An investigation on hyper S-posets over ordered semihypergroups

Definition 1.1

A hypergroupoid (S, ○) is a nonempty set S together with a hyperoperation, that is a map ○ :S × S → P*(S), where P*(S) denotes the set of all the nonempty subsets of S. The image of the pair (x, y) is denoted by x○ y.

Definition 1.2

A hypergroupoid (S, ○) is called a semihypergroup if the hyperoperation “ ○ ” is associative, that is, for all x, y, z ∊ S, (x ○ y) ○ z = x ○ (y ○ z), which means that

If x ∊ S and A, B are nonempty subsets of S, then

Generally, the singleton {x} is identified by its element x.

Definition 1.3

An algebraic hyperstructure (S, ○, ≤) is called an ordered semihypergroup (also called po-semihypergroup in [13]) if (S, ○) is a semihypergroup and (S, ≤) is a partially ordered set such that: for any x, y, a ∊ S, x ≤ y implies a ○ x ≤ a ○ y and x ○ a ≤ y ○ a. Here, if A, B ∊ P*(S), then we say that A ≤ B if for every a ∊ A there exists b ∊ B such that a ≤ b.

Clearly, every ordered semigroup can be regarded as an ordered semihypergroup, for instance, see [28].

Definition 1.4

A nonempty subset A of an ordered semihypergroup S is called a left (resp. right) hyperideal of S if

1. S ○ A ⊆ A (resp. A○ S ⊆ A).

2. If a ∊ A and S ∍ b ≤ a, then b ∊ A.

For more information on hyperstructure theory, ordered semigroup theory and the properties of S-acts, the reader is referred to [7], [30] and [22], correspondingly.

Definition 2.1

Let (S, ○) be a semihypergroup and A a nonempty set. If we have a mapping μ : A × S → P*(A) |(a, s)↦μ(a, s) := a ∗ s ∊ P*(A), called the hyper action of S (or the S-hyperaction) on A, such that a ∗ (s ○ t) = (a ∗ s) ∗ t, for all a ∊ A, s, t ∊ S, where

then we call A a right hyper S-act (also called right S-hypersystem in [10]), denoted by A_ℋ, or briefly A.

Left hyper S-acts are defined analogously and in this paper we will oten use the term hyper S-act to mean right hyper S-act.

Remark 2.2

Every S-act over a semigroup can be regarded as a hyper S-act over a semihypergroup. In fact, if A is an S-act over a semigroup (S, ⋅), the hyperoperation “ ∘ ” on S and the hyper S-action “∗” on A are defined respectively as s ○ t := {st}, a ∗ s := {as}, for any a ∊ A, s, t ∊ S, then, clearly A is a hyper S-act over a semihypergroup (S, ○).

Definition 2.3

Let A be a hyper S-act over a semihypergroup (S, ○) and B a nonempty subset of A. B is called a hyper S-subact of A if B is closed under the hyper S-action on A, i.e., b ∗ s ⊆ B for any b ∊ B, s ∊ S.

Clearly, for any a ∊ A, a ∗ s is a hyper S-subact of A, called cyclic hyper S-subact.

Let A be a hyper S-act over a semihypergroup (S, ○) and ρ an equivalence relation on A. If B and C are both nonempty subsets of A, then we write B ρ¯ C to denote that for every b ∊ B, there exists c ∊ C such that bρc and for every c ∊ C, there exists b ∊ B such that bρc. We write B ρ¯¯ C if for every b ∊ B and for every c ∊ C, we have bρc. The equivalence relation ρ is called congruence if for every (x, y) ∊ A × A, the implication xρy ⇒ x ∗ s ρ¯ y ∗ s, for all s ∊ S, is valid. ρ is called strong congruence if for every (x, y) ∊ A × A, from x ρ y, it follows that x ∗ s ρ¯¯ y ∗ s for all s ∊ S. We denote by C(A) (resp. SC(A)) the set of all congruences (resp. strong congruences) on a hyper S-act A.

Remark 2.4

The set C(A) of all congruences on a hyper S-act A is a complete lattice with respect to the intersection of set-theoretic and the union (also is called transitive product) defined as follows:

It is worth pointing out that the equality relation 1_A and the universal relation A × A on A are the minimum element and greatest element of C(A), respectively.

Theorem 2.5

Let A be a hyper S-act over a semihypergroup (S, ○) and ρ an equivalence relation on A. Then

1. If ρ is a congruence, then A/ρ is a hyper S-act with respect to the following hyper S-action: (a)ρ⊗s=⋃x∈a∗s(x)ρ, and it is called a factor hyper S-act.

2. If ρ is a strong congruence, then A/ρ is a hyper S-act with respect to the following (hyper) S-action: (a)_ρ ⊗ s = (x)_ρ for all x ∊ a ∗ s, and it is called a factor hyper S-act. In particular if S is a semigroup and the operation on S is defined by s ○ t:= {st} for all s, t ∊ S, then A/ρ is an S-act.

Proof

1. Let ρ be a congruence on A. Then the hyper S-action “ ⊗” is well defined. Indeed, let (a)_ρ, (b)_ρ ∊ A/ρ and s, t ∊ S be such that (a)_ρ = (b_)ρ, s = t. Then a ρ b. Since ρ is a congruence on A, we have a ∗ s ρ¯ b ∗ s. Hence for any x ∊ a ∗ s, there exists y ∊ b ∗ t such that xρ y, i.e., (x)_ρ = (y)_ρ. Thus (a)ρ⊗s=⋃x∈a∗s(x)ρ⊆⋃y∈b∗t(y)ρ=(b)ρ⊗t. In a similar way,it can be shown that (b)_ρ⊗ t ⊆(a)_ρ⊗ s. Therefore, (a)_ρ⊗ s = (b)_ρ⊗ t.

   Furthermore, let s, t∊ S and (a)_ρ∊ A/ρ. Then we have

   (a)ρ⊗(s∘t)=⋃u∈s∘t((a)ρ⊗u)=⋃u∈s∘t⋃x∈a∗u(x)ρ=⋃x∈a∗(s∘t)(x)ρ=⋃x∈(a∗s)∗t(x)ρ(SinceAis a hyperS−act)=⋃y∈a∗s⋃x∈y∗t(x)ρ=⋃y∈a∗s((y)ρ⊗t)=((a)ρ⊗s)⊗t.

   Thus A/ρ is a hyper S-act over S.

2. The proof is similar to that of (1), and hence we omit the details.

Let A be a hyper S-act over a semihypergroup (S, ○) and B a hyper S-subact of A. The relation ρ_B on S is defined as follows:

Clearly, ρ_B is an equivalence relation on A. Moreover, we have the following lemma.

Theorem 2.6

Let A be a hyper S-act over a semihypergroup (S, ○) and B a hyper S-subact of A. Then ρ_B is a congruence on A and it is called Rees congruence induced by B.

Proof

Let x, y ∊ A and x ρ_By. Then x = y ∊ A\B or x, y ∊ B. We consider the following cases:

Case 1. If x = y ∊ A\B, then, for any s ∊ S, x ∗ s = y ∗ s. Hence x∗sρ¯By∗s.

Case2. Let x, y ∊ B. Since B a hyper S-subact of A, we have x ∗ s⊆ B, y ∗ s ⊆ B for any s ∊ S. Thus, for any a ∊ x ∗ s, b ∊ y ∗ s, we have (a, b)∊ B × B ⊆ ρ_B. Thus x∗sρ¯By∗s.

Therefore, ρ_B is a congruence on A.

Remark 2.7

1. A/ρ_B = {{x}| x ∊ A\B} ∪ {B}, that is, for any (a)_ρB} ∊ A/ρ_B, we have (a)_ρB = {a}, a ∊ A\B or (a)_ρB = B.

2. By Theorems 2.5 and 2.6, (A/ ρ_B, ⊗_B) forms a factor hyper S-act, which is called Rees factor hyper S-act. Here the hyper S-action ⊗_B on A/ρ_B is defined by (a)ρB⊗Bs=⋃x∈a∗s(x)ρB,∀(a)ρB∈A/ρB,s∈S.

Definition 3.1

Let (S, ○, ≤_S) be an ordered semihypergroup. A right hyper S-poset (A, ≤_A), often denoted A_ℋ (or briefly A), is a right hyper S-act A equipped with a partial order ≤_A and, in addition, for all s, t ∊ S and a, b ∊ A, if s ≤_St then a ∗ s ≤_Aa ∗ t, and if a ≤_Ab then a ∗ s ≤_Ab ∗ s. Here, a ∗ s stands for the result of the hyper action of s on a, and if A₁, A₂ ∊ P ∗ (A), then we say that A₁ ≤_AA₂ if for every a₁ ∊ A₁ there exists a₂ ∊ A₂ such that a₁ ≤_A a₂.

Analogously we can define a left hyper S-poset _ℋ A. Throughout this paper we shall use the term hyper S-poset to mean right hyper S-poset.

Remark 3.2

1. Every S-poset over an ordered semigroup can be regarded as a hyper S-poset over an ordered semihypergroup.

2. An ordered semihypergroup S is a hyper S-poset with respect to the hyperoperation of S.

Let A be a hyper S-poset over an ordered semihypergroup (S, ○, ≤_S) and B a nonempty subset of A. B is called a hyper S-subposet of A if for any b ∊ B, s ∊ S, b ∗ s ⊆ B, denoted by B ≤A. For any a ∊ A, a ∗ S is clearly a hyper S-subposet of A, called cyclic hyper S-subposet. It is easily seen that a hyperideal of an ordered semihypergroup S is a hyper S-subposet of S.

Definition 3.3

Let (A, ≤_A) and (B, ≤_B) be two hyper S-posets over an ordered semihypergroup (S, ○, ≤), the“∗” and“ ⋄” are hyper S-actions on A and B, respectively f : A → B a mapping from A to B. f is called isotone if x ≤_A y implies f(x)≤_B f (y), for all x, y ∊ A. f is called reverse isotone if x, y ∊ A, f(x) ≤_Bf(y) implies x ≤_Ay. f is called homomorphism (resp. strong homomorphism) if it is isotone and satisfies f(a)⋄s=⋃x∈a∗sf(x) (resp. f(a)⋄s=f(x),∀x∈a∗s),foralla∈A,s∈S. f is called isomorphism (resp. strong isomorphism) if it is homomorphism (resp. strong homomorphism), onto and reverse isotone. The hyper S-posets A and B are called strongly isomorphic, in symbol A ≅ B, if there exists a strong isomorphism between them.

Remark 3.4

1. Suppose that (A, ≤_A) and (B, ≤_B) are two hyper S-posets over an ordered semihypergroup (S, ○, ≤). If f is a strong homomorphism and reverse isotone mapping from A to B, then A ≅ I m(f).

2. The class of right hyper S-posets and homomorphisms forms a category that we denote by H P O S-S. As usual, the monomorphisms of H P O S-S are exactly the injective homomorphisms.

By Definition 3.1, we can see that the congruences and strong congruences on hyper S-posets can be defined exactly as in the case of hyper S-acts. Thus it is unnecessary to repeat the concepts of congruences and strong congruences on hyper S-posets.

Let A be a hyper S-act and ρ a (strong) congruence on A. Then, by Theorem 2.5, the set A/ ρ := {(a)_ρ | a ∊ A} is a hyper S-act and the hyper S-action on A/ρ is defined via the hyper S-action on A. The following question is natural: If (A, ≤_A) is a hyper S-poset over an ordered semihypergroup (S, ○, ≤) and ρ a (strong) congruence on A, then is the set A/ρ a hyper S-poset? A probable order on A/ρ could be the relation“⪯” on A/ρ defined by means of the order “≤_A” on A, that is

But this relation is not an order, in general. We illustrate it by the following example.

Example 3.5

We consider a set S :={a, b, c, d, e} with the following hyperoperation “ ○” and the order “≤”:

We give the covering relation “≺” and the figure of S as follows:

Then (S, ○, ≤) is an ordered semihypergroup. We now consider the partially ordered set A = {c, d, e} defined by the order below:

We give the covering relation “≺_A” and the figure of A.

Then (A, ≤_A) is a hyper S-poset over S with respect to S-hyperaction on A as above hyperoperation table.

Let ρ be a (strong) congruence on A defined as follows:

Then A/ρ = {{d, c},{e}}.Moreover, the relation on A/ρ defined by

is not an order relation on A/ρ. In fact, since d ≤_A e, we have (d)_ρ ⪯(e)_ρ. Also, since e ≤ _A c, we have (e)/_ρ ⪯(c)_ρ = (d)_ρ. If “⪯” is an order relation on A/ρ, then (d)_ρ = (e)_ρ, which is impossible. Thus (A/ρ,⪯) is not a hyper S-poset.

The following question arises: Is there a (strong) congruence ρ on A for which A/ρ is a hyper S-poset? To solve the above question, we first introduce the following definition.

Definition 3.6

Let(A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S,∘, ≤). A relation ρ on A is called pseudoorder if it satisfies the following conditions:

1. ≤A⊆ρ.

2. aρb and bρc imply aρc, i. e., ρ 0 ρ ⊆ ρ.

3. aρb implies a∗sρ=⁡b∗s for all s ∈ S.

Note that an ordered semihypergroup S is a hyper S-poset with respect to the hyperoperation of S. Thus Definition 3.6 is a generalization of Definition 4.1 in [9]. For a similar definition about pseudoorders in ordered semigroups we refer the readers to Definition 1 in [16].

Theorem 3.7

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, 0, ≤) and ρ a pseudoorder on A. Then, there exists a strong congruence ρ_ on A such that A/ρ_ is a hyper S-poset over S.

Proof

We denote by ρ_ the relation on A defined by

First, we claim that ρ_ is a strong congruence on A. In fact, for any a ∈ A, clearly, (a, a)∈ ≤_A⊆ρ, so aρ_a. If (a,b)∈ρ_, then aρb and bρa. Thus (b,a)∈ρ_ Let (a,b)∈ρ_and(b,c)∈ρ_ Then aρb, bρa, bρc and cρb. Hence aρc and cρa, which imply that (a,c)∈ρ_ Thus ρ_ is an equivalence relation on A. Now, let aρ_bands∈S Then aρb and bρa. Since ρ is a pseudoorder on A, by condition (3) of Definition 3.6, we have

Thus, for every x ∈ a ∗ s and y ∈ b ∗ s, we have x ρ y and y ρ x. It implies that xρ_y Hence a∗sρ_¯¯b∗s Therefore, ρ_ is indeed a strong congruence on A. By Theorem 2.5, A/ρ_ is a hyper S-act over S.

Now, we define a relation ≺_ρonA/ρ_ as follows:

Then (A/ρ_,≺_ρ) is a poset. Indeed, suppose that (x)ρ_∈A/ρ_, where x ∈ A. Then (x, x)∈ ≤_A⊆ρ. Hence, (x)ρ_≺_ρ(x)ρ_. Let (x)ρ_≺_ρ(y)ρ_and(y)ρ_≺_ρ(x)ρ_. Then xρy and yρx. Thus xρ_y, and we have (x)ρ_=(y)ρ_. Now, if (x)ρ_≺_ρ(y)ρ_and(y)ρ_≺_ρ(z)ρ_, then x ρ y and y ρ z. Hence x ρ z, and we conclude that (x)ρ_≺_ρ(z)ρ_.

Furthermore, let (x)ρ_,(y)ρ_∈A/ρ_,(x)ρ_≺_ρ(y)ρ_ands∈S. Then x ρ y. By hypothesis and Definition 3.6, x∗sρ¯¯y∗s. Thus, for any a ∈ x ∗ s and b ∈ y ∗ s, we have a ρ b. This implies that (a)ρ_≺_ρ(b)ρ_. Hence we have

where the hyper S-action “⊗ ” on A/ρ_ is exactly that defined in Theorem 2.5. Moreover, let s, t ∈ S, s ≤ t and (a)ρ_∈A/ρ_. Then, similarly as discussed above, we have (a)ρ_⊗s≺_ρ(a)ρ_⊗t.

Therefore, A/ρ_ is a hyper S-poset over S.

Example 3.8

We consider a set S := {a, b, c, d, e} with the following hyperoperation“ ∘ ” and the order“ ≤ ”:

The covering relation “≺” and the figure of S are given by:

Then (S, 0, ≤) is an ordered semihypergroup (see [9]). We now consider the partially ordered set A = {a, d, e} defined by the order below:

We give the covering relation “ ≺_A ” and the figure of A.

Then (A, ≤ _A) is a hyper S-poset over S with respect to S-hyperaction on A as above hyperoperation table.

Let ρ be a pseudoorder on A defined as follows:

Applying Theorem 3.7, we get

Then A/ρ_={{a,d},{e}}.Moreover,(A/ρ_,≺_ρ) is a hyper S-poset over S, where the order relation ≺_ρonA/ρ_ is defined by

Theorem 3.9

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S,∘,≤) nd ρ a pseudoorder on A. Let

Let ℬ be the set of all pseudoorders on A/ρ_. Then, card card(𝒜) = card(ℬ).

Proof

For θ∈ 𝒜, we define a relation θ′ on A/ρ_ as follows:

First, we claim that θ′ is a pseudoorder on A/ρ_. To prove our claim, let ((x)ρ_,(y)ρ_)∈≺_ρ. Then, by Theorem 3.7, (x, y) ∈ ρ ⊆θ, which implies that ((x)ρ_,(y)ρ_)∈θ′.Thus,≺_ρ⊆θ′. Now, assume that ((x)ρ_,(y)ρ_)∈θ′ and ((y)ρ_,(z)ρ_)∈θ′. Then, (x, y) ∈θ and (y, z)∈θ. It implies that (x, z) ∈θ. Hence, ((x)ρ_,(z)ρ_)∈θ′. Also, let ((x)ρ_,(y)ρ_)∈θ′ and s ∈ S. Then, (x, y)∈θ and s ∈ S. Since θ is a pseudoorder on A, we have x∗sθ¯¯y∗s. Thus, for every a ∈ x ∗ s, b ∈ y ∗ s, we have (a,b) ∈θ. This implies that ((a)ρ_,(b)ρ_)∈θ′, and thus (x)ρ_⊗sθ′¯¯(y)ρ_⊗s. Therefore, θ′ is indeed a pseudoorder on A/ρ_.

Now, we define the mapping f:A→Bbyf(θ)=θ′,∀θ∈A. Then, f is a bijection from 𝒜 onto ℬ. In fact,

1. f is well defined. Indeed, let θ₁, θ₂ ∈ 𝒜 and θ₁ = θ₂. Then, for any ((x)ρ_,(y)ρ_)∈θ1′, we have (x, y) ∈ θ₁ = θ₂. It implies that ((x)ρ_,(y)ρ_)∈θ2′. Hence θ₁ ⊆θ₂ ′. By symmetry, it can be obtained that θ₂ ′⊆ θ₁′.

2. f is one to one. In fact, let θ₁, θ₂ ∈ 𝒜 and θ₁ ′ = θ₂ ′. Assume that (x, y) ∈θ₁. Then, ((x)ρ_,(y)ρ_)∈θ1′ , and thus ((x)ρ_,(y)ρ_)∈θ2′. This implies that (x, y) ∈θ₂. Thus, θ₁ ⊆θ₂. Similarly, we obtain θ₂ ⊆ θ₁.

3. f is onto. In fact, let δ ∈ ℬ. We define a relation θ on A as follows:

   θ:={(x,y)∈A×A|((x)ρ_,(y)ρ_)∈δ}.

   We show that θ is a pseudoorder on A and ρ ⊆ θ. Assume that (x, y) ∈ ρ. Then, by Theorem 3.7, ((x)ρ_,(y)ρ_)∈≺_ρ⊆δ, and thus (x, y) ∈θ. This implies that ρ ⊆θ. If (x, y) ∈ ≤_A, then (x, y) ∈ρ ⊆θ. Hence, ≤_A ⊆ θ. Let now (x, y) ∈θ and (y, z) ∈θ. Then ((x)ρ_,(y)ρ_)∈δand((y)ρ_,(z)ρ_)∈δ. Hence ((x)ρ_,(z)ρ_)∈δ, which implies that (x, z) ∈θ. Furthermore, let (x, y) ∈θ and s ∈ S. Then ((x)ρ_,(y)ρ_)∈δ and s ∈ S. Since δ is a pseudoorder on A/ρ_, we have (x)ρ_⊗sδ¯(y)ρ_⊗s,i.e.,⋃a∈x∗s(a)ρ_δ¯¯⋃b∈y∗s(b)ρ_. Thus, for every a ∈ x ∗ s and b∈y∗s,((a)ρ_,(b)ρ_)∈δ. It means that (a, b) ∈θ. Hence we conclude that x∗sθ¯¯y∗s. Moreover, clearly, θ′ = δ.

Corollary 3.10

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, 0, ≤), ρ, θ be pseudoorders on A such that ρ ⊆ θ. We define a relation θ/ρ on A/ρ_ as follows:

Then θ/ρ is a pseudoorder on A/ρ_.

Definition 4.1

Let} (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S,∘, ≤). A congruence (resp. strong congruence) ρ is called an order-congruence (resp. a strong order-congruence) if there exists an order relation “⪯” on A/ρ such that:

1. (A/ρ,⪯) is a hyper S-poset, where the S-hyperaction “ ⊗” on A/ρ is defined as one in Theorem 2.5.

2. The canonical epimorphism φ : → A/ρ, x ↦ (x)_ρ is isotone, that is, ↦ is a homomorphism (resp. strong homomorphism) from A onto A/ρ.

It is clear that the equality relation 1_A and the universal relation A × A on A are both order-congruences, but 1_A is not a strong order-congruence on A. In general, a strong order-congruence example is given as follows:

Example 4.2

We consider the ordered semihypergroup (S,∘, ≤) and the hyper S-poset (A, ≤_A) over S in Example 3.5. Let ρ be a strong congruence on A defined as follows:

Then S/ρ = {{c, e},{d}}. Moreover, ρ is a strong order-congruence on A. In fact, we define an order ⪯_ρ on A/ρ as follows:

Then (A/ρ,⪯_ρ) is a hyper S-poset and the mapping φ : A→ A/ρ, x ↦(x)_ρ is isotone. Hence ρ is a strong order-congruence on A.

Proposition 4.3

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S,∘, ≤) and ρ a pseudoorder on A. Then ρ_ is a strong order-congruence on A, where ρ_=ρ∩ρ−1.

Proof

By Theorem 3.7, (A/ρ_,≺_ρ) is a hyper S-poset over s, where the order relation ⪯_ρ is defined as follows:

Also, let x, y ∈ A and x ≤_A y. Then, since ρ is a pseudoorder on A, (x, y) ∈ ≤_A⊆ρ. Thus ((x)ρ_,(y)ρ_)∈≺_ρ, i.e.,(x)ρ_≺_ρ(y)ρ_. Therefore, ρ_ is a strong order-congruence on A.

In order to establish the relationship between strong order-congruences and pseudoorders on a hyper S-poset, the following lemma is essential.

Lemma 4.4

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S,∘, ≤) and σ a relation on A. Then the following statements are equivalent:

1. σ is a pseudoorder on A.

2. There exist a hyper S-poset (B, ≤_B) over S and a strong homomorphism φ: A → B such that

   ker→φ:={(a,b)∈A×A|φ(a)<_Bφ(b)}=σ,

   where ker→φ is called the directed kernel of φ

Proof

(1) ⇒(2). Let σ be a pseudoorder on A. We denote by σ_ the strong congruence on A defined by

Then, by Theorem 3.7, the set A/σ_:={(a)σ_|a∈A} with the S-hyperaction (a)σ_⊗s=(x)σ_,∀x∈a∗s, for all a ∈ A, s ∈ S and the order

is a hyper S-poset. Let B=(A/σ_,≺_σ) be the mapping of A onto A/σ_ defined by φ:A→A/σ_|a↦(a)σ_. Then, by Proposition 4.3, ↦ is a strong homomorphism from A onto A/σ_ and clearly, ker→φ=σ.

(2) ⇒(1). If there exist a hyper S-poset (B, ≤_B) over S and a strong homomorphism φ: A → B such that ker→φ=σ, then σ is a pseudoorder on A. Indeed, let (a, b)∈ ≤_A. Then, by hypothesis, φ(a) ≤_Bφ(b). Thus (a,b)∈ker→φ=σ, and we have ≤_A⊆σ. Now, let (a, b) ∈σ and (b, c) ∈ σ. Then φ(a) ≤_Bφ(b) ≤_Bφ(c). Hence φ(a)<_Bφ(c),i.e.,(a,c)∈ker→φ=σ. Also, if (a, b) ∈σ, then φ(a) ≤_Bφ(b). Since (B, ≤_B) is a hyper S-poset over s, for any s ∈ S we have φ(a)⋄ s ≤_Bφ(b)⋄ s, where “ ⋄” is the S-hyperaction on B. Since ↦ is a strong homomorphism from A to B, for every x ∈ a ∗ s and y ∈ b ∗ s, we have

Then (x,y)∈ker→φ=σ, and thus a∗sσ¯¯b∗s. Therefore, σ is a pseudoorder on A.

Theorem 4.5

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S,∘, ≤) and ρ ∈ SC(A). Then the following statements are equivalent:

1. ρ is a strong order-congruence on A.

2. There exists a pseudoorder σ on A such that ρ = σ∩σ⁻¹.

3. There exist a hyper S-poset B over s and a strong homomorphism φ: A → B such that ρ = ker(φ), where ker φ = {(a, b) ∈ A × A|φ(a) = φ(b) is the kernel of φ.

Proof

(1) ⇒(2). Let ρ be a strong order-congruence on A. Then there exist an order relation “⪯” on the factor hyper S-act A/ρ such that (A/ρ,⪯) is a hyper S-poset over S, and φ: A → A/ρ is a strong homomorphism. Let σ=ker→φ. By Lemma 4.4, σ is a pseudoorder on A and it is easy to check that ρ = σ∩σ⁻¹.

(2) ⇒ (3). For a pseudoorder σ on A, by Lemma 4.4, there exist a hyper S-poset B over s and a strong homomorphism φ A → B such that σ=ker→φ. Then we have

(3) ⇒(1). By hypothesis and Lemma 4.4, ker→φ is a pseudoorder on A. Then, by Theorem 3.7, ρ=ker→φ∩(ker→φ)−1 is a strong congruence on A. Thus, by the proof of Lemma 4.4, ρ is a strong order-congruence on A.

Remark 4.6

1. For a strong order-congruence ρ on A, since the order “⪯” such that (A/ρ,⪯) is a hyper S-poset is not unique in general we have the pseudoorder σ containing ρ such that ρ = σ∩σ⁻¹ is not unique.

2. If σ is a pseudoorder on a hyper S-poset A, then ρ = σ∩σ⁻¹ is the greatest strong order-congruence on A contained in σ. In fact, if ρ₁ is a strong order-congruence on A contained in σ, then ρ1=ρ1∩ρ1−1⊆σ∩σ−1=ρ.

Theorem 4.7

Let ρ be a strong order-congruence on a hyper S-poset (A, ≤_A). Then the least pseudoorder σ containing ρ is the transitive closure of relations ≤_A ∘ρ (resp. ρ∘ ≤_A), that is,

Proof

1. Let σ1=⋃n=1∞(≤A∘ρ)n. Clearly, ρ⊆ ≤_A∘ ρ⊆σ₁. Similarly, since ≤_A⊆ ≤_A∘ ρ, we have ≤_A⊆σ₁.

2. If (a, b) ∈σ₁, (b, c) ∈σ₁, then there exist m, n∈ Z⁺ such that (a, b)∈( ≤_A ∘ρ)^m and (b, c) ∈( ≤_A ∘ρ)ⁿ, where Z⁺ denotes the set of positive integers. Thus (a, c) ∈( ≤_A ∘ρ)^m+n ≤σ₁, i.e., σ₁ is transitive.

3. Let (a, b)∈σ₁ and s∈ S. Then there exists n ∈ Z⁺ such that (a, b)∈( ≤_A ∘ρ)ⁿ, that is, there exist a₁, b₁, a₂, b₂, …, a_n ∈ A such that

   a≤Aa1ρb1≤a2ρb2≤A⋯≤Aanρb.

   Since (A, ≤_A) is a hyper S-poset and ρ∈ SC(A), we have

   a∗s≤Aa1∗sρ¯¯b1∗sAa2∗sρ¯¯∗s≤⋯≤an∗sρ¯¯b∗s.

   Then, for any x ∈ a ∗ s, y ∈ b ∗ s, there exist x_i ∈ a_i ∗ s (i = 1,2,…, n), y_j ∈ b_j ∗ s (j = 1,2,…, n-1) such that

   x≤Ax1ρy1≤x2ρy2≤xnρy.

   It thus implies that (x, y) ∈(≤_A ∘ρ)ⁿ ⊆σ₁, and we obtain that a∗sσ¯¯1b∗s. Thus ⋃n=1∞(≤A∘ρ)n is a pseudoorder on A containing ρ.

   Furthermore, since σ is transitive, and ρ⊆σ, ≤_A ⊆ σ, we have ⋃n=1∞(≤A∘ρ)n⊆σ. Thus, by hypothesis, σ=⋃n=1∞(≤A∘ρ)n. In the same way, we can verify that σ=⋃n=1∞(∘ρ≤A)n. In the following, we shall give some characterizations of (strong) order-congruences on hyper S-posets. In order to obtain the main results, we first introduce the following concept.

Definition 4.8

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, ∘, ≤) and ρ an equivalence relation on A. A finite sequence of the form (x, a₁, b₁, a₂, b₂, …, a_n-1, b_n-1, a_n, y) of elements in A is called a ρ-chain if

1. (a₁, b₁) ∈ ρ, (a₂, b₂) ∈ ρ, …, (a_n-1, b_n-1) ∈ ρ, (a_n, y) ∈ ρ;

2. x ≤ _A a₁, b ≤_A a₂, b ≤_A a₃, …, b_n-2 ≤_A a_n-1, b_n-1 ≤_A a_n.

Briefly we write

The number n is called the length, x and y initial and terminal elements, respectively, of the ρ-chain. A ρ-chain is called close if its initial and terminal elements are equal, i.e. x = y.

We denote by ρ^C_xy the set of all ρ-chains with x as the initial and y as the terminal elements in the sequel.

Lemma 4.9

Let (A,≤_A) be a hyper S- poset over an ordered semihypergroup (S, 0, ≤) and ρ ∈ C(A). Then the following statements are true:

1. (x, y) ∈ (≤_A o ρ)ⁿ if and only if there exists a ρ-chain with length n in ρ^C_xy, i. e., ρ^C_xy ≠ ∅.

2. For any s ∈ S, if ρ^C_xy ≠ ∅ for some x, y ∈ A, then for every u ∈ x ∗ s, there exists v ∈ y ∗ s such that ρ^C_uv ≠ ∅.

Proof

1. The proof is straightforward by Definition 4.8, we omit it.

2. Let (x, a₁, b₁, a₂, b₂, …, a_n, y) ∈ ρ^C_xy and s ∈ S. Then

   x≤Aa1ρb1≤Aa2ρb2≤A⋯≤Aanρb.

   Since (A, ≤_A) is a hyper S-poset and ρ ∈ C(A), we have

   x∗s≤Aa1∗sρ¯b1∗s≤Aa2∗sρ¯b2∗s≤A⋯≤Aan∗sρ¯y∗s.

   Then, for any u ∈ x o s, there exist x_i ∈ a_i ∗ s(i = 1, 2, …, n), y_j ∈ b_j ∗ s(j=1,2, …, n− 1), v ∈ y ∗ s such that

   u≤Ax1ρy1≤Ax2ρy2≤Axnρv.

   It thus implies that (u, x₁, y₁, x₂, y₂, …, x_n, v) ∈ ρ^C_uv, i.e., ρ^C_uv ≠ ∅.

Lemma 4.10

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, o, ≤) and ρ ∈ S C(A). If ρ^C_xy ≠ ∅ for some x, y ∈ A, then, for any s ∈ S, we have ρ^C_uv ≠ ∅ for every u ∈ x ∗ s, v ∈ y ∗ s.

Proof

The proof is similar to that of Lemma 4.9 with a slight modification.

Lemma 4.11

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, o,≤) and ρ a (strong) congruence on A. If(x, y) ∈ ρ, (k, z) ∈ ρ, then ρ^C_xk ≠ ∅ if and only if ρ^C_yz ≠ ∅.

Proof

(⇒). If ρ^C_xk ≠ ∅, by Lemma 4.9 (1), there exists n ∈ Z⁺ such that (x, k) ∈ (≤_A oρ)ⁿ. Since (x, y) ∈ ρ, (z, k) ∈ ρ, we have

which implies that (y, z) ∈ (≤_A oρ)ⁿ⁺². By Lemma 4.9 (1), we have ρ^C_yz ≠ ∅.

(⇐). Similar to the proof of necessity, we omit it.

Now we shall give a characterization of order-congruences on a hyper S-poset.

Theorem 4.12

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, o,≤) and ρ ∈ C(A) . Then ρ is an order-congruence on A if and only if every close ρ-chain is contained in a single equivalent class of ρ.

Proof

Let ρ be an order-congruence on A. Then there exists an order ⪯ on the factor hyper S-act A/ρ such that (A/ρ,⪯) is a hyper S-poset over S and φ : A → A/ρ is a homomorphism. For any x ∈ A, and every close ρ-chain (x, a₁, b₁, …, a_n, x) in ρ^C_xx, we have

Then,

It implies that φ(x) = φ(a₁}) = φ(b₁) = φ(a₂) = φ(b₂) = ⋯ = φ(a_n). Consequently, (x, a₁, b₁, …, a_n, x) is contained in a single ρ-class.

Conversely, since ρ is a congruence on A, by Theorem 2.5, A/ρ is a hyper S-act. We define a relation “⪯” on the factor hyper S-act A/ρ as follows:

1. ⪯ is well-defined. In fact, let x₁, y₁ ∈ A be such that (x)_ρ = (x₁)_ρ, (y)_ρ = (y₁)_ρ. If (x)_ρ ⪯ (y)_ρ, then ρ^C_xy ≠ ∅. By Lemma 4.11, we have ρCx1y1≠∅, and (x₁)_ρ ⪯ (y₁)_ρ.

2. ⪯ is an ordered relation on A/ρ.

3. ⪯ is reflexive. In fact, since for any x ∈ A, x ≤_A xρx, and we have ρ^C_xx ≠ ∅ i.e., ((x)_ρ, (x)_ρ) ∈⪯.

4. ⪯ is transitive. Indeed, let ((x)_ρ, (y)_ρ)∈ ⪯, ((y)_ρ, (z)_ρ)∈ ⪯. Then we have ρ^C_xy ≠ ∅, ρ^C_yz ≠ ∅. By Lemma 4.9 (1), there exist m, n ∈ Z⁺ such that (x, y) ∈ (≤_A oρ)^m, (y, z) ∈ (≤_A oρ)ⁿ. Then we have

   (x,z)∈(≤Aoρ)mo(≤Aoρ)n=(≤Aoρ)m+n,

   i.e., ρ^C_XZ ≠ ∅. Thus ((x)_ρ, (z)_ρ)∈ ⪯.

5. ⪯ is anti-symmetric. In fact, if ((x)_ρ, (y)_ρ) ∈⪯, ((y)_ρ, (x)_ρ) ∈⪯, then ρ^C_xy ≠ ∅, ρ^C_yx ≠ ∅. Similar to the above proof, it can be obtained that ρ^C_xx ≠ ∅, i.e., there exists a close ρ-chain in ρ^C_xx containing x and y. By hypothesis, (x)_ρ = (y)_ρ.

6. (A/ρ,⪯) is a hyper S-poset over S. Indeed, let (x)_ρ ⪯ (y)_ρ and s ∈ S. Then ρ^C_xy ≠ ∅. By Lemma 4.9(2), for every u ∈ x ∗ s, there exists v ∈ y ∗ s such that ρ^C_uv ≠ ∅, i.e., (u)_ρ ⪯ (v)_ρ. Thus

   (x)ρ⊗s=⋃u∈x∗s(u)ρ⪯⋃v∈x∗s(v)ρ=(y)ρ⊗s.

   Also, let s, t ∈ S be such that s ≤ t. Then x ∗ s ≤_A x ∗ t for any x ∈ A. Thus, for every u′ ∈ x ∗ s, there exists v′ ∈ x ∗ t such that u′ ≤_A v′. It implies that (u′, v′) ∈≤_A oρ, and we have ρ^C_u′v′ ≠ ∅. Hence (u′)_ρ ⪯ (v′)_ρ, and we obtain

   (x)ρ⊗s=⋃u′∈x∗s(u′)ρ⪯⋃v′∈x∗t(v′)ρ=(x)ρ⊗t.

7. The mapping φ: A → A/ρ | x ↦(x)_ρ is isotone. In fact, let x, y ∈ A be such that x ≤_A y. Then (x, y) ∈≤_A oρ, we have ρ^C_xy ≠ ∅, i.e. (x)_ρ ⪯ (y)_ρ.

   Therefore, ρ is an order-congruence on A.

   Similarly, strong order-congruences on a hyper S-poset can be characterized as follows:

Theorem 4.13

Let (A, ≤_A) be a hyper S-poset over an ordered semihypergroup (S, o,≤) and ρ ∈ SC(A). Then ρ is a strong order-congruence on A if and only if every close ρ-chain is contained in a single equivalent class of ρ.

Proof

The proof is similar to that of Theorem 4.12 with suitable modification by using Lemma 4.10.

Recall that a nonempty subset B of a poset (A,≤) is called convex if a ≤ b ≤ c implies b ∈ B for all a, c ∈ B, b ∈ A; B is called strongly convex if a ∈ A, b ∈ B and a ≤ b imply a ∈ B. Any strongly convex subset of A is clearly convex, however, the converse does not hold in general.