RATIO MATHEMATICA 25 (2013), 3–14 ISSN:1592-7415 Class of Semihyperrings from Partitions of a Set A. Asokkumar Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur- 628216, Tamilnadu, India. ashok a58@yahoo.co.in Abstract In this paper we show that a partition {Pα : α ∈ Λ} of a non- empty set S, where Λ is an ordered set with the least element α0 and Pα0 is a singleton set, induces a hyperaddition + such that (S, +) is a commutative hypermonoid. Also by using a collection of subsets of S, induced by the partition of the set S, we define hypermultiplication on S so that (S, +, ·) is a semihyperring. Key words: hypermonoid, semihyperring, ∗-collection. MSC 2010: 20N20. 1 Introduction The theory of hyperstructures has been introduced by the French Math- ematician Marty [11] in 1934 at the age of 23 during the 8thcongress of Scan- dinavian Mathematicians held in Stockholm. Since then many researchers have worked on this new area and developed it. The theory of hyperstructure has been subsequently developed by Corsini [4, 5, 6], Mittas [13], Stratigopoulos [16] and various authors. Basic defini- tions and results about the hyperstructures are found in [5, 6]. Some re- searchers, namely, Davvaz [7], Massouros [12], Vougiouklis [18] and others developed the theory of algebraic hyperstructures. There are different notions of hyperrings (R, +, ·). If the addition + is a hyperoperation and the multiplication · is a binary operation then we say the hyperring is an Krasner (additive) hyperring [10]. Rota [15] introduced 3 A. Asokkumar a multiplicative hyperring, where + is a binary operation and · is a hy- peroperation. De Salvo [8] introduced a hyperring in which addition and multiplication are hyperoperations. These hyperrings are studied by Rahna- mani Barghi [14] and by Asokkumar and Velrajan [1, 2, 17]. Chvalina [3] and Hoskova [3, 9], studied hν-groups, Hν-rings. In this paper, by using different partitions of a set, we construct different semihyperrings (S, +, ·) where both + and · are hyperoperations. 2 Preliminaries This section explains some basic definitions that have been used in the sequel. A hyperoperation ◦ on a non-empty set H is a mapping of H × H into the family of non-empty subsets of H (i.e., x◦y ⊆ H, for every x,y ∈ H). A hypergroupoid is a non-empty set H equipped with a hyperoperation ◦. For any two subsets A, B of a hypergroupoid H, the set A◦B means ⋃ a∈A b∈B (a◦b). A hypergroupoid (H,◦) is called a semihypergroup if x◦(y◦z) = (x◦y)◦z for all x,y,z ∈ H(the associative axiom). A semihypergroup H is said to be regular (in the sense of Von Neumann) if a ∈ a◦H ◦a for every a ∈ H. A hypergroupoid (H,◦) is called a quasihypergroup if x ◦ H = H ◦ x = H for every x ∈ H(the reproductive axiom). A reproductive semihypergroup is called a hypergroup(in the sense of Marty).A comprehensive review of the theory of hypergroups appears in [5]. Definition 2.1. A semihyperring is a non-empty set R with two hyperop- erations + and · satisfying the following axioms: (1) (R, +) is a commutative hypermonoid, that is, (a) (x + y) + z = x + (y + z) for all x,y,z ∈ R, (b) there exists 0 ∈ R, such that x + 0 = 0 + x = {x} for all x ∈ R, (c) x + y = y + x for all x,y ∈ R. (2) (R, ·) is a semihypergroup, that is, x·(y·z) = (x·y)·z for all x,y,z ∈ R. (3) The hyperoperation · is distributive with respect to hyperoperation ’+’, that is, x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z for all x,y,z ∈ R. (4) There exists element 0 ∈ R, such that x · 0 = 0 ·x = 0 for all x ∈ R. Definition 2.2. Let S be a semihyperring, An element a ∈ S is said to be regular if there exists an element y ∈ S such that x ∈ xyx. A semihyperring S is said to be regular if each element of S is regular. 4 Class of Semihyperrings from Partitions of a Set 3 Semihyperring constructed from a ∗-collection. In this section, for a given commutative hypermonoid (S, +), we define hyperoperation · on S suitably so that (S, +, ·) is a regular semihyperring. Definition 3.1. Let S be a commutative hypermonoid. A collection of non- empty subsets {Sa : a ∈ S} of S satisfying the following conditions is called a ∗-collection if (i) Sa = {0} if and only if a = 0, (ii) if a 6= 0 then {0,a}⊆ Sa, (iii) ⋃ x∈Sa Sx = Sa for every a ∈ S, (iv) Sa + Sa = Sa for every a ∈ S and (v) ⋃ x∈a+b Sx = Sa + Sb for every a,b ∈ S. Example 3.2. Consider the set S = {0,a,b}. If we define a hyperoperation + on S as in the following table, then (S,+) is a commutative hypermonoid. + 0 a b 0 0 a b a a {a,b} {a,b} b b {a,b} {a,b} Now it is easy to see that S0 = {0}; Sa = S; Sb = S is a ∗-collection. Example 3.3. Consider the set S={0,a,b}. If we define a hyperoperation + on S as in the following table, then (S,+) is a commutative hypermonoid. + 0 a b 0 0 a b a a {a} {a,b} b b {a,b} {b} Now it is easy to see that S0 = {0}; Sa = S; Sb = S is a ∗-collection. Now we show that S0 = {0}; Sa = {a, 0}; Sb = {b, 0} is another ∗-collection. For each a ∈ S, ⋃ x∈Sa Sx = ⋃ x∈{a,0} Sx = Sa ⋃ S0 = {a, 0} ⋃ {0} = {a, 0} = Sa. Also S0 + S0 = {0} + {0} = {0} = S0; Sa + Sa = {0,a} + {0,a} = {0,a} = Sa; Sb + Sb = {0,b} + {0,b} = {0,b} = Sb. Further, for a,b ∈ S, we get ⋃ x∈a+b Sx = ⋃ x∈{a,b} Sx = Sa ⋃ Sb = {0,a,b} = Sa + Sb. Example 3.4. Consider the set S={0,a,b}. If we define a hyperoperation + on S as in the following table, then (S,+) is a commutative hypermonoid. + 0 a b 0 0 a b a a {0,a} S b b S {0,b} 5 A. Asokkumar It is easy to show that S0 = {0} ; Sa = S for every a 6= 0 ∈ S, is a ∗-collection and S0 = {0} ; Sa = {a, 0} for every a 6= 0 ∈ S is another ∗-collection Example 3.5. Consider the set S={0,a,b,c}. If we define a hyperoperation + on S as in the following table, then (S,+) is a commutative hypermonoid. + 0 a b c 0 {0} {a} {b} {c} a {a} {a} {a, b} {a, c} b {b} {a, b} {b} {b, c} c {c} {a, c} {b, c} {c} In this commutative hypermonoid, each one of the following is a ∗-collection. S0 = {0} ; Sa = {a, 0} for every a 6= 0 ∈ S, S0 = {0} ; Sa = S for every a 6= 0 ∈ S, S0 = {0}; Sa = {0,a}; Sb = {0,b,a}; Sc = {0,c,a}, S0 = {0}; Sa = {0,a,b}; Sb = {0,b}; Sc = {0,c,b}, S0 = {0}; Sa = {0,a,c}; Sb = {0,b,c}; Sc = {0,c}. Theorem 3.6. Let S be a commutative hypermonoid with the additive iden- tity 0 with the condition that x + y = {0} for x,y ∈ S implies either x = 0 or y = 0. Let {Sa : a ∈ S} be a ∗-collection on S. For a,b ∈ S, if we define a hypermultiplication on S as a · b = { Sa if a 6= 0,b 6= 0, 0 otherwise then (S, +, .) is a regular semihyperring. Proof. From the definition of the hypermultiplication, a ·0 = 0 ·a = 0 for all a ∈ S. Let a,b,c ∈ S. If any one of a,b,c is 0, then a ·(b ·c) = {0} = (a ·b) ·c. If a 6= 0,b 6= 0 and c 6= 0, then a·(b·c) = a·Sb = Sa. Also, (a·b)·c = Sa ·c =⋃ x∈Sa(x · c) = ⋃ x∈Sa Sx = Sa. Thus (a · b) · c = a · (b · c). Therefore, (S, ·) is a semihypergroup. Let a,b,c ∈ S. If a = 0 or b = 0 or c = 0, then it is obvious that a · (b + c) = a · b + a · c. Suppose a 6= 0,b 6= 0 and c 6= 0. If 0 ∈ b + c, then a · (b + c) = S0 ∪ Sa = Sa = Sa + Sa = a · b + a · c. If 0 /∈ b + c, then a · (b + c) = Sa = Sa + Sa = a · b + a · c. Thus a · (b + c) = a · b + a · c. Now we prove (a + b) · c = a · c + b · c. For, (a + b) · c = ⋃ x∈a+b x.c =⋃ x∈a+b Sx = Sa + Sb = a · c + b · c. Therefore, (a + b) · c = a · c + b · c. Thus (S, +, ·) is a semihyperring. Let x 6= 0 ∈ S. Now, for any y 6= 0 ∈ S, we have x ∈ Sx = x ·y ⊆ x ·Sy = x · (y ·x). Hence the semihyperring is regular. 6 Class of Semihyperrings from Partitions of a Set Example 3.7. The semihyperring obtained by using the Theorem 3.1 in the Example 3.1 is as follows. + 0 a b 0 0 a b a a {a,b} {a,b} b b {a,b} {a,b} . 0 a b 0 0 0 0 a 0 S S b 0 S S Example 3.8. The semihyperrings obtained by using the Theorem 3.1 in the Example 3.2 are as follows. + 0 a b 0 0 a b a a {a} {a,b} b b {a,b} {b} . 0 a b 0 0 0 0 a 0 S S b 0 S S . 0 a b 0 0 0 0 a 0 {0, a} {0,a} b 0 {0,b} {0,b} Example 3.9. The semihyperrings obtained by using the Theorem 3.1 in the Example 3.3 are as follows. + 0 a b 0 0 a b a a {0,a} S b b S {0,b} . 0 a b 0 0 0 0 a 0 S S b 0 S S . 0 a b 0 0 0 0 a 0 {0, a} {0,a} b 0 {0,b} {0,b} Theorem 3.10. Let S be a commutative hypermonoid with the additive iden- tity 0 with the condition that x + y = 0 for x,y ∈ S implies either x = 0 or y = 0. Let {Sa : a ∈ S} be a ∗-collection on S. For a,b ∈ S, if we define a hypermultiplication on S as a · b = { Sb if a 6= 0,b 6= 0, 0 otherwise then (S, +, .) is a regular semihyperring. Proof. The proof follows by the same lines as in the Theorem 3.1. Let x 6= 0 ∈ S. Now, for any y 6= 0 ∈ S, we have x ∈ Sx = y ·x ⊆ Sy ·x = (x ·y) ·x. Hence the semihyperring is regular. 7 A. Asokkumar Theorem 3.11. Let S be a commutative hypermonoid with the additive iden- tity 0 such that x + y = 0 for x,y ∈ S implies either x = 0 or y = 0. Let {Sa : a ∈ S} be a ∗-collection on S such that Sa ∩ Sb = X for all a 6= 0,b 6= 0 ∈ S where X is a subset of S such that X + X = X. For a,b ∈ S, if we define a hypermultiplication on S as a · b = { Sa ∩Sb = X if a 6= 0,b 6= 0, 0 otherwise then (S, +, .) is a regular semihyperring. Proof. Since 0 ∈ Sa and 0 ∈ Sb, we get 0 ∈ Sa∩Sb. This implies that the set X is non-empty. From the definition of hypermultiplication, a ·0 = 0 ·a = 0 for all a ∈ S. Let a,b,c ∈ S. If any one of a,b,c is 0, then a·(b·c) = {0} = (a·b)·c. If a 6= 0,b 6= 0 and c 6= 0, then a·(b·c) = X = (a·b)·c. Thus (a·b)·c = a·(b·c). Therefore, (S, ·) is a semihypergroup. If a = 0 or b = 0 or c = 0, then it is obvious that a · (b + c) = a ·b + a ·c. Suppose a 6= 0,b 6= 0 and c 6= 0 then, a · (b + c) = X = X + X = a ·b + a ·c. Similarly we have (a+b)·c = X = a·c+b·c. Thus (S, +, ·) is a semihyperring. Let x 6= 0 ∈ S. Since x ∈ Sx, we have x ∈ Sx = x ·x ⊆ x ·Sx = x · (x ·x). Hence the semihyperring is regular. Example 3.12. Using the Theorem 3.3 in the commutative hypermonoid given in the Example 3.4 and by using the following each ∗-collection S0 = {0}; Sa = {0,a}; Sb = {0,b,a}; Sc = {0,c,a} with X = {0,a}, S0 = {0}; Sa = {0,a,b}; Sb = {0,b}; Sc = {0,c,b} with X = {0,b}, S0 = {0}; Sa = {0,a,c}; Sb = {0,b,c}; Sc = {0,c} with X = {0,c}, we get three hypermultiplications so that we get three semihyperrings. 4 Semihyperrings induced by a Partition. In this section we show that a partition of a non-empty set S induces a hyperaddition + such that, (S, +) is a commutative hypermonoid and also the partition induces a ∗-collection. Using this ∗-collection,we define hyper- multiplication · on the set S, so that (S, +, .) a regular semihyperring. Theorem 4.1. Let S be any non-empty set and {Pα}α∈Λ be a partition of S, where Λ is an ordered set with the least element α0 ∈ Λ and Pα0 be a singleton set, say, {0}. Define a hyperaddition ”+” on S as follows: For all a ∈ S, 0 + a = a + 0 = {a}. For a 6= 0,b 6= 0 ∈ S, suppose a ∈ Pα and b ∈ Pβ and γ = max {α,β}, a + b = { Pγ if α 6= β, Pα = Pβ if α = β 8 Class of Semihyperrings from Partitions of a Set Then (i) (S, +) is a commutative monoid and (ii) the partition {Pα}α∈Λ induces a ∗-collection. Proof. It is clear that a + b = b + a for all a,b ∈ S. Let a,b,c ∈ S. Suppose that a ∈ Pα, b ∈ Pβ and c ∈ Pγ, where α,β,γ ∈ Λ. Case 1 : Suppose α < β < γ. Then a + (b + c) = a + Pγ = Pγ. Also, (a + b) + c = Pβ + c = Pγ. Therefore, a + (b + c) = (a + b) + c. Case 2 : Suppose β < α < γ. Then a + (b + c) = a + Pγ = Pγ. Also, (a + b) + c = Pα + c = Pγ. Therefore, a + (b + c) = (a + b) + c. Case 3 : Suppose α < γ < β. Then a + (b + c) = a + Pβ = Pβ. Also, (a + b) + c = Pβ + c = Pβ. Therefore, a + (b + c) = (a + b) + c. Case 4 : Suppose γ < α < β. Then a + (b + c) = a + Pβ = Pβ. Also, (a + b) + c = Pβ + c = Pβ. Therefore, a + (b + c) = (a + b) + c. Case 5 : Suppose γ < β < α. Then a + (b + c) = a + Pβ = Pα. Also, (a + b) + c = Pα + c = Pα. Therefore, a + (b + c) = (a + b) + c. Case 6 : Suppose β < γ < α. Then a + (b + c) = a + Pγ = Pα. Also, (a + b) + c = Pα + c = Pα. Therefore, a + (b + c) = (a + b) + c. Case 7 : Suppose α = β < γ. Then a + (b + c) = a + Pγ = Pγ. Also, (a + b) + c = Pβ + c = Pγ. Therefore, a + (b + c) = (a + b) + c. Case 8 : Suppose γ < α = β. Then a + (b + c) = a + Pα = Pα. Also, (a + b) + c = Pα + c = Pα. Therefore, a + (b + c) = (a + b) + c. Case 9 : Suppose α = γ < β. Then a + (b + c) = a + Pβ = Pβ. Also, (a + b) + c = Pβ + c = Pβ. Therefore, a + (b + c) = (a + b) + c. Case 10 : Suppose β < α = γ. Then a + (b + c) = a + Pα = Pα Also, (a + b) + c = Pα + c = Pα. Therefore, a + (b + c) = (a + b) + c. Case 11 : Suppose β = γ < α. Then a + (b + c) = a + Pγ = Pα. Also, (a + b) + c = Pα + c = Pα. Therefore, a + (b + c) = (a + b) + c. Case 12 : Suppose α < β = γ. Then a + (b + c) = a + Pγ = Pγ. Also, (a + b) + c = Pβ + c = Pγ. Therefore, a + (b + c) = (a + b) + c. 9 A. Asokkumar Case 13 : Suppose α = β = γ. Then a+(b+c) = Pα = Pβ = Pγ = (a+b)+c. Therefore, a+(b+c) = (a+b)+c. Thus the hyperoperation + is associative. So, (S, +) is a commutative hypermonoid. Let S0 = Pα0 = {0}. For a 6= 0 ∈ S, then Sa = ⋃ α0≤t≤α Pt where a ∈ Pα. It is clear that Sa = ⋃ x∈Sa Sx. For a 6= 0 ∈ S, and a ∈ Pα, then Sa + Sa = ⋃ α0≤t≤α Pt + ⋃ α0≤t≤α Pt = ⋃ α0≤t≤α Pt = Sa. Also S0 + S0 = {0} + {0} = {0} = S0. If either a = 0 or b = 0, then ⋃ x∈a+b Sx = Sa + Sb. Let a 6= 0,b 6= 0 ∈ S. Then a ∈ Pα and b ∈ Pβ for some α,β ∈ Λ. Case 1 : Suppose α 6= β, say α < β, then a + b = Pβ. Now x ∈ a + b implies x ∈ Pβ. Therefore, Sx = ⋃ α0≤t≤β Pt. Hence⋃ x∈a+b Sx = ⋃ x∈a+b ( ⋃ α0≤t≤β Pt) = ⋃ α0≤t≤α Pt = ⋃ α0≤t≤α Pt + ⋃ α0≤t≤β Pt = Sa + Sb. Case 2 : Suppose α = β then a + b = Pα. Therefore, ⋃ x∈a+b Sx =⋃ x∈Pα Sx = Sa + Sb. Therefore, ⋃ x∈a+b Sx = Sa + Sb. Thus {Sa : a ∈ S} is a ∗-collection. Remark 4.2. Let S be any non-empty set and x0 ∈ S. Let P0 = {x0} and {P1,P2,P3, · · ·,Pn, · · ·} be a partition of S \ {x0}. Then the partition {P0,P1,P2, ···,Pn, ···} of S induces a hyperoperation + on S so that (S, +) is a commutative hypermonoid and {P0,P1,P2, ···,Pn, ···} induces a ∗-collection. Theorem 4.3. Let S be any non-empty set and {Pα}α∈Λ be a partition of S, where Λ is an ordered set with the least element α0 and Pα0 is a singleton set.Then the partition induces a semihyperring. Proof. By the Theorem 4.1, the partition induces a hyperaddition + such that (S, +) is a commutative hypermonoid and it also induces a ∗-collection. Hence by the Theorem 3.1, we get a regular semihyperring. Example 4.4. We illustrate the construction of semihyperrings from the following examples. Let S = {0,a,b}. Consider a partition P1 = {0},P2 = {a},P3 = {b} of S. Here, the indexing set is Λ = {1, 2, 3} which is an ordered set. The commutative hypermonoid induced by this partition is given by the following Caley table. + 0 a b 0 0 a b a a {a} {b} b b {b} {b} 10 Class of Semihyperrings from Partitions of a Set The ∗-collection induced by this partition is S0 = {0},Sa = {0,a},Sb = {0,a,b} and the hypermultiplication induced by the ∗-collection is given in the Caley table. . 0 a b 0 0 0 0 a 0 {0,a} {0,a} b 0 {0,a,b} {0,a,b} Example 4.5. Let S = {0,a,b}. Consider a partition P1 = {0},P2 = {b},P3 = {a} of S. Here, the indexing set is Λ = {1, 2, 3} which is an ordered set. The commutative hypermonoid induced by this partition is given by the following Caley table. + 0 a b 0 0 a b a a {a} {a} b b {a} {b} The ∗-collection induced by this partition is S0 = {0},Sa = {0,a,b},Sb = {0,b} and the hypermultiplication induced by the ∗-collection is given in the Caley table. . 0 a b 0 0 0 0 a 0 {0,a,b} {0,a,b} b 0 {0,b} {0,b} Example 4.6. Let S = {0,a,b}. Consider a partition P1 = {0},P2 = {a,b} of S. Here, the indexing set is Λ = {1, 2} which is an ordered set. The commutative hypermonoid induced by this partition is given by the following Caley table. + 0 a b 0 0 a b a a {a,b} {a,b} b b {a,b} {a,b} The ∗-collection induced by this partition is S0 = {0},Sa = {0,a,b},Sb = {0,a,b} and the hypermultiplication induced by the ∗-collection is given in the Caley table. . 0 a b 0 0 0 0 a 0 {0,a,b} {0,a,b} b 0 {0,a,b} {0,a,b} Thus we have a regular semihyperring. 11 A. 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