¦πœ‘π‘₯. Let (𝑋,πœƒ)∢=(𝑋,πœ‘)β–‘(𝑋,πœ‘). Then π‘₯πœƒπ‘¦=βˆͺ{π‘Žπœ‘π‘π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯} for any π‘₯,π‘¦βˆˆπ‘‹. Since π‘₯πœ‘π‘¦=π‘¦πœ‘π‘₯, π‘Žπœ‘π‘βŠ†π‘₯πœƒπ‘¦ for any π‘Ž,π‘βˆˆπ‘₯πœ‘π‘¦. Since π‘₯,π‘¦βˆˆπ‘₯πœ‘π‘¦, π‘₯πœ‘π‘¦βŠ†π‘₯πœƒπ‘¦. We claim that π‘₯πœƒπ‘¦βŠ†π‘₯πœ‘π‘¦. If π›Όβˆˆπ‘₯πœƒπ‘¦, then π›Όβˆˆπ‘Žπœ‘π‘ for some π‘Žβˆˆπ‘₯πœ‘π‘¦ and π‘βˆˆπ‘¦πœ‘π‘₯. Since π‘₯πœ‘π‘¦=π‘¦πœ‘π‘₯, π›Όβˆˆπ‘Žπœ‘π‘ for some π‘Ž,π‘βˆˆπ‘₯πœ‘π‘¦, which shows that π›Όβˆˆπ‘₯πœ‘π‘¦. This proves that (𝑋,πœ‘)=(𝑋,πœƒ)=(𝑋,πœ‘)β–‘(𝑋,πœ‘), that is, (𝑋,πœ‘) is an idempotent hypergroupoid in (𝐻Bin(𝑋),β–‘).

Theorem 3.2. (𝐻Bin(𝑋),β–‘) is a supersemigroup of the semigroup (Bin(𝑋),β–‘) via the identification π‘₯↔{π‘₯}.

Proof. Suppose that (𝑋,πœ‘),(𝑋,πœ“) and (𝑋,πœ”) are hypergroupoids and let (𝑋,𝛼)∢=(𝑋,πœ“)β–‘(𝑋,πœ”) and (𝑋,𝛽)∢=(𝑋,πœ‘)β–‘(𝑋,πœ“). Then for any π‘₯,π‘¦βˆˆπ‘‹, we have π‘₯𝛼𝑦=(π‘₯πœ“π‘¦)πœ”(π‘¦πœ“π‘₯) and π‘₯𝛽𝑦=(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯). Let (𝑋,πœƒ)∢=[(𝑋,πœ‘)β–‘(𝑋,πœ“)]β–‘(𝑋,πœ”). Then (𝑋,πœƒ)=(𝑋,𝛽)β–‘(𝑋,πœ”) and hence we obtain the following =[]πœ”[].π‘₯πœƒπ‘¦=(π‘₯𝛽𝑦)πœ”(𝑦𝛽π‘₯)(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯)(π‘¦πœ‘π‘₯)πœ“(π‘₯πœ‘π‘¦)(3.6) If we let (𝑋,πœ‡)∢=(𝑋,πœ‘)β–‘[(𝑋,πœ“)W(𝑋,πœ”)], then (𝑋,πœ‡)=(𝑋,πœ‘)β–‘(𝑋,𝛼) and hence π‘₯πœ‡π‘¦=(π‘₯πœ‘π‘¦)𝛼(π‘¦πœ‘π‘₯) for any π‘₯,π‘¦βˆˆπ‘‹. Let π‘βˆΆ=π‘₯πœ‘π‘¦,π‘žβˆΆ=π‘¦πœ‘π‘₯. Then ==[]πœ”[].π‘₯πœ‡π‘¦=π‘π›Όπ‘ž(π‘πœ“π‘ž)πœ”(π‘žπœ“π‘)(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯)(π‘¦πœ‘π‘₯)πœ“(π‘₯πœ‘π‘¦)(3.7) This proves that (𝑋,πœƒ)=(𝑋,πœ‡), that is, (𝐻Bin(𝑋),β–‘) is a semigroup.

Proposition 3.3. The left-zero-semigroup (𝑋,βˆ—), that is, π‘₯βˆ—π‘¦=π‘₯ for any π‘₯,π‘¦βˆˆπ‘‹, is an identity of the semigroup (𝐻Bin(𝑋),β–‘).

Proof. Let (𝑋,βˆ—) be a left-zero-semigroup. Then (𝑋,βˆ—)∈Bin(𝑋). By the identification π‘₯↔{π‘₯}, we have (𝑋,βˆ—)∈(𝐻Bin(𝑋),β–‘). Given (𝑋,𝜈)∈𝐻Bin(𝑋), let (𝑋,πœƒ)∢=(𝑋,βˆ—)β–‘(𝑋,𝜈). Then for any π‘₯,π‘¦βˆˆπ‘‹, we have =π‘₯πœƒπ‘¦=(π‘₯βˆ—π‘¦)𝜈(π‘¦βˆ—π‘₯){π‘₯}𝜈{𝑦}=βˆͺ{π‘Žπœˆπ‘βˆ£π‘Žβˆˆ{π‘₯},π‘βˆˆ{𝑦}}=π‘₯πœˆπ‘¦,(3.8) that is, (𝑋,πœƒ)=(𝑋,𝜈). This proves that (𝑋,βˆ—) is a left identity on 𝐻Bin(𝑋).
Similarly, if we let (𝑋,πœƒ)=(𝑋,𝜈)β–‘(𝑋,βˆ—), then for any π‘₯,π‘¦βˆˆπ‘‹, =π‘₯πœƒπ‘¦=(π‘₯πœˆπ‘¦)βˆ—(π‘¦πœˆπ‘₯){π‘Žβˆ—π‘βˆ£π‘Žβˆˆπ‘₯πœˆπ‘¦,π‘βˆˆπ‘¦πœˆπ‘₯}={π‘Žβˆ£π‘Žβˆˆπ‘₯πœˆπ‘¦}=π‘₯πœˆπ‘¦,(3.9) that is, (𝑋,πœƒ)=(𝑋,𝜈). This proves that (𝑋,βˆ—) is a right identity on 𝐻Bin(𝑋).

Given an element (𝑋,πœ‘)∈𝐻Bin(𝑋), π‘₯πœ‘π‘¦βˆˆπ‘ƒβˆ—(𝑋), that is, βˆ…β‰ π‘₯πœ‘π‘¦βŠ†π‘‹. We extend (𝑋,πœ‘) to (π‘ƒβˆ—(𝑋),ξπœ‘) as ξπœ‘βˆΆπ‘ƒβˆ—(𝑋)Γ—π‘ƒβˆ—(𝑋)βŸΆπ‘ƒβˆ—ξ€·π‘ƒβˆ—ξ€Έ(𝑋)(3.10) by ξπœ‘(𝐴,𝐡)∢=π΄ξπœ‘π΅, where π΄ξπœ‘π΅=βˆͺ{π‘Žπœ‘π‘βˆ£π‘Žβˆˆπ΄,π‘βˆˆπ΅}. In particular, {π‘₯}ξπœ‘{𝑦}=βˆͺ{π‘Žπœ‘π‘βˆ£π‘Žβˆˆ{π‘₯},π‘βˆˆ{𝑦}}=π‘₯πœ‘π‘¦.(3.11) This produces a mapping πœ‹βˆΆπ»Bin(𝑋)β†’Binπ‘ƒβˆ—(𝑋). Let (𝑋,πœƒ)∢=(𝑋,πœ‘)β–‘(𝑋,πœ“). Then π‘₯πœƒπ‘¦=βˆͺ{π‘Žπœ“π‘βˆ£π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯} for any π‘₯,π‘¦βˆˆπ‘‹. Since π‘₯πœ‘π‘¦,π‘¦πœ‘π‘₯βˆˆπ‘ƒβˆ—(𝑋), we have (π‘₯πœ‘π‘¦)ξπœ“(π‘¦πœ‘π‘₯)=βˆͺ{π‘Žπœ“π‘βˆ£π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯}=π‘₯πœƒπ‘¦.(3.12) Since π‘₯πœ‘π‘¦={π‘₯}ξπœ‘{𝑦} via the identification π‘₯↔{π‘₯}, we obtain =ξ€·ξ€Έξ€·ξ€ΈΜ‚π‘₯πœƒπ‘¦=(π‘₯πœ‘π‘¦)ξπœ“(π‘¦πœ‘π‘₯){π‘₯}ξπœ‘{𝑦}ξπœ“{𝑦}ξπœ‘{π‘₯}=π‘₯πœƒπ‘¦,(3.13) where (π‘ƒβˆ—Μ‚(𝑋),πœƒ)=(π‘ƒβˆ—(𝑋),ξπœ‘)β–‘(π‘ƒβˆ—(𝑋),ξπœ“) in (Binπ‘ƒβˆ—(𝑋),β–‘). We claim that πœ‹ is a homomorphism. In fact, πœ‹((𝑋,πœ‘)β–‘(𝑋,πœ“))=πœ‹((𝑋,πœƒ))=(π‘ƒβˆ—Μ‚(𝑋),πœƒ)=(π‘ƒβˆ—(𝑋),ξπœ‘)β–‘(π‘ƒβˆ—(𝑋),ξπœ“)=πœ‹((𝑋,πœ‘))β–‘πœ‹((𝑋,πœ“)).

Given 𝐻Bin(𝑋), we may order it according to the rule (𝑋,πœ‘)≀(𝑋,πœ“)⟺π‘₯πœ‘π‘¦βŠ†π‘₯πœ“π‘¦,βˆ€π‘₯,π‘¦βˆˆπ‘‹.(3.14) We define a mapping [βˆ…]βˆΆπ‘‹Γ—π‘‹β†’π‘ƒ(𝑋) by [βˆ…](π‘₯,𝑦)∢=βˆ… for all π‘₯,π‘¦βˆˆπ‘‹. If we let 𝐻Binβˆ—(𝑋)∢=𝐻Bin(𝑋)βˆͺ{(𝑋,[βˆ…])}, then (𝑋,[βˆ…]) is the minimal element of (𝐻Binβˆ—(𝑋),≀).

Proposition 3.4. Let (𝑋,πœ‘)∈𝐻Bin(𝑋) and (𝑋,βˆ—)∈Bin(𝑋). If (𝑋,πœ‘)≀(𝑋,βˆ—), then (𝑋,πœ‘)=(𝑋,βˆ—).

Proof. If (𝑋,πœ‘)≀(𝑋,βˆ—), then βˆ…β‰ π‘₯πœ‘π‘¦βŠ†{π‘₯βˆ—π‘¦} for any π‘₯,π‘¦βˆˆπ‘‹. It follows that π‘₯πœ‘π‘¦={π‘₯βˆ—π‘¦}=π‘₯βˆ—π‘¦, proving that (𝑋,πœ‘)=(𝑋,βˆ—).

Proposition 3.5. Let (𝑋,βˆ—),(𝑋,β€’)∈Bin(𝑋). If (𝑋,βˆ—)≀(𝑋,β€’), then (𝑋,βˆ—)=(𝑋,β€’), that is, Bin(𝑋) is an antichain in (𝐻Binβˆ—(𝑋),≀).

Proof. If (𝑋,βˆ—)≀(𝑋,β€’), then {π‘₯βˆ—π‘¦}βŠ†{π‘₯‒𝑦} for any π‘₯,π‘¦βˆˆπ‘‹. It follows that π‘₯βˆ—π‘¦=π‘₯‒𝑦 for any π‘₯,π‘¦βˆˆπ‘‹, proving that (𝑋,βˆ—)=(𝑋,β€’).

4. 𝐡𝐢𝐾-Algebras on π»π΅π‘–π‘›βˆ—(𝑋)

In this section we discuss 𝐡𝐢𝐾-algebras on 𝐻Binβˆ—(𝑋) by introducing a binary operation as follows: given hypergroupoids (𝑋,πœ‘),(𝑋,πœ“)∈𝐻Binβˆ—(𝑋), we define a binary operation β€œβŠ–β€ by (𝑋,πœ‘)βŠ–(𝑋,πœ“)∢=(𝑋,πœ‘β§΅πœ“),(4.1) where π‘₯(πœ‘β§΅πœ“)π‘¦βˆΆ=π‘₯πœ‘π‘¦β§΅π‘₯πœ“π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹.

Theorem 4.1. (𝐻Binβˆ—(𝑋),βŠ–,[βˆ…]) is a 𝐡𝐢𝐾-algebra.

Proof. For any (𝑋,πœ‘)∈𝐻Binβˆ—(𝑋), since π‘₯[βˆ…]𝑦⧡π‘₯πœ‘π‘¦=βˆ… for any π‘₯,π‘¦βˆˆπ‘‹, we have (𝑋,[βˆ…])βŠ–(𝑋,πœ‘)=(𝑋,[βˆ…]).
Given (𝑋,πœ‘)∈𝐻Binβˆ—(𝑋), since π‘₯πœ‘π‘¦β§΅π‘₯πœ‘π‘¦=βˆ… for any π‘₯,π‘¦βˆˆπ‘‹, we have (𝑋,πœ‘)βŠ–(𝑋,πœ‘)=(𝑋,[βˆ…]).
Assume that (𝑋,πœ‘)βŠ–(𝑋,πœ“)=(𝑋,[βˆ…])=(𝑋,πœ“)βŠ–(𝑋,πœ‘). Then π‘₯πœ‘π‘¦β§΅π‘₯πœ“π‘¦=βˆ…,π‘₯πœ“π‘¦β§΅π‘₯πœ‘π‘¦=βˆ… for any π‘₯,π‘¦βˆˆπ‘‹, which shows that π‘₯πœ‘π‘¦=π‘₯πœ“π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹, that is, (𝑋,πœ‘)=(𝑋,πœ“).
Given (𝑋,πœ‘),(𝑋,πœ“)∈𝐻Binβˆ—(𝑋), since [π‘₯πœ‘π‘¦β§΅[π‘₯πœ‘π‘¦β§΅π‘₯πœ“π‘¦]]⧡π‘₯πœ“π‘¦=βˆ… for any π‘₯,π‘¦βˆˆπ‘‹, we obtain [(𝑋,πœ‘)βŠ–[(𝑋,πœ‘)βŠ–(𝑋,πœ“)]]βŠ–(𝑋,πœ“)=(𝑋,[βˆ…]).
Given (𝑋,πœ‘),(𝑋,πœ“),(𝑋,𝛿)∈𝐻Binβˆ—(𝑋), since [(π‘₯πœ‘π‘¦β§΅π‘₯πœ“π‘¦)⧡(π‘₯πœ‘π‘¦β§΅π‘₯𝛿𝑦)]⧡(π‘₯𝛿𝑦⧡π‘₯πœ“π‘¦)=βˆ… for any π‘₯,π‘¦βˆˆπ‘‹, we obtain [((𝑋,πœ‘)βŠ–(𝑋,πœ“))βŠ–((𝑋,πœ‘)βŠ–(𝑋,𝛿)]βŠ–[(𝑋,𝛿)βŠ–(𝑋,πœ“)]=(𝑋,[βˆ…]). This proves the theorem.

5. Several Properties on 𝐻𝐡𝑖𝑛(𝑋)

In this section, we discuss some properties on 𝐻Bin(𝑋).

Proposition 5.1. The product β€œβ–‘β€ is order-preserving, that is, if (𝑋,πœ‘)≀(𝑋,πœ‰),(𝑋,πœ“)≀(𝑋,πœ”), then (𝑋,πœ‘)β–‘(𝑋,πœ“)≀(𝑋,πœ‰)β–‘(𝑋,πœ”).

Proof. Let (𝑋,πœ‘)≀(𝑋,πœ‰),(𝑋,πœ“)≀(𝑋,πœ”) in 𝐻Bin(𝑋). If we let (𝑋,πœƒ)∢=(𝑋,πœ‘)β–‘(𝑋,πœ“) and (𝑋,𝜌)∢=(𝑋,πœ‰)β–‘(𝑋,πœ”), then for any π‘₯,π‘¦βˆˆπ‘‹, βŠ†π‘₯πœƒπ‘¦=(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯)(π‘₯πœ‰π‘¦)πœ“(π‘¦πœ‰π‘₯)βŠ†(π‘₯πœ‰π‘¦)πœ”(π‘¦πœ‰π‘₯)=π‘₯πœŒπ‘¦,(5.1) proving that (𝑋,πœƒ)≀(𝑋,𝜌).

We define a mapping [𝑋]βˆΆπ‘‹Γ—π‘‹β†’π‘ƒ(𝑋) by [𝑋](π‘₯,𝑦)∢=𝑋 for all π‘₯,π‘¦βˆˆπ‘‹. Then (𝑋,[𝑋]) is the maximal element of (𝐻Binβˆ—(𝑋),≀). Given (𝑋,πœ‘)∈𝐻Bin(𝑋), if we let (𝑋,πœƒ)∢=(𝑋,[𝑋])β–‘(𝑋,πœ‘), then π‘₯πœƒπ‘¦=(π‘₯[𝑋]𝑦)πœ‘(𝑦[𝑋]π‘₯)=π‘‹πœ‘π‘‹=βˆͺ{π‘Žπœ‘π‘βˆ£π‘Ž,π‘βˆˆπ‘‹} for any π‘₯,π‘¦βˆˆπ‘‹.

Proposition 5.2. If (𝑋,πœ‘)∈𝐻Bin(𝑋), then (𝑋,πœ‘)β–‘(𝑋,[𝑋])=(𝑋,[𝑋]).

Proof. Let (𝑋,πœƒ)∢=(𝑋,πœ‘)β–‘(𝑋,[𝑋]). Then, for any π‘₯,π‘¦βˆˆπ‘‹, we have [𝑋][𝑋]}π‘₯πœƒπ‘¦=(π‘₯πœ‘π‘¦)(π‘¦πœ‘π‘₯)=βˆͺ{π‘Žπ‘βˆ£π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯=𝑋,(5.2) proving that (𝑋,πœƒ)=(𝑋,[𝑋]).

Given (𝑋,πœ‘)∈𝐻Binβˆ—(𝑋), we define a hypergroupoid (𝑋,πœ‘πΆ) by π‘₯πœ‘πΆπ‘¦βˆΆ=𝑋⧡π‘₯πœ‘π‘¦, for any π‘₯,π‘¦βˆˆπ‘‹. We call it the complementary hypergroupoid of (𝑋,πœ‘πΆ).

For example, if (𝑋,β‹…,𝑒) is a group, then π‘₯⋅𝐢𝑦=𝑋⧡{π‘₯⋅𝑦}, where π‘₯,π‘¦βˆˆπ‘‹. It follows that π‘₯⋅𝐢𝑒=𝑒⋅𝐢π‘₯=𝑋⧡{π‘₯} and π‘₯⋅𝐢π‘₯βˆ’1=π‘₯βˆ’1⋅𝐢π‘₯=𝑋⧡{𝑒} for any π‘₯βˆˆπ‘‹.

A hypergroupoid (𝑋,πœ‘) is said to be a complementary 𝑑-algebra if there exists 0βˆˆπ‘‹ such that (i) π‘₯πœ‘π‘₯=𝑋⧡{0}; (ii) 0πœ‘π‘₯=𝑋⧡{0}; (iii) π‘₯πœ‘π‘¦=π‘¦πœ‘π‘₯=𝑋⧡{π‘₯} implies π‘₯=𝑦, for any π‘₯,π‘¦βˆˆπ‘‹.

The following proposition can be easily seen.

Proposition 5.3. Given (𝑋,πœ‘)∈𝐻Binβˆ—(𝑋), (𝑋,πœ‘) is a 𝑑-algebra if and only if (𝑋,πœ‘πΆ) is a complementary 𝑑-algebra.

Example 5.4. Let π‘‹βˆΆ=𝐑 be the set of all real numbers and π‘“βˆΆπ‘‹β†’π‘‹ be a mapping. Define a map πœ‘π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘ƒβˆ—(𝑋) by πœ‘π‘“(π‘₯,𝑦)∢=[π‘₯βˆ’|𝑓(𝑦)|,π‘₯+|𝑓(𝑦)|]. Then (𝑋,πœ‘π‘“) be a hypergroupoid for which π‘₯πœ‘π‘“π‘¦=[π‘₯βˆ’|𝑓(𝑦)|,π‘₯+|𝑓(𝑦)|] has a midpoint π‘₯ where π‘₯,π‘¦βˆˆπ‘‹.
In particular, let 𝑓(π‘₯)∢=π‘₯2 for any π‘₯βˆˆπ‘‹ and let (𝑋,πœƒ)∢=(𝑋,πœ‘π‘“)β–‘(𝑋,πœ‘π‘“). Then π‘₯πœƒπ‘¦=(π‘₯πœ‘π‘“π‘¦)πœ‘π‘“(π‘¦πœ‘π‘“π‘₯) = βˆͺ{π‘Žπœ‘π‘“π‘|π‘Žβˆˆ[π‘₯βˆ’|𝑓(𝑦)|,π‘₯+|𝑓(𝑦)|],π‘βˆˆ[π‘¦βˆ’|𝑓(π‘₯)|,𝑦+|𝑓(π‘₯)|]} = βˆͺ{[π‘Žβˆ’π‘2,π‘Ž+𝑏2]|π‘Žβˆˆ[π‘₯βˆ’π‘¦2,π‘₯+𝑦2],π‘βˆˆ[π‘¦βˆ’π‘₯2,𝑦+π‘₯2]} = [π‘₯βˆ’2𝑦(𝑦+π‘₯2)βˆ’π‘₯4,π‘₯+2𝑦(𝑦+π‘₯2)+π‘₯4], an interval of length 𝑦2+(𝑦+π‘₯2)2β‰₯0, where π‘₯=𝑦=0 implies 0πœƒ0=[0,0]={0}, corresponding to 0 in the identification.

A hypergroupoid (𝑋,πœ‘) is said to be left inclusive if π‘₯∈π‘₯πœ‘π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹.

Note that the only left inclusive hypergroupoid which is a groupoid is the left-zero-semigroup. In fact, let (𝑋,βˆ—) be a left inclusive hypergroupoid which is a groupoid. Then π‘₯∈{π‘₯βˆ—π‘¦} for any π‘₯,π‘¦βˆˆπ‘‹. It follows that π‘₯=π‘₯βˆ—π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹, that is, (𝑋,βˆ—) is a left-zero-semigroup.

Proposition 5.5. The left inclusive hypergroupoids on 𝑋 relative to the product β€œβ–‘β€ on 𝐻Bin(𝑋) form a subsemigroup of (𝐻Bin(𝑋),β–‘).

Proof. Let (𝑋,πœ‘),(𝑋,πœ“) be left inclusive hypergroupoids and let (𝑋,πœƒ)∢=(𝑋,πœ‘)β–‘(𝑋,πœ“). Then π‘₯πœƒπ‘¦=(π‘₯πœ‘π‘¦)πœ“(π‘¦πœ‘π‘₯)=βˆͺ{π‘Žπœ“π‘|π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯} for any π‘₯,π‘¦βˆˆπ‘‹. Since (𝑋,πœ‘) is left inclusive, π‘₯∈π‘₯πœ‘π‘¦,π‘¦βˆˆπ‘¦πœ‘π‘₯, and hence π‘₯πœ“π‘¦βŠ†π‘₯πœƒπ‘¦ for any π‘₯,π‘¦βˆˆπ‘‹. Moreover, (𝑋,πœ“) is left inclusive implies that π‘₯∈π‘₯πœ“π‘¦, which proves that π‘₯∈π‘₯πœƒπ‘¦.

Proposition 5.6. Let (𝑋,πœ‘)≀(𝑋,πœ“) in 𝐻Bin(𝑋). If (𝑋,πœ‘) is left inclusive, then (𝑋,πœ“) is also left inclusive.

Proof. Let (𝑋,πœ‘)≀(𝑋,πœ“). Then π‘₯πœ‘π‘¦βŠ†π‘₯πœ“π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹. Since (𝑋,πœ‘) is left inclusive, we have π‘₯∈π‘₯πœ‘π‘¦βŠ†π‘₯πœ“π‘¦, proving the proposition.

Proposition 5.6 means that the collection of all left inclusive hypergroupoids is a filter in the poset (𝐻Bin(𝑋),≀).

A hypergroupoid (𝑋,πœ‘) is said to be left-self-avoiding if π‘₯βˆ‰π‘₯πœ‘π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹.

Proposition 5.7. The complementary hypergroupoid (𝑋,πœ‘πΆ) of a left inclusive hypergroupoid (𝑋,πœ‘) is left-self-avoiding.

Proof. Let (𝑋,πœ‘πΆ) be the complementary hypergroupoid of a left inclusive hypergroupoid (𝑋,πœ‘). Then π‘₯πœ‘πΆπ‘¦=𝑋⧡π‘₯πœ‘π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹. Since (𝑋,πœ‘) is left inclusive, π‘₯∈π‘₯πœ‘π‘¦ for any π‘₯,π‘¦βˆˆπ‘‹, and hence π‘₯βˆ‰π‘₯πœ‘πΆπ‘¦, proving the proposition.

Proposition 5.8. The complementary hypergroupoid (𝑋,πœ‘πΆ) of a left-self-avoiding hypergroupoid (𝑋,πœ‘) is left inclusive.

Proof. Straightforward.

Proposition 5.9. Let (𝑋,πœƒ)=(𝑋,πœ‘)β–‘(𝑋,πœ“) where (𝑋,πœ‘) is left inclusive and (𝑋,πœƒ) is left-self-avoiding. Then (𝑋,πœ“) is left-self-avoiding.

Proof. Let (𝑋,πœƒ) be a left-self-avoiding hypergroupoid. Then (𝑋,πœƒπΆ) is left inclusive by Proposition 5.8. It follows that π‘₯∈π‘₯πœƒπΆπ‘¦=𝑋⧡βˆͺ{π‘Žπœ“π‘βˆ£π‘Žβˆˆπ‘₯πœ‘π‘¦,π‘βˆˆπ‘¦πœ‘π‘₯}. This means that π‘₯βˆ‰π‘Žπœ“π‘ for any π‘Žβˆˆπ‘₯πœ‘π‘¦ and π‘βˆˆπ‘¦πœ‘π‘₯ where π‘₯,π‘¦βˆˆπ‘‹. Since (𝑋,πœ‘) is left inclusive, π‘₯∈π‘₯πœ‘π‘¦,π‘¦βˆˆπ‘¦πœ‘π‘₯. Hence π‘₯βˆ‰π‘₯πœ“π‘¦, proving that (𝑋,πœ“) 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),β–‘) can still benefit from more detailed investigation it follows that the same is even more true for (𝐻Bin(𝑋),β–‘). In the latter case one must rely on proper adaptations obtained from (Bin(𝑋),β–‘) 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.

References

  1. H. S. Kim and J. Neggers, β€œThe semigroups of binary systems and some perspectives,” Bulletin of the Korean Mathematical Society, vol. 45, no. 4, pp. 651–661, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. H. F. Fayoumi, β€œLocally-zero groupoids and the center of Bin (X),” Korean Mathematical Society. Communications, vol. 26, no. 2, pp. 163–168, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. K. Iséki, β€œOn BCI-algebras,” Mathematics Seminar Notes, vol. 8, no. 1, pp. 125–130, 1980. View at Google Scholar Β· View at Zentralblatt MATH
  4. K. Iséki and S. Tanaka, β€œAn introduction to the theory of BCK-algebras,” Mathematica Japonica, vol. 23, no. 1, pp. 1–26, 1978/79. View at Google Scholar
  5. J. Neggers and H. S. Kim, β€œOn d-algebras,” Mathematica Slovaca, vol. 49, no. 1, pp. 19–26, 1999. View at Google Scholar
  6. J. S. Han, H. S. Kim, and J. Neggers, β€œStrong and ordinary d-algebras,” Journal of Multiple-Valued Logic and Soft Computing, vol. 16, no. 3–5, pp. 331–339, 2010. View at Google Scholar
  7. J. Zhan, B. Davvaz, and K. P. Shum, β€œOn probabilistic n-ary hypergroups,” Information Sciences, vol. 180, no. 7, pp. 1159–1166, 2010. View at Publisher Β· View at Google Scholar
  8. J. Zhan and Y. L. Liu, β€œOn f-derivations of BCI-algebras,” Mathematica Slovaca, vol. 49, pp. 19–26, 1999. View at Google Scholar
  9. J. Zhan, S. Sh. Mousavi, and M. Jafarpour, β€œOn hyperactions of hypergroups,” University of Bucharest. Scientific Bulletin A, vol. 73, no. 1, pp. 117–128, 2011. View at Google Scholar Β· View at Zentralblatt MATH
  10. B. Davvaz and V. Leoreanu, Hyperring Theory and Applications, International Academic Press, Palm Harbor, Fla, USA, 2007.