Soft hypergroups and soft hypergroup homomorphism

Soft hypergroups and soft hypergroup homomorphism Ganeshsree Selvachandran and Abdul Razak Salleh Citation: AIP Conf. Proc. 1522, 821 (2013); doi: 10.1063/1.4801211 View online: http://dx.doi.org/10.1063/1.4801211 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1522&Issue=1 Published by the American Institute of Physics.

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Soft Hypergroups and Soft Hypergroup Homomorphism Ganeshsree Selvachandran and Abdul Razak Salleh School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor DE, Malaysia Abstract. Soft set is a general mathematical tool for dealing with uncertainties and imprecision. Many complicated problems in economics, engineering, environment, social science, medical science and other fields involve uncertain data. These problems cannot be solved using existing classical methods such as fuzzy set theory and rough set theory. Hence the concept of soft sets was introduced as a new mathematical tool for dealing with uncertainties that is free from the difficulties affecting the existing methods. The theory of hyperstructures and hyperalgebra on the other hand, was introduced as an extension of the classical theory of sets as well as the theory of abstract algebra. This theory states that in an algebraic hyperstructure, the composition of two elements is a set. The study of soft hyperstructures which is an extension of the theory of soft sets and the theory of algebraic hyperstructures began with the introduction of the notion of soft hypergroupoids and soft subhypergroupoids. In this paper, we study the theory of soft hyperalgebra by introducing the notion of soft hypergroups and soft subhypergroups as an extension of the notion of soft sets, algebraic hyperstructures and soft hyperstructures. This is followed by the introduction of the notion of soft homomorphism of hypergroups. Furthermore the basic properties and structural characteristics of these notions are studied and discussed. Keywords: Soft Hypergroups, Soft Hyperstructures, Soft Hypergroup Homomorphism PACS: 02.10.De, 02.10.Hh, 01.30.Cc

INTRODUCTION Hyperstructure theory was first introduced in 1934 by a French mathematician, Marty at the 8th Congress of Scandinavian Mathematicians. In a classical algebraic structure, the composition of two elements is an element while in an algebraic hyperstructure, the composition of two elements is a set. Since the introduction of the notion of hyperstructure, comprehensive research has been done on this topic and the notions of hypergroupoid, hypergroup, hyperring and hypermodule have been introduced. A recent book by Corsini and Leoreanu [2] expounds on the applications of hyperstructures in the areas of geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automation, cryptography, combinatorics, codes, artificial intelligence and probability theory. The concept of ‫ܪ‬ఔ - structures introduced by Vougiouklis [6] constitute a generalization of the well-known algebraic hyperstructures such as hypergroups, hyperrings and hypermodules. Some axioms pertaining to the abovementioned hyperstructures such as the associative law and the distributive law are replaced by their corresponding weak axioms. The study of soft hyperstructures was initiated by Yamak et al. [8] as an extension to the theory of soft sets and the theory of hyperstructures. They defined the notion of soft hypergroupoids and soft subhypergroupoids and proceeded to study some of the basic properties of these concepts. In this paper we define the notion of soft hypergroups and soft subhypergroups as a generalization of the notion of soft sets, hypergroups and soft hyperstructures. This is followed by the introduction of the notion of soft homomorphism of hypergroups. Some properties of these notions are then studied and discussed.

Preliminaries In this section, some well-known and useful definitions pertaining to hypergroup theory, soft hyperstructures, soft sets and vague soft sets will be presented. These definitions and concepts will be used throughout this chapter. Throughout this chapter, let ‫ ܪ‬be a hypergroup, ‫ ܧ‬be a set of parameters and ‫ܧ ؿ ܣ‬. Definition 1 (Marty [4]) A hypergroup ‫ܪۃ‬ǡ ‫ ۄל‬is a set ‫ ܪ‬equipped with an associative hyperoperation ሺ‫ל‬ሻǣ ‫ ܪ‬ൈ ‫ ܪ‬՜ ܲሺ‫ܪ‬ሻ which satisfies the reproduction axiom given by ‫ ܪ ל ݔ‬ൌ ‫ ݔ ל ܪ‬ൌ ‫ ܪ‬for all ‫ ݔ‬in ‫ܪ‬. Proceedings of the 20th National Symposium on Mathematical Sciences AIP Conf. Proc. 1522, 821-827 (2013); doi: 10.1063/1.4801211 © 2013 AIP Publishing LLC 978-0-7354-1150-0/$30.00

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Definition 2 (Marty [4]) A subset ‫ ܭ‬of ‫ ܪ‬is called a subhypergroup if ‫ܭۃ‬ǡ ‫ ۄל‬is a hypergroup. Definition 3 (Molodtsov [5]) Let ܷ be an initial universe set and let ‫ ܣ‬be the set of parameters. Let ܲሺܷሻ denote the power set of ܷǤ A pair ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is called a soft set over ܷ, where ‫ ܨ‬is a mapping given by ‫ܨ‬ǣ ‫ ܣ‬՜ ܲሺܷሻ. Definition 4 (Xu et al.[7]) Let ܷ be a universe, ‫ ܧ‬be a set of parameters, ܸሺܷሻ be the power set of vague sets on ܷ and ‫ܧ ك ܣ‬Ǥ A pair ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ is called a vague soft set over ܷ where ‫ܨ‬෠ is a mapping given by ‫ܨ‬෠ ǣ ‫ ܣ‬՜ ܸሺܷሻ and ܸሺܷሻ is the power set of ܷ. In other words, a vague soft set over ܷ is a parametrized family of vague sets of the universe ܷ. Every set ‫ܨ‬෠ ሺ݁ሻ for all ݁߳‫ܣ‬, from this family may be considered as the set of ݁ െapproximate elements of the vague soft set ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ. Hence the vague soft set ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ can be viewed as consisting of a collection of approximations of the following form: ൫‫ܨ‬෠ ǡ ‫ܣ‬൯ ൌ ൛‫ܨ‬෠ ሺ݁௜ ሻǣ݅ ൌ ͳǡ ʹǡ ͵ǡ ǥ ൟ ൌ ሼൣ‫ݐ‬ி෠ሺ௘೔ሻ ሺ‫ݔ‬௜ ሻǡ ͳ െ ݂ி෠ሺ௘೔ ሻ ሺ‫ݔ‬௜ ሻ൧Ȁ‫ݔ‬௜ ‫ ݅ ׷‬ൌ ͳǡ ʹǡ ͵ǡ ǥ ሽ for all ݁߳‫ ܣ‬and for all ‫ܷ߳ݔ‬. Definition 5 (Molodtsov [5]) Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻand ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be soft sets over ܷǤ Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is a soft subset of ሺ‫ܩ‬ǡ ‫ܤ‬ሻ if (i) (ii)

‫ ܤ ؿ ܣ‬and ‫ܣ߳߳׊‬ǡ ‫ܨ‬ሺߝሻ‫ܩ‬ሺߝሻ are the same approximations.

෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ In this case, ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is said to be soft superset of ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and this is This relationship is denoted as ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ؿ‬ ෥ ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ denoted as ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ‫ـ‬ Definition 6 (Ali et al. [9]) The extended intersection (union resp.) of two soft sets ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ over a common universe ܷ is the soft set ሺ‫ܪ‬ǡ ‫ܥ‬ሻ, where ‫ ܥ‬ൌ ‫ܤ ׫ ܣ‬ǡ‫ܥ߳ߝ׊‬ǡ ‫ܨ‬ሺߝሻǡߝ ‫ ܣ א‬െ ‫ܤ‬ ‫ܪ‬ሺߝሻ ൌ ቐ ‫ܩ‬ሺߝሻǡߝ ‫ ܤ א‬െ ‫ܣ‬ ‫ܨ‬ሺߝሻ ‫ ת‬ሺ‫ ׫‬Ǥ ሻ‫ܩ‬ሺߝሻǡߝ ‫ܤ ת ܣ א‬Ǥ ෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ܥ‬ሻ. This is denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ת‬ Definition 7 (Maji et al. [10]) If ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are two soft sets then ̶ሺ‫ܨ‬ǡ ‫ܣ‬ሻሺǤሻሺ‫ܩ‬ǡ ‫ܤ‬ሻ̶ is a soft set denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ ר‬ሺ‫ ש‬Ǥ ሻሺ‫ܩ‬ǡ ‫ܤ‬ሻ and is defined by: ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ ר‬ሺ‫ ש‬Ǥ ሻሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻǡ where ‫ܪ‬ሺߙǡ ߚሻ ൌ ‫ܨ‬ሺߙሻ ‫ ת‬ሺ‫ ׫‬Ǥ ሻ‫ܩ‬ሺߚሻǡ ‫׊‬ሺߙǡ ߚሻ߳‫ ܣ‬ൈ ‫ܤ‬. Definition 8 (Ali et al. [9]) Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft sets over a common universe ܷ such that ‫׎ ് ܤ ת ܣ‬. The restricted intersection (restricted union resp.) of ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ ڑ‬ሺ‫ ࣬׫‬Ǥ ሻሺ‫ܩ‬ǡ ‫ܤ‬ሻ and is defined as ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ ڑ‬ሺ‫ ࣬׫‬Ǥ ሻሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ܥ‬ሻ , where ‫ ܥ‬ൌ ‫ ܤ ת ܣ‬and for all ߝ߳‫ ܥ‬, ‫ܪ‬ሺܿሻ ൌ ‫ܨ‬ሺܿሻ ‫ת‬ ሺ‫ ׫‬Ǥ ሻ‫ܩ‬ሺܿሻ. Soft Hypergroups In this section, the notion of soft hypergroups are introduced. This is an extension to the concepts of soft sets, hypergroups and soft hypergroupoids. Using the definition of soft hypergroups that are introduced here, some new properties of soft hypergroups are obtained. Lastly some fundamental properties of soft hypergroups are studied and investigated. From now on, let ‫ ܪ‬be a hypergroup ‫ܪۃ‬ǡ ‫( ۄל‬or ‫ܪ‬ఔ -group), ‫ ܧ‬be a set of parameters and ‫ܧ ك ܣ‬.


Definition 9 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ be a non-null soft set over ‫ܪ‬Ǥ Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is called a soft hypergroup over ‫ ܪ‬if ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ ܪ‬for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ. Definition 10 (Davvaz [3]) Let ‫ ܪ‬be a set and ‫ܨ‬෠௔ be a subset of ‫ܪ‬Ǥ The hyperoperation ‫ ܪ ׷ל‬ൈ ‫ ܪ‬՜ ܲሺ‫ܪ‬ሻ is defined as follows: Let ‫ݔ‬ǡ ‫ܪ߳ݕ‬Ǥ If ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻǡ then ‫ ݕ ל ݔ‬ൌ ‫ ݔ ל ݕ‬ൌ ൛‫ܪ߳ݐ ׷ ݐ‬ǡ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݐ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻൟǤ Definition 11 Let ܺ be a group and ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ be a vague soft set over ܺǤ Then ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ is a vague soft group over ܺ if and only if for all ܽ߳‫ ܣ‬and ‫ݔ‬ǡ ‫ܺ߳ݕ‬ǡ the following conditions are satisfied: (i) (ii)

‫ݐ‬ி෠ೌ ሺ‫ݕݔ‬ሻ ൒ ൛‫ݐ‬ி෠ೌ ሺ‫ݔ‬ሻǡ ‫ݐ‬ி෠ೌ ሺ‫ݕ‬ሻൟ and ͳ െ ݂ி෠ೌ ሺ‫ݕݔ‬ሻ ൒ ൛ͳ െ ݂ி෠ೌ ሺ‫ݔ‬ሻǡ ͳ െ ݂ி෠ೌ ሺ‫ݕ‬ሻൟǡ i.e. ‫ܨ‬෠௔ ሺ‫ݕݔ‬ሻ ൒ ቀ‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻቁǡ

‫ݐ‬ி෠ೌ ሺ‫ି ݔ‬ଵ ሻ ൒ ‫ݐ‬ி෠ೌ ሺ‫ݔ‬ሻ and ͳ െ ݂ி෠ೌ ሺ‫ି ݔ‬ଵ ሻ ൒ ͳ െ ݂ி෠ೌ ሺ‫ݔ‬ሻǡ i.e. ‫ܨ‬෠௔ ሺ‫ି ݔ‬ଵ ሻ ൒ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ

i.e. for each ܽ߳‫ܣ‬ǡ ‫ܨ‬෠௔ is a vague subgroup of ܺ in Rosenfeld’s sense. Definition 12 Let ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ be a non-empty vague soft set over ‫ܪ‬Ǥ Then ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ is a vague soft hypergroup over ‫ ܪ‬if for each ܽ߳ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ the following conditions are satisfied: (i) (ii) (iii)

For all ‫ݔ‬ǡ ‫ܪ߳ݕ‬ǡ ሼ‫ݐ‬ி෠ೌ ሺ‫ݔ‬ሻǡ ‫ݐ‬ி෠ೌ ሺ‫ݕ‬ሻሽ ൑ ሼ‫ݐ‬ி෠ೌ ሺ‫ݖ‬ሻǣ ‫ݕ ל ݔ߳ݖ‬ሽ and ሼͳ െ ݂ி෠ೌ ሺ‫ݔ‬ሻǡ ͳ െ ݂ி෠ೌ ሺ‫ݕ‬ሻሽ ൑ ሼͳ െ ݂ி෠ೌ ሺ‫ݖ‬ሻǣ ‫ݕ ל ݔ߳ݖ‬ሽǡ i.e. ൛‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻൟ ൑ ൛‫ܨ‬෠௔ ሺ‫ݖ‬ሻǣ ‫ݕ ל ݔ߳ݖ‬ൟǡ For all ‫ݓ‬ǡ ‫ ܪ߳ݔ‬there exists ‫ ܪ߳ݕ‬such that ‫ ݕ ל ݓ߳ݔ‬and ൛‫ݐ‬ி෠ೌ ሺ‫ݓ‬ሻǡ ‫ݐ‬ி෠ೌ ሺ‫ݔ‬ሻൟ ൑ ‫ݐ‬ி෠ೌ ሺ‫ݕ‬ሻ and ൛ͳ െ ݂ி෠ೌ ሺ‫ݓ‬ሻǡ ͳ െ ݂ி෠ೌ ሺ‫ݔ‬ሻሽ ൑ ͳ െ ݂ி෠ೌ ሺ‫ݕ‬ሻǡ i.e. ൛‫ܨ‬෠௔ ሺ‫ݓ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻൟ ൑ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻǡ For all ‫ݓ‬ǡ ‫ ܪ߳ݔ‬there exists ‫ ܪ߳ݖ‬such that ‫ ݓ ל ݖ߳ݔ‬and ൛‫ݐ‬ி෠ೌ ሺ‫ݓ‬ሻǡ ‫ݐ‬ி෠ೌ ሺ‫ݔ‬ሻൟ ൑ ‫ݐ‬ி෠ೌ ሺ‫ݖ‬ሻ and ൛ͳ െ ݂ி෠ೌ ሺ‫ݓ‬ሻǡ ͳ െ ݂ி෠ೌ ሺ‫ݔ‬ሻሽ ൑ ͳ െ ݂ி෠ೌ ሺ‫ݖ‬ሻǡ i.e. ൛‫ܨ‬෠௔ ሺ‫ݓ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻൟ ൑ ‫ܨ‬෠௔ ሺ‫ݖ‬ሻǤ

Conditions (ii) and (iii) represent the left and right reproduction axioms respectively. Therefore ‫ܨ‬෠௔ is a non-empty vague subhypergroup of ‫ ܪ‬in Rosenfeld’s sense for each ܽ߳൫‫ܨ‬෠ ǡ ‫ܣ‬൯Ǥ Proposition 1 Let ሺ‫ܪ‬ǡ ‫כ‬ሻ be a group. If ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ is a vague soft group over ሺ‫ܪ‬ǡ ‫כ‬ሻǡ then ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ is a soft hypergroup over ሺ‫ܪ‬ǡ ‫ל‬ሻǤ Proof. Let ሺ‫ܨ‬෠ ǡ ‫ܣ‬ሻ be a vague soft group over ሺ‫ܪ‬ǡ ‫כ‬ሻ and ሺ‫ܪ‬ǡ ‫ל‬ሻ be a hypergroup with hyperoperation as given in Definition 10. Then for all ‫ݔ‬ǡ ‫߳ݕ‬ሺ‫ܪ‬ǡ ‫כ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݕݔ‬ሻ ൒ ൛‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻൟ and ‫ܨ‬෠௔ ሺ‫ି ݔ‬ଵ ሻ ൒ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻǤ Thus for each ‫ݔ‬ǡ ‫߳ݕ‬ሺ‫ܪ‬ǡ ‫כ‬ሻǡ the following is obtained: ൛‫ܨ‬෠௔ ሺ‫ݖ‬ሻ ‫ݕ ל ݔ߳ݖ ׷‬ൟ ൌ ൛‫ܨ‬෠௔ ሺ‫ݖ‬ሻ ‫ܨ ׷‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݖ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻൟ ෠ ൌ ‫ܨ‬௔ ሺ‫ݔ‬ሻ ൒ ൛‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻൟǡ which means that ൛‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻൟ ൑ ൛‫ܨ‬෠௔ ሺ‫ݖ‬ሻ ‫ݕ ל ݔ߳ݖ ׷‬ൟǤ Therefore condition (i) of Definition 12 has been verified. Now let ‫ݓ‬ǡ ‫ܪ߳ݔ‬Ǥ Thus if ‫ܨ‬෠௔ ሺ‫ݓ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ then ‫ܨ‬෠௔ ሺ‫ݓ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ which implies that ‫( ݔ ל ݓ߳ݔ‬by Definition 12 and if ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݓ‬ሻǡ then we have ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݔ‬ሻ ൑ ‫ܨ‬෠௔ ሺ‫ݓ‬ሻ which also implies that ‫ݓ ל ݔ߳ݔ‬Ǥ As such, we have ‫ ܽ ל ݔ߳ݔ‬ൌ ܽ ‫ݔ ל‬Ǥ Hence if we let ‫ ݕ‬ൌ ‫ݔ‬ǡ then for any case, we obtain ൛‫ܨ‬෠௔ ሺ‫ݔ‬ሻǡ ‫ܨ‬෠௔ ሺ‫ݓ‬ሻൟ ൑ ‫ܨ‬෠௔ ሺ‫ݕ‬ሻǤ Therefore condition (ii) of Definition 12 has been verified. Thus it has been proven that ‫ܨ‬෠௔ is a subhypergroup of ሺ‫ܪ‬ǡ ‫ל‬ሻǤ Hence ൫‫ܨ‬෠ ǡ ‫ܣ‬൯ is a soft hypergroup over ሺ‫ܪ‬ǡ ‫ל‬ሻǤ ‫ז‬ Theorem 1 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬Ǥ Then the extended intersection of ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft hypergroup over ‫ ܪ‬if it is non-null. ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ת‬


෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ܥ‬ሻǡ where ‫ ܥ‬ൌ ‫ ܤ ׫ ܣ‬and for all ‫߳ݔ‬ሺ‫ܪ‬ǡ ‫ܥ‬ሻǡ Proof. By Definition 6, ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ת‬ ‫ܨ‬ሺ‫ݔ‬ሻ‫ ܣ א ݔ‬െ ‫ܤ‬ǡ ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ቐ‫ܩ‬ሺ‫ݔ‬ሻ‫ ܤ א ݔ‬െ ‫ܣ‬ǡ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ת‬ሺ‫ݔ‬ሻ‫ܤ ת ܣ א ݔ‬Ǥ Since ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are both non-null soft sets over ‫ܪ‬, it follows that ሺ‫ܪ‬ǡ ‫ܥ‬ሻ is also a non-null soft set over ‫ ܪ‬for all the three possible cases for ‫ݔ‬. If ‫ ܣ߳ݔ‬െ ‫ܤ‬, then ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ is a non-null subhypergroup of ‫ ܪ‬because ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is a soft hypergroup over ‫ܪ‬. If ‫ ܤ߳ݔ‬െ ‫ܣ‬, then ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ‫ܩ‬ሺ‫ݔ‬ሻ is a non-null subhypergroup of ‫ ܪ‬since ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft hypergroup over ‫ܪ‬Ǥ If ‫ ܤ ת ܣ߳ݔ‬, then ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ת‬ሺ‫ݔ‬ሻ ് ‫ ׎‬since ‫ܨ‬ሺ‫ݔ‬ሻ ് ‫ ׎‬and ‫ܩ‬ሺ‫ݔ‬ሻ ് ‫ ׎‬are both subhypergroups of ‫ ܪ‬and therefore their intersection must be non-null too. Thus in any case, ‫ܪ‬ሺ‫ݔ‬ሻ is a ෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft subhypergroup of ‫ܪ‬Ǥ Hence it follows that ሺ‫ܪ‬ǡ ‫ܥ‬ሻ is a soft hypergroup over ‫ ܪ‬i.e. ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ת‬ hypergroup over ‫ܪ‬Ǥ ‫ז‬ ෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft hypergroup over Corollary 1 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬Ǥ Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ש‬ ‫ ܪ‬if it is non-null. ෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻǡ where ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ׫‬ሺ‫ݕ‬ሻ for all ሺ‫ݔ‬ǡ ‫ݕ‬ሻ߳ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ Proof. By Definition 7, ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ש‬ ‫ܤ‬ሻǤ Given that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are both soft hypergroups over ‫ܪ‬, by Definition 9, both ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ must be non-null. Thus ‫ܨ‬ሺ‫ݔ‬ሻ is a non-null subhypergroup of ‫ ܪ‬for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ‫ܩ‬ሺ‫ݕ‬ሻ is a non-null subhypergroup of ‫ ܪ‬for all ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ Therefore for all ሺ‫ݔ‬ǡ ‫ݕ‬ሻ߳ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻ, ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ׫‬ሺ‫ݕ‬ሻ ് ‫׎‬, where ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ is a non-null subhypergroup of ‫ ܪ‬for all ሺ‫ݔ‬ǡ ‫ݕ‬ሻ߳ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻ. Hence, by Definition 9, ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻ is a soft ෥ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft hypergroup over ‫ܪ‬Ǥ hypergroup over ‫ܪ‬, which means that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ש‬ ‫ז‬ Definition 13 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬ଵ and ‫ܪ‬ଶ respectively. The product of ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is defined as ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൈ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻ, where ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ൈ ‫ܩ‬ሺ‫ݕ‬ሻ for all ሺ‫ݔ‬ǡ ‫ݕ‬ሻ߳ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻǤ Corollary 2 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬ଵ and ‫ܪ‬ଶ respectively. Then the product of ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൈ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft hypergroup over ‫ܪ‬ଵ ൈ ‫ܪ‬ଶ . Proof. Assume that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are two soft hypergroups over ‫ܪ‬ଵ and ‫ܪ‬ଶ respectively. Thus, ‫ܨ‬ሺ‫ݔ‬ሻ ് ‫ ׎‬is a subhypergroup of ‫ܪ‬ଵ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ‫ܩ‬ሺ‫ݕ‬ሻ ് ‫ ׎‬is a subhypergroup of ‫ܪ‬ଶ for all ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ As such, ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ൈ ‫ܩ‬ሺ‫ݕ‬ሻ ് ‫ ׎‬is a subhypergroup of ‫ܪ‬ଵ ൈ ‫ܪ‬ଶ for all ሺ‫ݔ‬ǡ ‫ݕ‬ሻ߳ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ܤ‬ሻǤ Hence ሺ‫ܪ‬ǡ ‫ ܣ‬ൈ ‫ז‬ ‫ܤ‬ሻ is a soft hypergroup over ‫ܪ‬ଵ ൈ ‫ܪ‬ଶ i.e. ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൈ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft hypergroup over ‫ܪ‬ଵ ൈ ‫ܪ‬ଶ . Soft Hypergroup Homomorphism We begin this section by presenting the notion of good and inclusion homomorphism of hypergroups introduced by Corsini [1] and the notion of soft functions introduced by Yamak et al. [8]. These notions are then used in the context of soft hypergroups to define the notion of soft hypergroup homomorphism. We then proceed to study some properties of this notion. Definition 14 (Corsini [1]) Let ሺ‫ܪ‬ଵ ǡ ‫ڄ‬ሻ and ሺ‫ܪ‬ଶ ǡ ‫כ‬ሻ be two hypergroups. The map ݂ ‫ܪ ׷‬ଵ ՜ ‫ܪ‬ଶ is called a good (resp. inclusion) homomorphism if for all ‫ݔ‬ǡ ‫ܪ߳ݕ‬ଵ ǡ the following relationship holds: ݂ሺ‫ݕ ڄ ݔ‬ሻ ൌ ݂ሺ‫ݔ‬ሻ ‫݂ כ‬ሺ‫ݕ‬ሻ ൫Ǥ ݂ሺ‫ݕ ڄ ݔ‬ሻ ‫݂ ك‬ሺ‫ݔ‬ሻ ‫݂ כ‬ሺ‫ݕ‬ሻ൯. Definition 15 (Yamak et al. [8]) Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft sets over ܷ and ܷԢ respectively, ݂ ‫ ܷ ׷‬՜ ܷԢ, ݃ ‫ ܣ ׷‬՜ ‫ ܤ‬be two functions. Then we say that the pair ሺ݂ǡ ݃ሻ is a soft function from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ denoted by ሺ݂ǡ ݃ሻǣ ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ՜ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ if it satisfies ݂൫‫ܨ‬ሺ‫ݔ‬ሻ൯ ൌ ‫ܩ‬൫݃ሺ‫ݔ‬ሻ൯ for all ‫ܣ߳ݔ‬Ǥ If ݂ and ݃ are injective (resp. surjective, bijective), then ሺ݂ǡ ݃ሻ is said to be injective (resp. surjective, bijective). The notion of soft hypergroup homomorphism is now introduced using the notion of good and inclusion homomorphism and soft functions given above.


Definition 16 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬ଵ and ‫ܪ‬ଶ respectively and let ሺ߮ǡ ߰ሻ be a soft function from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. Then ሺ߮ǡ ߰ሻis called a soft inclusion (resp. good) homomorphism of hypergroups if the following conditions are satisfied: (i) (ii) (iii)

߮ is a an inclusion (resp. good) homomorphism from ‫ܪ‬ଵ to ‫ܪ‬ଶ Ǥ ߰ is a function from ‫ ܣ‬into ‫ܤ‬Ǥ ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ ൌ ‫ܩ‬൫߰ሺ‫ݔ‬ሻ൯ቀǤ ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ ‫ܩ ك‬൫߰ሺ‫ݔ‬ሻ൯ቁ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ.

Theorem 2 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ሺ‫ܪ‬ଵ ǡ ‫ל‬ሻ and ሺ‫ܪ‬ଶ ǡ ‫כ‬ሻ respectively. Let ሺ߮ǡ ߰ሻ be a soft good homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ and let ߰ ‫ ܣ ׷‬՜ ‫ ܤ‬be an injective mapping. Then ൫߮ሺ‫ܨ‬ሻǡ ߰ሺ‫ܣ‬ሻ൯ is a soft hypergroup over ‫ܪ‬ଶ . Proof. Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ሺ‫ܪ‬ଵ ǡ ‫ל‬ሻ and ሺ‫ܪ‬ଶ ǡ ‫כ‬ሻ respectively. Then ‫ܨ‬ሺ‫ݔ‬ሻ is a nonnull subhypergroup of ‫ܪ‬ଵ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ Since ሺ߮ǡ ߰ሻ is a soft good homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, the following relationship holds true: ൫߮ሺ‫ܨ‬ሻǡ ߰ሺ‫ܣ‬ሻ൯ ൌ ߰൫ሺ‫ܨ‬ǡ ‫ܣ‬ሻ൯ Now let ‫߳ݕ‬൫߮ሺ‫ܨ‬ሻǡ ߰ሺ‫ܣ‬ሻ൯Ǥ Then there exists ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ such that ‫ ݕ‬ൌ ߰ሺ‫ݔ‬ሻǤ Thus, ߮ሺ‫ܨ‬ሻ൫߮ሺ‫ݔ‬ሻ൯ is a nonnull subhypergroup of ‫ܪ‬ଶ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ However, ߮ሺ‫ܨ‬ሻ൫߮ሺ‫ݔ‬ሻ൯ ൌ ൫߮ሺ‫ܨ‬ሻ൯ሺ‫ݕ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ‫߳ݕ‬൫߮ሺ‫ܨ‬ሻǡ ߰ሺ‫ܣ‬ሻ൯Ǥ As such, ൫߮ሺ‫ܨ‬ሻ൯ሺ‫ݕ‬ሻ is a non-null subhypergroup of ‫ܪ‬ଶ for all ‫߳ݕ‬൫߮ሺ‫ܨ‬ሻǡ ߰ሺ‫ܣ‬ሻ൯Ǥ ‫ז‬ Hence ൫߮ሺ‫ܨ‬ሻǡ ߰ሺ‫ܣ‬ሻ൯ is a soft hypergroup over ‫ܪ‬ଶ Ǥ Theorem 3 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ሺ‫ܪ‬ଵ ǡ ‫ל‬ሻ and ሺ‫ܪ‬ଶ ǡ ‫כ‬ሻ respectively. Let ሺ߮ǡ ߰ሻ be a soft inclusion homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. Then ቀ߮ ିଵ ൫‫ܩ‬෠ ൯ǡ ߰ ିଵ ሺ‫ܤ‬ሻቁ is a soft hypergroup over ‫ܪ‬ଵ if it is non-null. Proof. Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ሺ‫ܪ‬ଵ ǡ ‫ל‬ሻ and ሺ‫ܪ‬ଶ ǡ ‫כ‬ሻ respectively. Then ‫ܩ‬ሺ‫ݕ‬ሻ is a nonnull subhypergroup of ‫ܪ‬ଶ for all ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. Due to the fact that ሺ߮ǡ ߰ሻ is a soft inclusion homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, the following relationship holds: ൫߮ ିଵ ሺ‫ܩ‬ሻǡ ߰ ିଵ ሺ‫ܤ‬ሻ൯ ‫ି ߰ ك‬ଵ ൫ሺ‫ܩ‬ǡ ‫ܤ‬ሻ൯ Now let ‫߳ݔ‬൫߮ ିଵ ሺ‫ܩ‬ሻǡ ߰ ିଵ ሺ‫ܤ‬ሻ൯Ǥ Then there exists ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ such that ߰ሺ‫ݔ‬ሻ ൌ ‫ݕ‬Ǥ As such, ߮ ିଵ ሺ‫ܩ‬ሻ൫߰ ିଵ ሺ‫ݕ‬ሻ൯ is a non-null subhypergroup of ‫ܪ‬ଵ for all ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ Thus, ߮ ିଵ ሺ‫ܩ‬ሻ൫߰ ିଵ ሺ‫ݕ‬ሻ൯ ‫ك‬ ൫߮ ିଵ ሺ‫ܩ‬ሻ൯ሺ‫ݔ‬ሻ since ߮ is an inclusion homomorphism. Therefore, ൫߮ ିଵ ሺ‫ܩ‬ሻ൯ሺ‫ݔ‬ሻ is a also a non-null subhypergroup of ‫ܪ‬ଵ for all ‫߳ݔ‬൫߮ ିଵ ሺ‫ܩ‬ሻǡ ߰ ିଵ ሺ‫ܤ‬ሻ൯Ǥ Hence ൫߮ ିଵ ሺ‫ܩ‬ሻǡ ߰ ିଵ ሺ‫ܤ‬ሻ൯ is a soft hypergroup over ‫ܪ‬ଵ Ǥ ‫ז‬ Theorem 2 and 3 proves that the image and pre-image of a soft hypergroup is also a soft hypergroup. Theorem 4 Let ‫ܪ‬ଵ ǡ ‫ܪ‬ଶ and ‫ܪ‬ଷ be non-null hypergroups and let ሺ‫ܨ‬ǡ ‫ܣ‬ሻǡ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ and ሺ‫ܬ‬ǡ ‫ܥ‬ሻ be soft hypergroups over ‫ܪ‬ଵ ǡ ‫ܪ‬ଶ and ‫ܪ‬ଷ respectively. Let ሺ߮ǡ ߰ሻ ‫ ׷‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ՜ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ and ሺ߮ ᇱ ǡ ߰ ᇱ ሻ ‫ ׷‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ՜ ሺ‫ܬ‬ǡ ‫ܥ‬ሻ be two soft good (resp. inclusion) homomorphisms respectively. Then ሺ߮ ᇱ ‫߮ ל‬ǡ ߰ ᇱ ‫߰ ל‬ሻ is a soft good (resp. inclusion) homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܬ‬ǡ ‫ܥ‬ሻ. Proof. Suppose that ሺ߮ǡ ߰ሻ is a soft good (resp. inclusion) homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. Then ߮ ‫ܪ ׷‬ଵ ՜ ‫ܪ‬ଶ is a good (resp. inclusion) homomorphism from ‫ܪ‬ଵ to ‫ܪ‬ଶ , ߰ ‫ ܣ ׷‬՜ ‫ ܤ‬is a function from ‫ ܣ‬into ‫ ܤ‬and it satisfies the condition ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ ൌ ‫ܩ‬൫߰ሺ‫ݔ‬ሻ൯ ቀǤ ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ ‫ܩ ك‬൫߰ሺ‫ݔ‬ሻ൯ቁ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ Now let ሺ߮ ᇱ ǡ ߰ ᇱ ሻ be a soft good (resp. inclusion) homomorphism from ሺ‫ܩ‬ǡ ‫ܤ‬ሻ to ሺ‫ܬ‬ǡ ‫ܥ‬ሻǤ Similarly, ߮ ᇱ ‫ܪ ׷‬ଶ ՜ ‫ܪ‬ଷ is a good (resp. inclusion) homomorphism from ‫ܪ‬ଶ to ‫ܪ‬ଷ and ߰ ᇱ ‫ ܤ ׷‬՜ ‫ ܥ‬is a function from ‫ ܤ‬into ‫ ܥ‬and for all ‫߳ݔ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ the following condition is satisfied:


߮ ᇱ ൫‫ܩ‬ሺ‫ݔ‬ሻ൯ ൌ ‫ܬ‬൫߰ ᇱ ሺ‫ݔ‬ሻ൯ ቀǤ ߮ ᇱ ൫‫ܩ‬ሺ‫ݔ‬ሻ൯ ‫ܬ ك‬൫߰ ᇱ ሺ‫ݔ‬ሻ൯ቁ. Therefore, ߮ ᇱ ‫ܪ ׷ ߮ ל‬ଵ ՜ ‫ܪ‬ଷ is a good (resp. inclusion) homomorphism from ‫ܪ‬ଵ to ‫ܪ‬ଷ and ߰ ᇱ ‫ ܣ ׷ ߰ ל‬՜ ‫ ܥ‬is a function from ‫ ܣ‬into ‫ܥ‬Ǥ Thus, ሺ߮ ᇱ ‫߮ ל‬ǡ ߰ ᇱ ‫߰ ל‬ሻ is a soft function by Definition 15. Furthermore, for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ሺ߮ ᇱ ‫߮ ל‬ሻ൫‫ܨ‬ሺ‫ݔ‬ሻ൯ ൌ ߮ ᇱ ቀ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ቁ ൌ ߮ ᇱ ሺ‫ܩ‬ሺ߰ሺ‫ݔ‬ሻሻሻ ൌ ‫ܬ‬ሺ߰ ᇱ ሺ߰ሺ‫ݔ‬ሻሻሻ

൬Ǥ ‫ ߮ ك‬ᇱ ቀ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ቁ൰ ൫Ǥ ‫ ߮ ك‬ᇱ ሺ‫ܩ‬ሺ߰ሺ‫ݔ‬ሻሻሻ൯ ൫Ǥ ‫ܬ ك‬ሺ߰ ᇱ ሺ߰ሺ‫ݔ‬ሻሻሻ൯

ൌ ‫ܬ‬൫ሺ߰ ᇱ ‫߰ ל‬ሻሺ‫ݔ‬ሻ൯. ቀǤ ‫ܬ ك‬൫ሺ߰ ᇱ ‫߰ ל‬ሻሺ‫ݔ‬ሻ൯ቁǤ Thus by Definition 16, ሺ߮ ᇱ ‫߮ ל‬ǡ ߰ ᇱ ‫߰ ל‬ሻ satisfies all the conditions required for a soft function to be a soft good (resp. inclusion) homomorphism of hypergroups. Hence ሺ߮ ᇱ ‫߮ ל‬ǡ ߰ ᇱ ‫߰ ל‬ሻ is a soft good (resp. inclusion) homomorphism from ሺ‫ܨ‬ǡ ‫ܣ‬ሻ to ሺ‫ܬ‬ǡ ‫ܥ‬ሻ. ‫ז‬ Soft Subhypergroups In this section, the notion of soft subhypergroups over a hypergroup is introduced. This notion is an extension to the notion of soft hypergroups and soft subsets. We then proceed to study and investigate some basic properties of this concept. Definition 17 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬. Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is called a soft subhypergroup of ሺ‫ܩ‬ǡ ‫ܤ‬ሻǡ denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൑ௌ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, if the following conditions are satisfied: (i) ‫ܤ ك ܣ‬ǡ (ii) ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ Theorem 5 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ ܪ‬and let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ ك‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൑ௌ ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ Proof. Suppose that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are two soft hypergroups over ‫ܪ‬Ǥ Then ‫ܨ‬ሺ‫ݔ‬ሻ and ‫ܩ‬ሺ‫ݔ‬ሻ are non-null subhypergroups of ‫ܪ‬. If ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is a soft subset of ሺ‫ܩ‬ǡ ‫ܤ‬ሻ denoted by ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ ك‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, then by Definition 5, ‫ܤ ك ܣ‬ and ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ك‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ. Thus, ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ. Therefore, since ‫ ܤ ك ܣ‬and ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ, by Definition 17, it can be concluded that ‫ז‬ ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is a soft subhypergroup of ሺ‫ܩ‬ǡ ‫ܤ‬ሻ i.e. ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൑ௌ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. Theorem 6 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ ܪ‬and let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ be a soft subhypergroup of ሺ‫ܩ‬ǡ ‫ܤ‬ሻ. ෩ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft subhypergroup of ሺ‫ܩ‬ǡ ‫ܤ‬ሻ if it is non-null. Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ڑ‬ Proof. Suppose that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are soft hypergroups over ‫ܪ‬. Then ‫ܨ‬ሺ‫ݔ‬ሻ is a non-null subhypergroup of ‫ ܪ‬for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ‫ܩ‬ሺ‫ݕ‬ሻ is a non-null subhypergroup of ‫ ܪ‬for all ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ Furthermore, assume that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ is a soft subhypergroup over ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ Then by Definition 17, ‫ ܤ ك ܣ‬and ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ for ෩ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ ൌ ሺ‫ܪ‬ǡ ‫ܥ‬ሻǡ where ‫ ܥ‬ൌ ‫ ׎ ് ܤ ת ܣ‬and ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ת‬ all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ . Now by Definition 8, ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ڑ‬ ‫ܩ‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܪ‬ǡ ‫ܥ‬ሻǤ As such, it can be concluded that ‫ ܥ‬ൌ ‫ ܤ ك ܤ ת ܣ‬i.e. ‫ܤ ك ܥ‬Ǥ Since ‫ܨ‬ሺ‫ݔ‬ሻ ് ‫ ׎‬and ‫ܩ‬ሺ‫ݔ‬ሻ ് ‫׎‬ǡ it follows that ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ת‬ሺ‫ݔ‬ሻ ് ‫ ׎‬too. Thus for all ‫߳ݔ‬ሺ‫ܪ‬ǡ ‫ܥ‬ሻǡ ‫ܪ‬ሺ‫ݔ‬ሻ ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ‫ܩ ת‬ሺ‫ݔ‬ሻ is a non-null subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ since ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻǤ Hence ሺ‫ܪ‬ǡ ‫ܥ‬ሻ is a soft subhypergroup over ෩ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft subhypergroup over ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ i.e. ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫ڑ‬ ‫ז‬ Corollary 3 Let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ ܪ‬and let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ be a soft subhypergroup of ሺ‫ܩ‬ǡ ‫ܤ‬ሻǤ ෥ ࣬ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ is a soft subhypergroup of ሺ‫ܩ‬ǡ ‫ܤ‬ሻ if it is non-null. Then ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ‫׫‬ Proof. The proof is similar to that of Theorem 6 and is straightforward using Definition 8.

‫ז‬

Theorem 7 Let ߮ ‫ܪ ׷‬ଵ ՜ ‫ܪ‬ଶ be a good homomorphism of hypergroups and let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬ଵ Ǥ If ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൑ௌ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, then ሺ߮ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻ ൑ௌ ሺ߮ሺ‫ܩ‬ሻǡ ‫ܤ‬ሻǤ


Proof. Suppose that ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ are two soft hypergroups over ‫ܪ‬ଵ Ǥ Then ‫ܨ‬ሺ‫ݔ‬ሻ and ‫ܩ‬ሺ‫ݕ‬ሻ are non-null subhypergroups of ‫ܪ‬ଵ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ‫߳ݕ‬ሺ‫ܩ‬ǡ ‫ܤ‬ሻ respectively. Now if ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൑ௌ ሺ‫ܩ‬ǡ ‫ܤ‬ሻǡ then by Definition 17, ‫ ܤ ك ܣ‬and ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ Let ߮ be a good homomorphism of hypergroups from ‫ܪ‬ଵ to ‫ܪ‬ଶ Ǥ As such, ߮ሺ‫ܨ‬ሻሺ‫ݔ‬ሻ ൌ ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ and ߮ሺ‫ܩ‬ሻሺ‫ݔ‬ሻ ൌ ߮൫‫ܩ‬ሺ‫ݔ‬ሻ൯ are both non-null subhypergroups of ሺ߮ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻ and ሺ߮ሺ‫ܩ‬ሻǡ ‫ܤ‬ሻ respectively. Furthermore, since ‫ܨ‬ሺ‫ݔ‬ሻ is a subhypergroup of ‫ܩ‬ሺ‫ݔ‬ሻ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǡ ߮ሺ‫ܨ‬ሻሺ‫ݔ‬ሻ ൌ ߮൫‫ܨ‬ሺ‫ݔ‬ሻ൯ is a subhypergroup of ߮ሺ‫ܩ‬ሻሺ‫ݔ‬ሻ ൌ ߮൫‫ܩ‬ሺ‫ݔ‬ሻ൯ for all ‫߳ݔ‬ሺ‫ܨ‬ǡ ‫ܣ‬ሻǤ Thus ሺ߮ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻ is a soft hypergroup over ሺ߮ሺ‫ܩ‬ሻǡ ‫ܤ‬ሻ for all ‫߳ݔ‬ሺ߮ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻǤ Hence by ‫ז‬ Definition 17, ሺ߮ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻ is a soft subhypergroup of ሺ߮ሺ‫ܩ‬ሻǡ ‫ܤ‬ሻ, denoted by ሺ߮ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻ ൑ௌ ሺ߮ሺ‫ܩ‬ሻǡ ‫ܤ‬ሻǤ Theorem 8 Let ߮ǣ‫ܪ‬ଵ ՜ ‫ܪ‬ଶ be a good homomorphism of hypergroups and let ሺ‫ܨ‬ǡ ‫ܣ‬ሻ and ሺ‫ܩ‬ǡ ‫ܤ‬ሻ be two soft hypergroups over ‫ܪ‬ଶ . If ሺ‫ܨ‬ǡ ‫ܣ‬ሻ ൑ௌ ሺ‫ܩ‬ǡ ‫ܤ‬ሻ, then ሺ߮ ିଵ ሺ‫ܨ‬ሻǡ ‫ܣ‬ሻ ൑ௌ ሺ߮ ିଵ ሺ‫ܩ‬ሻǡ ‫ܤ‬ሻǤ ‫ז‬

Proof. The proof is similar to that of Theorem 7.

CONCLUSIONS In this paper, we have introduced the notion of soft hypergroups as an extension of the notion of soft sets, soft hyperstructures and hypergroups. Other related concepts such as the homomorphism of soft hypergroups and soft subhypergroups were also introduced. Several fundamental properties of soft hypergroups were also presented and studied. To extend our work in this paper, other concepts related to the notion of soft hypergroups such as soft quotient hypergroups and soft cyclic hypergroups can be studied and explored.

ACKNOWLEDGEMENTS The financial support received from Universiti Kebangsaan Malaysia, Faculty of Science and Technology under grant No. UKM–DLP–2011–038 is gratefully acknowledged.

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