An Introduction to the Theory of Hv-Semilattices

An introduction to the theory of Hv-semilattices

A. Dehghan Nezhad and B. Davvaz Department of Mathematics, Yazd University, Yazd, Iran [email protected] [email protected]

Abstract

In this paper, we introduce the concept of Hv-semilattice and obtain some char- acterizations of it. We give the deﬁnitions of ideal and of hyperorder on an Hv- semilattice. We also study some of their related properties.

Key words: hyperoperation, Hv-semilattice, ideal of Hv-semilattice, hyperorder of Hv- semilattice, fundamental relation. 2000 Mathematics Subject Classiﬁcation: 06A06, 54A10.

1 Introduction

Hyperstructure theory was born in 1934 when Marty [32] deﬁned hypergroups as a generalization of groups. A hypergroupoid is a non-empty set H together with a map ∗ : H × H → P∗(H) which is called hyperoperation, where P∗(H) denotes the set of all non-empty subsets of H. The image of pair (x, y) is denoted by x ∗ y. If x ∈ H and A, B are non-empty subsets of H, then by A ∗ B, A ∗ x and x ∗ B we mean [ A ∗ B = a ∗ b, A ∗ x = A ∗ {x} and x ∗ B = {x} ∗ B, a∈A,b∈B respectively. A hypergroupoid (H, ∗) is called a hypergroup if for all x, y, z ∈ H the following two conditions hold:

(i) x ∗ (y ∗ z) = (x ∗ y) ∗ z,

(ii) x ∗ H = H ∗ x = H.

Seventy years have elapsed since Marty’s pioneer paper. During this period, numerous papers on algebraic hyperstructures have been published, the field has experimented an enormous growth. A recent book [6] contains a wealth of applications. There are applications to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, combinatorics, codes, artificial intelligence, and probabilistic. Hv-structures were for the first time introduced by Vougiouklis in Fourth AHA congress (1990)[49]. The concept of Hv-structures constitute a generalization of the well-known algebraic hyperstructures (hypergroup, hyperring, hypermodule and so on).

1 Actually some axioms concerning the above hyperstructures such as the associative law, the distributive law and so on are replaced by their corresponding weak axioms. Since the quotients of the Hv-structures with respect to the fundamental equivalence relations (β∗, γ∗, ∗, ets.) are always ordinary structures, we can say that they are by virtue structures and this is why they are called Hv-structures. Many authors have published papers relating different “Hv-structures”. In particular a variety of Hv-structures theory have been defined such as: Partial abelian Hv-monoids [12], Hv-semigroups [5,38], Hv-groups [2,5,10,13,20,33,37,47,52], Hv-rings [7,8,16,17,20,21,36,39-44,49], Hv-modules [9,11,15,19,51], Hv-vector spaces [50], Hv-fields [53], and other papers on Hv-structures are [6,14,24,31,45,46,48,54,55]. The reader will find in [43] some basic definitions and theorems about the Hv-structures. In [14], Davvaz surveyed the theory of Hv-structures. Hyperlattices were for the first time introduced by Konstantinidou and Mittas [29]. The concept of hyperlattice is a generalization of the concept of lattice [3]. Other contribu- tor to the development of hyperlattice theory were Konstantinidou [25-30], Ashrafi [1], Rahnamai-Barghi [34,35], Xiao and Zhao [56].

2 About the deﬁnitions of semilattice

A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. In mathematical order theory, a semilattice is a partially ordered set (poset) closed under one of two binary operations, either supremum (join) or infimum (meet). Hence we speak of either a join-semilattice or a meet-semilattice. If an ordered set is both a meet- and join-semilattice, it is also a lattice. Semilattices as posets: Let S be a set partially ordered by the binary relation ≤. (S, ≤) is a meet-semilattice if for all elements x and y of S, the greatest lower bound of the set {x, y} exists. The greatest lower bound of the set {x, y} is called the meet of x and y, denoted x ∧ y. Replacing “greatest lower bound” with “least upper bound” results in the dual concept of a join-semilattice. The least upper bound of {x, y} is called the join of x and y, denoted x ∨ y. Meet and join are binary operations on S. A simple induction argument shows that the existence of all possible pairwise suprema (infima), for each the definition, implies the existence of all non-empty finite suprema (infima). A join-semilattice is bounded if it has a least element, the join of the empty set. Dually, a meet-semilattice is bounded if it has a greatest element, the meet of the empty set. Other properties may be assumed; see the article on completeness in order theory for more discussion on this subject. That article also discusses how we may rephrase the above definition in terms of the existence of suitable Galois connections between related posets, an approach of special interest for category theoretic investigations of the concept. Semilattices as algebraic structures: A “meet-semilattice” is an algebraic structure consisting of a set S with the binary operation ∧, called meet, such that for all members x, y, and z of S, the following identities hold: (i) x ∧ x = x (Idempotency),

(ii) x ∧ y = y ∧ x (Commutativity),

(iii) x ∧ (y ∧ z) = (x ∧ y) ∧ z (Associativity).

2 If ∨, denoting join, replaces ∧ in the definition just given, a join-semilattice results. Meet and join form a dual pair of binary operations, and meet-semilattice and join-semilattice are dual algebraic structures. A meet-semilattice is bounded if (S, ∧) includes the distin- guished element 1 such that for all x in S, x∧1 = x. 1 is the greatest element of S. Dually, (S, ∨, 0) is a join-semilattice with least element 0 if ∨ and 0 replace ∧ and 1, respectively, in the definition just given. A semilattice is an idempotent, commutative semigroup, and a bounded semilattice is an idempotent commutative monoid. Alternatively, a semilattice is a commutative band. Hence semilattices are magmas. Connection between both definitions: An order theoretic meet-semilattice (S, ≤) gives rise to a binary operation ∧ such that (S, ∧) is an algebraic meet-semilattice. Con- versely, the meet-semilattice (S, ∧) gives rise to a binary relation ≤ that partially orders S in the following way. For all elements x and y in S,

x ≤ y ⇐⇒ x = x ∧ y.

The relation ≤ introduced in this way defines a partial ordering from which the binary operation ∧ may be recovered. Conversely, the order induced by the algebraically defined semilattice (S, ∧) coincides with that induced by ≤. Hence both definitions may be used interchangeably, depending on which one is more convenient for a particular purpose. A similar conclusion holds for join-semilattices and the dual ordering ≥.

3 Hv-semilattices

In this section, we introduce the notion of Hv-semilattice. The notion of Hv-semilattice is a generalization of the semilattice notion in classical theory as well as hypersemilattice.

Deﬁnition 3.1. Let L be a nonempty set with a binary hyperoperation ∗ on L such that, for all a, b, c ∈ L, the following conditions hold:

(i) a ∈ a ∗ a (idempotent)

(ii) a ∗ b = b ∗ a ( commutative )

(iii) (a ∗ b) ∗ c ∩ a ∗ (b ∗ c) 6= φ (weak associative).

Then (L, ∗) is called an Hv-semilattice. When in the condition (iii) we have equality, then (L, ∗) is called a hypersemilattice [56] Now, we present some examples of Hv-semilattices.

Example 3.2.

(i) Consider L = {a, b, c} and deﬁne hyperoperation ∗ on L by the following table:

∗ a b c a {a}{c, a}{b, a} b {a, c}{b, a}{a} c {b, a}{a}{c, a}

3 Then (L, ∗) is an Hv-semilattice which is not a hypersemilattice. Indeed, we have b ∗ (c ∗ a) = {a, b, c} and (b ∗ c) ∗ a = {a}. Therefore ∗ is not associative, but ∗ is weak associative for all a, b, c ∈ L.

∞ (ii) We consider the classical differential ring of real functions f ∈ C (J),J = (a, b) ⊆ R ∞ ( not excluding the case J = R ) with the usual differentiation. For any f, g ∈ C (J) we define a hyperoperation ∗ on the ring C∞(J) by, for all x ∈ J, (f ∗ g)(x) = {f(x), g(x), f 0 (x), g0 (x)}. For any f, g, h ∈ C∞(J), we have

0 0 0 f ∈ f ∗ f = {f, f } and f ∗ g = {f, g, f , g } = g ∗ f.

Also (f ∗ g) ∗ h = {f, g, f 0 , g0 } ∗ h = {f, g, hf 0 , g0 , h0 , f 00 , g00 } 6= f ∗ (g ∗ h) = f ∗ {g, h, g0 , h0 } = {f, g, h, f 0 , g0 , h0 , g00 , h00 }. But f ∗ (g ∗ h) ∩ (f ∗ g) ∗ h = {f, g, h, f 0 , g0 , h0 , g00 } 6= φ. Therefore (C∞(J), ∗) is an Hv-semilattice.

∗ (iii) Let f be a function from L into P (R). We deﬁne the hyperoperation ∗f as follows: a ∗f b = {x ∈ L | f(x) ⊆ f(a) ∪ f(b)} for a, b ∈ L. It is clear that (L, ∗f ) is an Hv-semilattice.

All properties of Hv-semilattices are also true for subsets. So we have:

Proposition 3.3. Let L be a nonempty set and let ∗ be a binary hyperoperation on ∗ L. Then (L, ∗) is an Hv-semilattice if and only if for all A, B, C ∈ P (L) the following conditions hold:

(i) A ⊆ A ∗ A,

(ii) A ∗ B = B ∗ A,

(iii) (A ∗ B) ∗ C ∩ A ∗ (B ∗ C) 6= φ.

Proof. The proof is straightforward.

To each binary relation R on a set L, a partial hyperoperation LR = (L, ) is associated, as follows: for all x, y ∈ L, x x = {y ∈ L | (x, y) ∈ R}, x z = x x ∪ z z.

Proposition 3.4. If R is a reﬂexive relation on L, then LR is an Hv-semilattice.

Proof. For arbitrary x, y, z ∈ L, we have,

(i) (x, x) ∈ R, so x ∈ {y ∈ L|(x, x) ∈ R}, i.e, x ∈ x ∗ x,

4 (ii) x z = (x x) ∪ (z z) = (z z) ∪ (x x) = z x, and we suppose that Q = (x y) z, Q0 = x (y z). By the deﬁnition of above hyperoperation , we obtain

Q = [(x x) (x x)] ∪ (z z) ∪ [(y y) (y y)], and Q0 = (x x) ∪ [y y) (y y)] ∪ [(z z) (z z)]. 0 By (i), it is easy to see that x, y, z ∈ Q ∩ Q . Therefore, this completes the proof.

Proposition 3.5. Let (Z, +, 6) be the additive group of all integers with a usual or- ∗ dering “ ≤ ”. Then, the hyperstructure (N, ∗), where ∗ : N × N −→ P (N), k ∗ l = {u ∈ N | k + l ≤ 2u} is an Hv-semilattice.

Proof. Here we only prove the week associativity and the other conditions (idempotency, commutativity) are trivial. For any k, l, s ∈ N, we have 2k+l+s+i 2k+l+s+i k ∗ (l ∗ s) = [ 4 , +∞) ∩ N, where i = 0, 1, 2, 3 is chosen that 4 ∈ N, and

k+l+2s+i k+l+2s+i (k ∗ l) ∗ s = [ 4 , +∞) ∩ N, where i = 0, 1, 2, 3 is chosen that 4 ∈ N. It is easy to see that (k ∗ l) ∗ s ∩ k ∗ (l ∗ s) 6= ∅. But, it is not associative. Because, for example, we have (1 ∗ 2) ∗ 5 = [4, +∞) ∩ N 6= [3, +∞) ∩ N = 1 ∗ (2 ∗ 5).

Deﬁnition 3.6. Let (•) and (∗) be two hyperoperations on H. We call (∗) the dual of (•) if and only if for all x, y ∈ H, x • y = y ∗ x.

Proposition 3.7. Let (∗) be the dual of (•). Therefore (H, •) is an Hv-semilattice if and only if (H, ∗) is an Hv-semilattice.

Proof. The proof is straightforward.

Let (L, ⊕), (S, ⊗) be two Hv-semilattices. A map f : L −→ S is called a weak homomorphism if f(x ⊕ y) ∩ (f(x) ⊗ f(y)) 6= φ for all x, y ∈ L. f is called an inclusion homomorphism if f(x⊕y) ⊆ (f(x)⊗f(y)) for all x, y ∈ L. Finally, f is called a strong homomorphism ( preserving binary hyperoperation) if f(x⊕y) = f(x)⊗f(y) for all x, y ∈ L. Let α be a strong homomorphism from an Hv-semilattice L onto Hv-semilattice S. The relation α−1oα is an equivalence ρ on L(aρb if and only if α(a) = α(b) ) known as the kernel of α. The natural mapping associated with ρ is φ : L −→ L/kerα where φ(a) = ρ(a). The mapping ψ : L/ρ −→ S, where ψ(ρ(a)) = α(a) is then the unique bijection that makes the following diagram commute: ψ : L/ρ −→ S, φ : L −→ L/kerα , α : L −→ S. If f is an onto, one to one and strong homomorphism, then it is called isomorphism, if moreover f defined on the same Hv-semilattice then it is called automorphism, if moreover f defined on the same Hv-semilattice then it is called automorphism. It is easy verification that the

5 set of all automorphism in L, written AutL, is a group. If f is injective as a map of sets, then f is said to be a monomorphism. If f is surjective, then f is called an epimorphism. If f : L −→ S and g : S −→ T are homomorphisms of Hv-semilattices, it is easy to see that gof : L −→ T , is also a homomorphism. Like wise the composition of monomor- phisms is also a monomorphism, similarly to epimorphisms and isomorphisms.

Proposition 3.8. Let (L, ⊕) be an Hv-semilattice and S be a nonempty set with a binary hyperoperation ⊗. If a function f : L −→ S is surjective and preserving binary weak hperoperation, then (S, ⊗) is an Hv-semilattice.

Proof. For all a1, b1, c1 ∈ S we have

(i) a1 = f(a) ∈ f(a ⊕ a) = f(a) ⊗ f(a) = a1 ⊗ a1, i.e. a1 ∈ a1 ⊗ a1

(ii) a1 ⊗ b1 = f(a) ⊗ f(b) = f(a ⊕ b) = f(b ⊕ a) = f(b) ⊗ f(a) = b1 ⊗ a1.

(iii) [(a1 ⊗ b1) ⊗ c1] ∩ [a1 ⊗ (b1 ⊗ c1)] = [(f(a) ⊗ f(b)) ⊗ f(c)] ∩ [f(a) ⊗ (f(b) ⊗ f(c))] = [f((a ⊕ b) ⊕ c)] ∩ [f(a ⊕ (b ⊕ c))] 6= φ, since we have (a ∗ b) ∗ c ∩ a ∗ (b ∗ c) 6= φ for all a, b, c, ∈ L.

Therefore (S, ⊗) is an Hv-semilattice.

On a set L several Hv-semilattices can be deﬁned. A partial order on those Hv- semilattices is introduced as follows.

Deﬁnition 3.9. Let (L, ⊕), (L, ⊗) be two Hv-semilattices deﬁned on the same set L. We call ⊕ less than or equal to ⊗, and write ⊕ ≤ ⊗, if there is f ∈ Aut(L, ⊗) such that x ⊕ y ⊆ f(x ⊗ y) for all x, y ∈ L.

Definition 3.10. Let (L, ∗) be an Hv-semilattice. An element a ∈ L is called an absorbent element of L if it satisfies c ∈ a ∗ c for all c ∈ L. An element b ∈ L is called a fixed element of L if it satisfies b ∗ c = {b} for all c ∈ L.

Example 3.11. Let L be a non-empty set and deﬁne a binary hyperoperation on L by a ∗ b = L for all a, b ∈ L. It is easy to verify that (L, ∗) is an Hv-semilattice, and we call it trivial Hv-semilattice.

Let (L, +) be an Hv-semilattice, θ an equivalence relation on L and θ(x) the θ - equivalence class of the element x ∈ L. In L/θ consider the hyperoperation ⊕ deﬁned on the usual manner: θ(x) ⊕ θ(y) = {θ(z) | z ∈ θ(x) + θ(y)} for all x, y ∈ L.

Proposition 3.12. (L/θ, ⊕) is an Hv-semilattice.

Proof. For all x ∈ L we have θ(x) = {y ∈ L | xθy}. It is easy to verify that the hyperoperation ⊕ is idempotent, commutative and weak associative. For example, we show that this is weak associative. For all x, y, z ∈ L, we have (x + y) + z ⊆ (θ(x) ⊕

6 θ(y)) ⊕ θ(z), x + (y + z) ⊆ θ(x) ⊕ (θ(y) ⊕ θ(z)). Since x + (y + z) ∩ (x + y) + z 6= φ, so (θ(x) ⊕ θ(y)) ⊕ θ(z) ∩ θ(x) ⊕ (θ(y) ⊕ θ(z)) 6= φ. Thus ⊕ is weak associative.

Proposition 3.13. Let φ1 and φ2 be two strong homomorphisms of Hv-semilattice L −1 −1 upon Hv-semilattices L1 and L2 respectively, such that φ1 ◦ φ1 ⊆ φ2 ◦ φ2. Then, a unique strong homomorphism θ of L1 upon L2 such that θ ◦ φ1 = φ2, exists.

Proof. We show that θ is a strong homomorphism of L1 upon L2. For all a1, b1 ∈ L1 we have θ(a1 ⊕ b1) = θ(φ1(a) ⊕ φ1(b)) = θ(φ1(a + b)) = φ2(a + b) = φ2(a) ⊗ φ2(b) = θ(φ1(a)) ⊗ θ(φ1(b)) = θ(a1) ⊗ θ(b1).

Let (L, ⊕) be an Hv-semilattice. By a congruence on L we mean an equivalence relation ρ such that xρy if and only if for every a ∈ L and for every u ∈ x ⊕ a, there exists v ∈ y ⊕ a such that uρv.

Proposition 3.14. Let α : L −→ S be a strong homomorphism of Hv-semilattices. Then ρ = kerα is a congruence and a strong homomorphism f : L/ρ −→ S exists such that f ◦ φ = α ( note that L/ρ is an Hv-semilattice) .

Proof. The proof is straightforward.

Proposition 3.15. If ρ1 and ρ2 are congruences on an Hv-semilattice L such that ρ1 ⊆ ρ2, then a strong homomorphism from L/ρ1 upon L/ρ2, exists .

Proof. The proof is straightforward.

Proposition 3.16. Let f : L −→ S be an epimorphism of Hv-semilattices. If a is an absorbent element of L, then f(a) is also an absorbent element of S. An analogous result holds for a ﬁxed element.

Proof. There exists c ∈ L such that f(c) = s for any s ∈ S because f is surjective, a is an absorbent element provided c ∈ a ⊕ c for any c ∈ L. Then we have since s = f(c) ∈ f(a ⊕ c) = f(a) ⊗ f(c) = f(a) ⊕ s for all s ∈ S. Therefore f(a) is an absorbent element of S. The proof for a ﬁxed element is analogous.

The product operation plays an important role as a means of constructing new construction from the old. Now, we consider the product of Hv-semilattices (L, ⊕) and (S, ⊗). Let (L, ⊕), and (S, ⊗) be two Hv-semilattices. Deﬁne a binary hyperoperation on the carte- sian product L × S as follows: (a1, b1) × (a2, b2) = {(c, d)|c ∈ a1 ⊕ a2, d ∈ b1 ⊗ b2} for all (a1, b1), (a2, b2) ∈ L × S, then (L × S, ×) is called the direct product of Hv-semilattices (L, ⊕), and (S, ⊗).

Proposition 3.17. The direct product of two Hv-semilattices is also an Hv-semilattice.

Proof. Let (L, ⊕), and (S, ⊗) be two Hv-semilattices. Then

7 (i) For each (a, b) ∈ L × S, we have that a ∈ a ⊕ a and b ∈ b ⊕ b, then (a, b) ∈ {(c, d)|c ∈ a ⊕ a, d ∈ b ⊗ b} = (a, b) × (a, b)

(ii) For all (a1, b1), (a2, b2) ∈ L × S, we have a1 ⊕ a2 = a2 ⊕ a1 and b1 ⊕ b2 = b2 ⊕ b1 by the deﬁnition of Hv-semilattice. Then it is clear that (a1, b1) × (a2, b2) = {(c, d)|c ∈ a1 ⊕ a2, d ∈ b1 ⊗ b2} = {(c, d)|c ∈ a2 ⊕ a1, d ∈ b2 ⊗ b1} = (a2, b2) × (a1, b1).

(iii) For all (a1, b1), (a2, b2), (a3, b3) ∈ L × S (a1 ⊕ a2) ⊕ a3 ∩ a1 ⊕ (a2 ⊕ a3) 6= φ and (b1 ⊕ b2) ⊕ b3 ∩ b1 ⊕ (b2 ⊕ b3) 6= φ hold. Then, we may obtain that M = ((a1, b1) × (a2, b2)) × (a3, b3) = {(c, d)|c ∈ a1 ⊕ a2, d ∈ b1 ⊗ b2} × (a3, b3) = {(e, f)|e ∈ c ⊕ a3, f ∈ d ⊗ b3, c ∈ a1 ⊕ a2, d ∈ b1 ⊗ b2} = {(e, f)|e ∈ (a1 ⊕ a2) ⊕ a3, f ∈ (b1 ⊗ b2) ⊗ b3} and N = (a1, b1) × ((a2, b2) × (a3, b3)) = (a1, b1) × {(c, d)|c ∈ a2 ⊕ a3, d ∈ b2 ⊗ b3} = {(e, f)|e ∈ a1 ⊕ (a2, a3), f ∈ b1 ⊗ (b2, b3)}. Since L, S are Hv-semilattices, so we have (a1 ⊕ a2) ⊕ a3 ∩ a1 ⊕ (a2 ⊕ a3) 6= φ and (b1 ⊕ b2) ⊕ b3 ∩ b1 ⊕ (b2 ⊕ b3) 6= φ.

Therefore (L × S, ×) is an Hv-semilattice.

Proposition 3.18. Let (L, ⊕) and (S, ⊗) be two Hv-semilattices. If a and b are absorbent elements of L and S respectively, then (a, b) is an absorbent element of (L×S, ×).

Proof. Since a and b are absorbent elements, we have that c ∈ a ⊕ c, for all c ∈ L and d ∈ b ⊗ d for all d ∈ S. Then for all (c, d) ∈ L × S, (c, d) ∈ {(e, f) | e ∈ a ⊕ c, f ∈ b ⊗ d} = (a, b) × (c, d). So (a, b) is an absorbent element of (L × S, ×).

Proposition 3.19. Let (L, ⊕) be an Hv-semilattice, and N be a non-empty subset of L. Then there exists a well-deﬁned binary operation on L/N = {N ⊕ a | a ∈ L} given by (N ⊕ a) (N ⊕ b) = {N ⊕ n | n ∈ a ⊕ b} for all a, b ∈ L, such that (L/N, ) is an Hv-semilattice.

Proof. To begin, let us show is a binary hyperoperation on L/N. Obviously, is a function L/N × L/N −→ P∗(L/N). Then, for all N ⊕ a, N ⊕ b, N ⊕ c ∈ L/N and all A, B ∈ P ∗(L/N), we have

∗ (N ⊕ a) (N ⊕ a) = {(N ⊕ n) | n ∈ (a ⊕ a)} ∈ P (L/N), ∗ (N ⊕ c) A = ∪N⊕a∈A((N ⊕ c) (N ⊕ a)) ∈ P (L/N), ∗ A (N ⊕ c) = ∪N⊕a∈A((N ⊕ a) (N ⊕ c)) ∈ P (L/N), ∗ A B = ∪N⊕a∈A,N⊕b∈B((N ⊕ a) (N ⊕ b)) ∈ P (L/N).

That is, can be seen as a binary hyperoperation on L/N. Now, we prove that (L/N, ) is an Hv-semilattice. We have

(i) Since a ∈ a ⊕ a, then N ⊕ a ∈ {N ⊕ n | n ∈ a ⊕ a} = (N ⊕ a) (N ⊕ a).

(ii) Since a ⊕ b = b ⊕ a, then (N ⊕ a) (N ⊕ b) = {N ⊕ n | n ∈ a ⊕ b} = {N ⊕ n | n ∈ b ⊕ a} = (N ⊕ b) (N ⊕ a).

8 (iii) Since (a ⊕ b) ⊕ c ∩ a ⊕ (b ⊕ c) 6= φ, then

Q = ((N ⊕ a) (N ⊕ b)) (N ⊕ c) = {N ⊕ n | n ∈ a ⊕ b} (N ⊕ c) = ∪n∈a⊕b((N ⊕ n) (N ⊕ c)) = ∪n∈(a⊕b){N ⊕ m | m ∈ n ⊕ c} = {N ⊕ m | m ∈ (a ⊕ b) ⊕ c},

and 0 Q = (N ⊕ a) ((N ⊕ b) (N ⊕ c)) = (N ⊕ c) {N ⊕ l | l ∈ b ⊕ c} = ∪l∈(b⊕c)((N ⊕ a) (N ⊕ l)) = ∪n∈(b⊕c{N ⊕ s | s ∈ a ⊕ l} = {N ⊕ s | s ∈ a ⊕ (b ⊕ c)}. Since for all a, b, c ∈ L we have (a ⊕ b) ⊕ c ∩ a ⊕ (b ⊕ c) 6= φ, then Q ∩ Q0 6= φ.

Therefore hyperoperation ⊕ is weak associative.

4 Generalized action and Hv-semilattices

In this section, we generalize in a certain sense the classical concept of action of a hypergroup on a given phase space. This subject plays a very important role in the current progress of concrete mathematics (especially geometry). Now, let us recall the deﬁnition of a generalized action of G (as a hypergroup ) on set X.

Deﬁnition 4.1.[23]. Let (G, ) be a hypergroup and X be a set. The map φ : G × X −→ X is called a generalized action of G on X, if the following axiom hold: for all g, h ∈ G and x ∈ X, φ(g, φ(h, x)) ∈ φ(g h, x), where φ(g h, x) = {φ(r, x) | r ∈ g h} for all x ∈ X, g, h ∈ G. Then the triple τ = (X, G, φ) is called an action of the hypergroup G on the phase set X. Let θ be a tolerance relation(i.e., reﬂexive and symmetric binary relation). Then the pair (X, θ) is a tolerance space.

Deﬁnition 4.2.[23]. Let (X, θ) be a tolerance space (so called phase tolerance space), (G, ) be a semihypergroup and φ : G × X −→ X a mapping such that

(i) φ(g, φ(h, x)) ∈ φ(g h, x), where φ(g h, x) = {φ(r, x); r ∈ g h} for each x ∈ X, g, h ∈ G;

(ii) if x, y ∈ X are such that xθy, then φ(g, x) θ φ(g, y) holds for any g ∈ G.

Then the triple τ = (X, G, φ) is called an action of the semihypergroup G with phase tolerance space. Moreover if, the triple τ = (X, G, φ) is an action of the hypergroup with phase tolerance space, in case the tolerance θ is trivial, in fact the preceding deﬁnition coincides with the above deﬁnition.

Let us deﬁne for arbitrary pair of elements x, y ∈ X a binary hyperoperation ⊗ :

9 X × X −→ P ∗(X), in this way: x ⊗ y = φ(G, x) ∪ φ(G, y) ∪ {x, y}, where φ(G, x) = {φ(g, x) | g ∈ G} and similarly for φ(G, y). For all A ⊆ X, φ(g, A) = {φ(g, a) | a ∈ A}.

Proposition 4.3. The pair (X, ⊗) is an Hv-subsemilattice.

Proof. For arbitrary x, y, z ∈ X, we have

(i) x ∈ x ⊗ x = φ(G, x) ∪ {x},

(ii) x ⊗ y = φ(G, x) ∪ φ(G, y) ∪ {x, y} = φ(G, y) ∪ φ(G, x) ∪ {y, x} = y ⊗ x,

(iii) For the weak associativity axiom, we have

(x ⊗ y) ⊗ z = φ(G, x ⊗ y) ∪ φ(G, z) ∪ {x ⊗ y, z} = φ(G, φ(G, x)) ∪ φ(G, φ(G, y)) ∪ φ(G, {x, y})φ(G, z) ∪ {φ(G, x) ∪ φ(G, y) ∪ {x, y}, z} = φ(G, φ(G, x)) ∪ φ(G, φ(G, y)) ∪ φ(G, x) ∪ φ(G, y) ∪ φ(G, z) ∪ {φ(G, x) ∪ φ(G, y) ∪ {x, y}, z}

and x ⊗ (y ⊗ z) = φ(G, x) ∪ φ(G, y ⊗ z) ∪ {x, y ⊗ z} = φ(G, x) ∪ φ(G, φ(G, y)) ∪ φ(G, φ(G, z)) ∪ φ(G, {y, z}) ∪ {x, φ(G, y) ∪ φ(G, z) ∪ {x, y}} = φ(G, x) ∪ φ(G, φ(G, y)) ∪ φ(G, φ(G, z)) ∪ φ(G, y) ∪ φ(G, z) ∪ {x, φ(G, y) ∪ φ(G, z) ∪ {x, y}}.

Thus the weak associative condition is satisﬁed, since ∅ 6= φ(G, x) ∪ φ(G, y) ∪ φ(G, z) ∈ (x ⊗ y) ⊗ z ∩ x ⊗ (y ⊗ z).

5 Hv-subsemilattices

Let (L, ⊕), be a Hv-semilattice, and M be a non-empty subset of L. Then M is called ∗ Hv-subsemilattice of (L, ⊕) if a ⊕ b ∈ P (M) for all a, b ∈ M. That is to say, M is an Hv-subsemilattice of (L, ⊕) if and only if M is closed under the binary hyperoperation on L. An Hv-subsemilattice M is a single point Hv-subsemilattice if |M| = 1, and Hv- subsemilattice M such that M 6= L is called proper Hv-subsemilattice. We may easily get the conclusion as follows: M is an Hv-subsemilattice of (L, ⊕) if and only if M ⊕ M = M.

Example 5.1. Let L be a non-empty set, and deﬁne a binary hyperoperation on L as follows: a ⊕ b = {a, b} for all a, b ∈ L. Then (L, ⊕) is an Hv-semilattice. Each nonempty subset of L is an Hv-subsemilattice of (L, ⊕).

Proposition 5.2. Let L −→ S be a homomorphism of Hv-semilattices. Then the following conditions hold:

(i) If M is an Hv-subsemilattice of (L, ⊕), then f(M) is an Hv-subsemilattice of (S, ⊗).

−1 (ii) If f is surjective and N is an Hv-subsemilattice of (S, ⊗), then f (N), which is −1 deﬁned by f (N) = {a ∈ L | f(a) ∈ N}, is also an Hv-subsemilattice of (L, ⊕).

10 Proof. (i) Since f is a homomorphism of Hv-semilattice, there exist a, b ∈ M such that f(a) = a1, f(b) = b1 for all a1, b1 ∈ f(M). By the deﬁnition of Hv-subsemilattice a⊕b ⊆ M holds. So we have a1 ⊗ b1 = f(a) ⊗ f(b) = f(a ⊕ b) ⊆ f(M). Then f(M) is an Hv- subsemilattice of (S, ⊗). (ii) Since f is a surjective function, f −1(N) always exists. For all a, b ∈ f −1(N), f(a) ∈ −1 N, f(b) ∈ N, we have f(a ⊕ b) = f(a) ⊕ f(b) ⊆ N. So f (N) is an Hv-semilattice of (L, ⊕).

Proposition 5.3. Let (L, ⊕) be an Hv-semilattice and let M and N be Hv-subsemilattices of (L, ⊕). Then M ∩ N is also an Hv-subsemilattice of (L, ⊕) if M ∩ N is non-empty.

Proof. The proof is straightforward.

Note that, we can easily verify that the union of an Hv-subsemilattice of Hv-semilattice (L, ⊕), may not be the Hv-subsemilattice of (L, ⊕), because it is not closed under the binary hyperoperation on L.

Example 5.4. Let L = {a, b, c, d} and deﬁne a binary hyperoperation ♦ on L with help of the following table:

♦ a b c d a {a}{a}{a}{a} b {a}{a, b}{a}{a} c {a}{a}{a, c}{a, b, c} d {a}{a}{a, b, c}{a, d}

It is easy to prove that (L, ♦) is an Hv-semilattice. If M1 = {a, c},M2 = {a, d}, then they are Hv-subsemilattices of (L, ♦). But we can easily verify that M1 ∪ M2 = {a, c, d} is not an Hv-subsemilattice of L because it isn’t closed under the binary hyperoperation on L.

By the deﬁnitions of an Hv-semilattice and the product of Hv-semilattices, we have the following. Let (L, ⊕) and (S, ⊗) be Hv-semilattices and let M and N be Hv-subsemilattices of (L, ⊕) and (S, ⊗), respectively. Then M × N is also an Hv-subsemilattice of (L × S, ×).

6 Deﬁnition of an ideal of an Hv-semilattice

Ideal play an important role in the study of algebraic structures. In this section, we introduce the deﬁnition of an ideal of an Hv-semilattice and discuss some basic properties of it.

Deﬁnition 6.1. Let (L, ⊕) be an Hv-semilattice, and N be a nonempty subset of L. We say N is an ideal of (L, ⊕) if a ⊕ N ⊆ N for all a ∈ L. If N 6= L, then N is called a proper ideal of (L, ⊕).

11 Obviously, any Hv-semilattice is an Hv-subsemilattice and ideal of itself. If N is an ideal of (L, ⊕), then N is an Hv-subsemilattice of (L, ⊕). But the converse is not true. We would like to give an example to illustrate it.

Example 6.2. Let us recall from Example 4.2, the set N1 = {a, b} is an Hv-subsemilattice, but not an ideal of (L, ⊕). And N2 = {a, d, c} is an ideal of (L, ⊕), but N2 is not an Hv- subsemilattice of (L, ⊕).

Next, we want to give some equivalent statements about ideal of Hv-semilattice and introduce some special ideals.

Proposition 6.3. Let I be an ideal and M an Hv-subsemilattice of an Hv-semilattice L. Then I ∩ M is an ideal of M,I ∪ M is an Hv-subsemilattice of L, and there is an inclusion M I∪M homomorphism from I∩M upon I .

Proof. The proof is straightforward.

Proposition 6.4. Let I be an ideal of an Hv-semilattice L. We consider the Rees relation on L as follows: xρy ⇐⇒ x = y or (x ∈ I and y ∈ I). Then ρ is congruence on L.

Proof. The proof is straightforward.

Proposition 6.5. Let (L, ⊕), be an Hv-semilattice and let N be a non-empty subset of L. Then the following conditions are equivalent: (i) N is an ideal of (L, ⊕).

(ii) a ⊕ n ∈ P ∗(N) for all a ∈ L and all n ∈ N.

(iii) L ⊕ N ⊆ N.

Proof. The proof is straightforward.

Proposition 6.6. Let N be an ideal of an Hv-semilattice (L, ⊕). If a is an absorbent element of L, then the following condition hold: (i) N = L if and only if a ∈ N.

(ii) N is a proper ideal of (L, ⊕) if and only if a isn’t belong to N.

Proof. The proof is straightforward.

Proposition 6.7. Let a be an element of an Hv-semilattice (L, ⊕). Then {a} is an ideal of (L, ⊕).

Proof. The proof is straightforward.

12 Proposition 6.8. Let M and N be ideals of an Hv-semilattice (L, ⊕). Then, we have the following conclusions:

(i) M ∩ N is an ideal of (L, ⊕) and M ∩ N = M ⊕ N.

(ii) M ∪ N is also an ideal of (L, ⊕).

Proof. (i) Let us prove that M ∩ N 6= φ. Suppose that m ∈ M, n ∈ N. Then m ⊕ n ⊆ M, m ⊕ n ⊆ N by Proposition 6.5 (ii), that is m ⊕ n ⊆ M ∩ N. So, we have M ∩ N 6= φ. For all n ∈ M ∩ N, i.e., n ∈ M and n ∈ N, and for all a ∈ L, we have a ⊕ n ⊆ M and a ⊕ n ⊆ N, i.e. a ⊕ n ∈ p∗(M ∪ N). Therefore M ∩ N is an ideal of (L, ⊕). By Proposition 6.5 (iii), we can easily get that M ⊕ N ⊆ M ∩ N. For all a ∈ M ∩ N, a ∈ a ⊕ a ⊆ M ∩ N, i.e., M ∩ N ⊆ M ⊕ N. So M ∩ N = M ⊕ N. (ii) For all n ∈ M ∩ N and for all a ∈ L, we have that a ⊕ n ⊆ M or a ⊕ n ⊆ N, then ∗ a ⊕ n ⊆ M ∪ N, i.e., a ⊕ n ∈ P (M ∪ N). So M ∪ N is an ideal of (L, ⊕).

Proposition 6.9. Let M be an Hv-subsemilattice of an Hv-semilattice (L, ⊕) and let I be an ideal of (L, ⊕). If M ∩ I is non-empty, then M ∩ I is an ideal of (N, ⊕).

Proof. The proof is straightforward.

Proposition 6.10. Let I and J be ideals of an Hv-semilattices (L, ⊕) and (S, ⊗), respectively. Then I × J is also an ideal of (L × S, ∗).

Proof. The proof is straightforward.

Proposition 6.11. Let L −→ S be a homomorphism of Hv-semilattices. If a is a ﬁxed element of S, then f −1(a) = {n ∈ L | f(n) = a} is an ideal of (L, ⊕).

Proof. For all m ∈ L and all n ∈ f −1(a), f(m ⊕ n) = f(m) ⊗ f(n) = f(m) ⊗ a = {a}, i.e., −1 −1 m ⊕ n ⊆ f (a). Therefore f (a) is an ideal of (L, ⊗).

Proposition 6.12. Let L −→ S be an epimorphism of Hv-semilattices. Then, we can get the following results:

(i) If I is an ideal of (L, ⊕), then f(I) is also an ideal of (S, ⊗).

(ii) If J is an ideal of (S, ⊗), then f −1(J), which is denoted by f −1(J) = {a ∈ L | f(a) ∈ J} is also an ideal of (L, ⊕).

Proof. The proof is straightforward.

7 Hyperorder on an Hv-semilattice

An ordered semilattice is a triple (S, ·, ≤), where (S, ·) is a semilattice and “ ≤ ” is an ordering on the set S with subsituation property on (S, ·) (i.e., for an arbitrary quadruple of elements a, b, c, d ∈ S for which a ≤ b, c ≤ d the relation a · c ≤ b · d holds). It should be

13 noticed that the subsituation property is equivalent to a simpler condition for an arbitrary triple of elements a, b, c ∈ S such that a ≤ b the relations , a · c ≤ b · c and c · a ≤ c · b hold. Below we will need the following result in which we denote for t from an ordered set H; [t)≤ = {x ∈ H | t ≤ x} (principal upper and determind by t).

Proposition 7.1. Let (H, ·, ≤) be an ordered commutative semilattice and deﬁne a binary hyperoperation ∗ on H in this way; a ∗ b = [a · b)≤ for any a, b ∈ H. Then (H, ∗) is an Hv-semilattice.

Proof. See [22].

Now, we deﬁne a hyperorder ≤L on an Hv-semilattice (L, ⊕) as follows :

Deﬁnition 7.2. Let (L, ⊕), be an Hv-semilattice and a, b ∈ L. We say that a ≤L b if a ⊕ c ⊆ b ⊕ c for all c ∈ L, and ≤L is called the hyperorder on Hv-semilattice L.

Proposition 7.3. Let (L, ⊕) be an Hv-semilattice and let I be an ideal of (L, ⊕). If a ∈ I and b ≤L a, then b ∈ I.

Proof. If b ≤L a, we have b ⊕ c ∩ a ⊕ c 6= φ for all c ∈ L. Let c = b. Then b ∈ b ⊕ b ⊆ I, i.e., b ∈ I.

Proposition 7.4. Let (L, ⊕) be a poset together with a binary hyperoperation ⊕ de- ﬁned by a ⊕ b = {c | c ≤ a, c ≤ b, c ∈ L} ∈ P ∗(L) for all a, b ∈ L. Then, the following conditions hold:

(i) (L, ⊕) is a Hv-semilattice.

(ii) For all a, b ∈ L, a ≤L b if and only if a ≤ b. (iii) Let J be a nonempty subset of L. Then J is an ideal of (L, ⊕) if and only if for all j ∈ J, x ∈ L, if x ≤L j, then x ∈ J. (iv) J is an ideal of (L, ≤) if and only if J is an ideal of (L, ⊕).

Proof. The proof is straightforward.

Deﬁnition 7.5. Let (L, ⊕), be an Hv-semilattice and a, b ∈ L. If a ≤L b and b ≤L a, then say a is hyperequal to b which is denoted by a =L b.

Corollary 7.6. Let (L, ⊕) be an Hv-semilattice and a, b ∈ L. Then a =L b if and only if a ⊕ b = b ⊕ c for all c ∈ L.

Proposition 7.7. Let (L, ⊕) be an Hv-semilattice. Then (L, ⊕) is a trivial Hv-semilattice if and only if a =L b for all a, b ∈ L.

Proof. Suppose that (L, ⊕) is a trivial Hv-semilattice, then for all a, b ∈ L, a ⊕ c = L

14 and b ⊕ c = L for all c ∈ L, i.e., a =L b. Conversely, a =L x for all x ∈ L, then a ⊕ b = x ⊕ b by Corollary 6.5. Therefore, we have a ⊕ b = ∪x∈L(x ⊕ b). So a ⊕ b = ∪x ∈ L(x ⊕ b) = ∪x ∈ l(∪y ∈ Lx ⊕ y) = L ⊕ L = L. That is to say (L, ⊕) is a trivial Hv-semilattice.

Proposition 7.8. Let (L, ⊕) be an Hv-semilattice. Then =L is an equivalence relation on L.

Proof. The proof is straightforward.

Proposition 7.9. Let (L, ⊕) be an Hv-semilattice, [a] = {x ∈ L | x =L a} and CL = {[a] | a ∈ L}. We may deﬁne a binary hyperoperation on CL by [a] ∗ [b] = {[n]|n ∈ a ⊕ b}, then (CL, ∗) is also an Hv-semilattice.

Proof. For all [a], [b], (c] ∈ CL, we have: (i) Since a ∈ a ⊕ a, then [a] ∈ {[n]|n ∈ a ⊕ a} = [a] ⊕ [a]. (ii) Since a ⊕ b = b ⊕ a, then [a] ∗ [b] = {[n]|n ∈ a ⊕ b} = {[n]|n ∈ b ⊕ a} = [a] ⊕ [b]. (iii) Since (a ⊕ b) ⊕ c ∩ a ⊕ (b ⊕ c) 6= ∅, then

Q = ([a] ∗ [b]) ∗ [c] = {[n]|n ∈ a ⊕ b} ∗ [c] = ∪n∈a⊕b([n] ∗ [c]) = ∪n∈a⊕{[m]|m ∈ n ⊕ c} = {[m]|m ∈ (a ⊕ b) ⊕ c}, and Q0 = {[m]|m ∈ a ⊕ (b ⊕ c)} = ∪n∈b⊕c([a] ∗ [c]) = [a] ∗ {[n]|n ∈ b ⊕ c} = [a] ∗ ([b] ∗ [c]) ∗ [c]). 0 and we have Q ∩ Q 6= ∅. Therefore (CL, ∗) is an Hv-semilattice.

8 Semilattices obtained from Hv-semilattices

∗ Let (L, ∗) be an Hv-semilattice. We define the relation β as the smallest equivalence relation on L such that the quotient L/β∗, the set of all equivalence classes is a semilattice. The β∗ is called the fundamental equivalence relation, and L/β∗ is called the fundamental semilattice. Suppose that β∗(a) is the equivalence class containing a ∈ L. Then the product on the semilattice L/β∗ is defined as follows: β∗(a) β∗(b) = β∗(c) for all c ∈ β∗(a) ∗ β∗(b). Let U denote the set of all products of elements of L. We define the relation β on L as follows:

15 a β b if and only if {a, b} ⊆ u for some u ∈ U.

Let us denote βˆ the transitive closure of β. Then we can rewrite the deﬁnition of βˆ on L as follows:

a βˆ b if and only if there exist z1, . . . , zn+1 ∈ L with z1 = a, zn+1 = b and u1, . . . , un ∈ U such that {zi, zi+1} ⊆ ui (i = 1, . . . , n). Then we have the following theorem.

Theorem 8.1. The fundamental relation β∗ is the transitive closure of the relation β.

Proof. The proof is similar to the proof of Theorem 1 [43].

Acknowledgement The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper.

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