Equational characterizations for some subclasses of domains

Fan Feng; Xiangrui Li

doi:10.1515/math-2025-0219

Artikel Open Access

Equational characterizations for some subclasses of domains

Fan Feng und Xiangrui Li

Veröffentlicht/Copyright: 19. Dezember 2025

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Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

It is well known that the continuity of a poset can be seen as a special distributivity. There is an open problem: is there an equational characterization for continuous semilattices? Based on equational characterizations of continuous lattices, bounded complete domains and L-domains, we prove that a special class of domains can be characterized by an equation. As an application, an equational characterization for a subclass of continuous semilattices is given. Moreover, by using ideals instead of directed sets, we obtain a unified equational characterization for more subclasses of domains, including that of domains mentioned above. Unfortunately, even if using ideals, we still can not characterize all of the domains. Some examples are provided to illustrate it.

Keywords: domains; continuous semilattices; equational characterization; ideals; family of dcpos

MSC 2020: 06A06; 06B35

1 Introduction

In the late 1960s, Dana Scott found the semantic structure in computer science being close to partial order structures. Based on this observation, he established Domain Theory, which plays a central role in the field of theoretical computer science. It is well known that the continuity and the quasicontinity of posets are important concept in Domain Theory. These can be used to describe convergence and approximation in order theory [1], [2], [3]. Similar to the study of universal algebra, it raises the question as to whether domains are maintained under subalgebras, products and homomorphic images. For this question, it was shown in [4] that two special subclass of domains that continuous lattices and bounded complete domains can be characterized by the distributivity. This kind of characterization is called an equational characterization. The continuity on complete lattice can be viewed as an infinite distributive law. In [5], Marcel Erné shows the relationship between continuity and the other laws of infinite distribution. In fact, many infinite distributive laws can be applied to a broader range. For example, Wei Luan and Qingguo Li showed that quasi-continuity complete semilattice can be characterized by an equation in [6] and Paul Taylor provided an equational characterization for L-domains in [7]. There is an open problem about equational characterization in [8]: is there an equational characterization for continuous semilattices?

In this paper, we first introduce a concept of an A D M dcpo. Then we prove that continuous A D M dcpos can be characterized by an equation for some given A a family of some subsets of the dcpo. In particular, a subclass of continuous semilattices can be characterized in this way. This partially solves the problem presented in [8]. Moreover, we obtain an equational characterization for a special class of domains including A D M domains, by using ideals instead of directed sets. However, we can not characterize general domains in this way, especially for FS-domains. At last, we give some examples about these characterizations.

2 Preliminaries

In this section, we recall some definitions and results related to the paper. A partially ordered set is a nonempty set equipped with a partial order ≤, where the partial order is a binary relation satisfying transitive, reflexive and antisymmetric. The term poset will be used to denote a partially ordered set.

Definition 2.1.

[9]

Let L be a set equipped a partial order ≤ and A be a subset of L.

A partial order ≤_A on A is called the hereditary order, if ≤_A = ≤∩(A × A).
An element x ∈ L is called an upper bound of A, if a ≤ x for all a ∈ A. Respectively, an element y ∈ L is called a lower bound of A, if y ≤ a for all a ∈ A. For x ∈ L, we write ↓ x = {y ∈ L : y ≤ x} and ↑ x = {z ∈ L : x ≤ z}.
An element x ∈ L is called the least upper bound of A, if x ≤ y for each upper bound y of A. And we write it as ⋁A or sup A. Respectively, p is called the greatest lower bound of A, if q ≤ p for each lower bound q of A. The greatest lower bound is written as ⋀A or inf A.
A is called a directed set, if A is nonempty and every finite subset of A has an upper bound in A. Respectively, A is called a filtered set, if A is nonempty and every finite subset of A has a lower bound in A.
A is called a lower set, if A = {x ∈ L : x ≤ a for some a ∈ A}. Respectively, A is called an upper set, if A = {y ∈ L : a ≤ y for some a ∈ A}.
A is called a ideal, if A is a directed lower set. A is called a filter, if A is a filtered upper set.
A is called a principal idea, if A is an ideal with ⋁A ∈ A. A is called a principal filter, if A is a filter with ⋀A ∈ A.

Definition 2.2.

[10]

A complete lattice is a poset in which every subset has a sup and an inf.
A poset is called a complete semilattice if every nonempty subset has an inf and every directed subset has a sup.
A poset is called a dcpo if every directed subset has a sup.
A dcpo is called an L-dcpo if every principal ideal equipped with its hereditary order is a complete lattice.

Definition 2.3.

[10]

Let L be a poset. We say that x is way-below y, in symbols x ≪ y, iff for all directed subsets D ⊆ L for which sup D exists, the relation y ≤ sup D always implies the existence of a d ∈ D with x ≤ d. For each x ∈ L, we denote by ↡ x the set of all elements are way-below x.
A poset L is called continuous if it satisfies the axiom of approximation:
( ∀ x ∈ L ) x = ⋁ ↑ ↡ x ,
i.e. for all x ∈ L, the set ↡x = {u ∈ L∣u ≪ x} is directed and x = ⋁{u ∈ L∣u ≪ x}.
A dcpo which is continuous is called a domain.
A domain which is a semilattice is called a continuous semilattice.
A domain which is also an L-dcpo is called an L-domain.
A domain which is a complete lattice is called a continuous lattice.
A domain which is a complete semilattice is called a bounded complete domain.
An element x of L is called a compact element, if for each directed set D of L with ⋁D exists, x ≤ ⋁D always implies x ≤ d for some d ∈ D (i.e., x ≪ x). Denotes K(L) as the set of all compact elements.
A poset L is called algebraic if it satisfies the axiom of approximation:
( ∀ x ∈ L ) x = ⋁ ↑ ( ↡ x ∩ K ( L ) ) ,
i.e. for all x ∈ L, the set (↡ x ∩ K(L)) is directed and x = ⋁(↡ x ∩ K(L)).

Definition 2.4.

[11] Let L be a poset. A topology τ on L is called the Alexandrov topology, if τ is the set of all upper set of L.

For a poset L and x ∈ L, let J ( x ) = { I ∈ I d ( L ) ∣ x ≤ sup I } where Id(L) is the set of all ideals of L. The following propositions are excerpted from [8], Proposition I-1.5, Proposition I-4.3].

Proposition 2.5.

If L is an algebraic domain, then L is a domain.

Proposition 2.6.

Let L be a poset. Then the following conditions are equivalent:

y ≪ x;
y ∈ ⋂ J ( x ) .

Theorem 2.7.

[4] Let L be a complete semilattice. Then the following conditions are equivalent.

L is continuous.
Let {x _j,k∣j ∈ J, k ∈ K(j)} be a nonempty family of elements in L such that {x _j,k∣k ∈ K(j)} is directed for each j ∈ J. Then the following identity holds:
⋀ j ∈ J ⋁ k ∈ K ( j ) ↑ x j , k = ⋁ f ∈ M ↑ ⋀ j ∈ J x j , f ( j ) ,
where K(j) is a index set for any j ∈ J and M is the set of all choice functions f : J → ⋃ _j∈J K(j) with f(j) ∈ K(j) for all j ∈ J.

If L is a complete lattice, then these conditions are also equivalent to
Let {x _j,k ∈ J × K} be any family in L. Then the following identity holds:
⋀ j ∈ J ⋁ k ∈ K x j , k = ⋁ f ∈ N ↑ ⋀ j ∈ J ⋁ k ∈ f ( j ) x j , k ,
where N denotes the set of all choice functions f from J into the finite subsets of K, i.e., f : J → fin(K).

Next, we recall the definition of connectedness in order theory.

Definition 2.8.

[12] Let P be a poset. Then P is called connected, if every two elements x, y can be connected by a zigzag in P, i.e. there is n ∈ N and there are x ₀, …, x _n, y ₀, …, y _n ∈ P such that x = x ₀, y = x _n and x _i ≤ y _j whenever 0 ≤ j − i⩽1. A subset A of P is called a connected set, if A is connected as a subspace of P.

The following theorem is an excerpt of [7], Proposition 1.3.3].

Theorem 2.9.

Let L be an L-dcpo. Then the following conditions are equivalent.

L is continuous.
Let { D j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J be a family of directed subsets of L such that {⋁D _j∣j ∈ J} is a connected set. Then the following identity holds:
⋀ j ∈ J ⋁ k ∈ K ( j ) ↑ x j , k = ⋁ f ∈ M ↑ ⋀ j ∈ J x j , f ( j ) ,
where M is the set of all choice functions f : J → ⋃ _j∈J K(j) with f(j) ∈ K(j) for all j ∈ J.

3 Main results

First, we propose some families of dcpos. For convenience, we denote by PFS(L) the family of all principal filters and singleton sets of a poset L.

Definition 3.1.

Let L be a dcpo and A be a family of sets with P F S ( L ) ⊆ A .

L is called an A M dcpo if inf A exists for each A ∈ A .
L is called an A D dcpo if for each family
{ D j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J
of directed subsets of L with { ⋁ D j ∣ j ∈ J } ∈ A , then there exists a choice function f : J → ⋃ _j∈J K(j) such that f(j) ∈ K(j) for all j ∈ J and { x j , f ( j ) ∣ j ∈ J } ∈ A .
L is called an A D M dcpo, if L is both an A M dcpo and an A D dcpo.

Example 3.2.

Let L be a complete lattice and A be the family of all subsets of L. Then L is an A D M dcpo.
Let L be a complete semilattice and A be the family of all nonempty subsets of L. Then L is an A D M dcpo.
Let L be an L-dcpo and A be the family of all connected sets of L. Then L is an A D M dcpo.
Let L be a dcpo and A be the family of all compact subsets of L with respect to Alexandrov topology. Then L is an A M dcpo if and only if L is a semilattice.

Now we give the main result of this paper.

Theorem 3.3.

Let L be a dcpo and A be a family of sets with P F S ( L ) ⊆ A . If L is an A D M dcpo, then the following are equivalent.

L is continuous.
If { D j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J is a family of directed sets with { ⋁ D j ∣ j ∈ J } ∈ A , then the following identity holds:
⋁ f ∈ M ↑ ⋀ j ∈ J x j , f ( j ) = ⋀ j ∈ J ⋁ k ∈ K ( j ) ↑ x j , k ,
where M is the set of all choice functions f : J → ⋃ _j∈J K(j) such that f(j) ∈ K(j) for all j ∈ J and { x j , f ( j ) ∣ j ∈ J } ∈ A .

Proof.

We prove that (1) implies (2).

To this end, we first claim that {⋀_j∈J x _j,f(j)∣f ∈ M} is directed. Assume f ₁, f ₂ ∈ M. For each j ∈ J, there is x j , k j ∈ D j with x j , f 1 ( j ) , x j , f 1 ( j ) ≤ x j , k j . Then { ↑ x j , k j ∩ D j } j ∈ J is a family of directed sets with

⋁ ( ↑ x j , k j ∩ D j ) ∣ j ∈ J ∈ A .

Take a choice function f : J → ⋃ _j∈J K(j) such that x j , f ( j ) ∈ ↑ x j , k j ∩ D j and { x j , f ( j ) ∣ j ∈ J } ∈ A . Then ⋀ j ∈ J x j , f 1 ( j ) , ⋀ j ∈ J x j , f 2 ( j ) ≤ ⋀ j ∈ J x j , f ( j ) and f ∈ M. Thus {⋀_j∈J x _j,f(j)∣f ∈ M} is directed.

Assume that { D j } j ∈ J = { x j , k ∣ k ∈ K ( j ) } j ∈ J is a family of directed sets with { ⋁ D j ∣ j ∈ J } ∈ A . For convenience, let lhs denote the left side of the equation of (2) and rhs denote the right one. For each f ₀ ∈ M and j ₀ ∈ J, x j 0 , f 0 ( j 0 ) ≤ ⋁ k ∈ K ( j 0 ) ↑ x j 0 , k . Thus it is clear that lhs ≤ rhs.

Assume y ≪rhs. Given j ∈ J, there is k _j ∈ K(j) such that y ≤ x j , k j . Because { ↑ x j , k j ∩ D j ∣ j ∈ J } is a family of directed sets with

⋁ ( ↑ x j , k j ∩ D j ) ∣ j ∈ J = ⋁ D j ∣ j ∈ J ∈ A ,

there is a choice function f ∈ M such that x j , k j ≤ x j , f ( j ) for each j ∈ J and { x j , f ( j ) ∣ j ∈ J } ∈ A . It follows that y ≤ ⋀_j∈J x _j,f(j). So rhs ≤ lhs.

Now we show that (2) implies (1).

Fix x ∈ L, let { I j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J denote all of ideals with sup in ↑ x. The equation of (2) holds, because { ⋁ I j ∣ j ∈ J } = ↑ x ∈ A . We claim that

⋀ j ∈ J x j , f ( j ) ∣ f ∈ M = ↡ x .

Since z ∈ ↡ x implies z ∈ ⋂_j∈J I _j, there is f ∈ M such that x _j,f(j) = z for each j ∈ J, due to the fact that A includes all the singleton sets. That is ↡ x ⊆{⋀_j∈J x _j,f(j)∣f ∈ M}. Conversely, for each j ₀ ∈ J and all f ∈ M, we have ⋀ j ∈ J x j , f ( j ) ≤ x j 0 , f ( j 0 ) . It follows that ⋀ j ∈ J x j , f ( j ) ∈ I j 0 . Hence, ⋀_j∈J x _j,f(j) ∈ ∩_j∈J I _j = ↡ x. In the equation, lhs = ⋁^↑↡ x and rhs = ⋀ ↑ x. Thus, x = ⋁^↑↡ x. Therefore, L is continuous.

Remark 3.4.

Note that we can characterize many domains by changing A . From the examples in Example 3.2, we obtain that Theorem 2.7 and Theorem 2.9 are two special cases of Theorem 3.3.

Taking bounded complete domains as an example, let L be a dcpo and A be the set of all nonempty subsets. Then L is a bounded complete domain if and only if L is an A D M dcpo satisfying Theorem 3.3.

For a special subclass of continuous semilattices, we have the following equational characterization.

Corollary 3.5.

Suppose that dcpo L is also a semilattice, and A is the family of all compact subsets of L with respect to Alexandrov topology. If L is an A D dcpo, then the following are equivalent.

L is continuous.
If { D j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J is a family of directed sets with { ⋁ D j ∣ j ∈ J } ∈ A , then the following identity holds:
⋁ f ∈ M ↑ ⋀ j ∈ J x j , f ( j ) = ⋀ j ∈ J ⋁ k ∈ K ( j ) ↑ x j , k ,
where M is the set of all choice functions f : J → ⋃ _j∈J K(j) such that f(j) ∈ K(j) for all j ∈ J and { x j , f ( j ) ∣ j ∈ J } ∈ A .

Proof.

Since a compact set of the semilattice L with the Alexandrov topology always has an inf, L is an A M dcpo. Thus, we conclude that (1) equivalent to (2) by Theorem 3.3.

Meet continuous is a distributivity on directed complete semilattice. In [13], Hui Kou stated that the property can be extended to general dcpos and the characterization uses sets to replace points. By using ideals instead of directed sets, we will obtain an equational characterization for a subclass of domains, including that of domains mentioned above. For convenience, we denote by PF(L) the family of all principal filters of a poset L.

Definition 3.6.

Let L be a dcpo and A be a family of sets with P F ( L ) ⊆ A . Then L is called an A I M dcpo, if

⋂ j ∈ J I j ∈ I d ( L )

for each { I j } j ∈ J ⊆ I d ( L ) with { ⋁ I j ∣ j ∈ J } ∈ A .

Proposition 3.7.

Let L be a dcpo and A be a family of sets with P F S ( L ) ⊆ A . If L is an A D M dcpo, then L is an A I M dcpo.

Proof.

Assume that { I j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J is a family of ideals with { ⋁ I j ∣ j ∈ J } ∈ A . Since L is an A D dcpo, there exists f : J → ⋃ _j∈J K(j) such that f(j) ∈ K(j) for all j ∈ J and { x j , f ( j ) ∣ j ∈ J } ∈ A . Thus, ⋀_j∈J x _j,f(j) exists and ⋀_j∈J x _j,f(j) ∈ ⋂_j∈J I _j. That is ⋂_j∈J I _j ≠ ∅.

Fix a, b ∈ ⋂_j∈J I _j. Then there exists k _j ∈ K(j) such that a , b ≤ x j , k j for each j ∈ J. So { ↑ x j , k j ∩ I j ∣ j ∈ J } is a family of directed sets with { ⋁ ( ↑ x j , k j ∩ I j ) ∣ j ∈ J } ∈ A . And there exists a choice function f : J → ⋃ _j∈J K(j) such that x j , f ( j ) ∈ ( ↑ x j , k j ∩ I j ) for each j ∈ J and { x j , f ( j ) ∣ j ∈ J } ∈ A . It follows that ⋀_j∈J x _j,f(j) ∈ ⋂_j∈J I _j and a, b ≤ ⋀_j∈J x _j,f(j). Therefore, ⋂_j∈J I _j is an ideal.

Theorem 3.8.

Let L be a dcpo and A be a family of sets with P F ( L ) ⊆ A . If L is an A I M dcpo, then the following conditions are equivalent.

L is continuous.
If { I j } j ∈ J ⊆ I d ( L ) with {⋁I _j∣j ∈ J} = ↑x for some x ∈ L, then the following identity holds:
↓ ( ⋁ ⋂ j ∈ J I j ) = ⋂ j ∈ J ↓ ( ⋁ I j ) .

Proof.

(1) implies (2): Let { I j } j ∈ J ⊆ I d ( L ) with { ⋁ I j ∣ j ∈ J } ∈ A . We have ↓(⋁⋂_j∈J I _j) ⊆⋂_j∈J ↓(⋁I _j), since ⋁ ⋂ j ∈ J I j ≤ ⋁ I j 0 for each j ₀ ∈ J. Conversely, for each y ∈ ⋂_j∈J ↓(⋁I _j). Assume z ∈ ↡ y. Then z ∈ I _j for each j ∈ J, due to y ≤ ⋁I _j. It follows that ⋁_z∈↡y z ≤ ⋁⋂_j∈J I _j. Since L is a domain, we have that y = ⋁↡ y, Thus, y ∈ ↓(⋁⋂_j∈J I _j).

(2) implies (1): Assume x ∈ L. Let { I j } j ∈ J = { I ∈ I d ( L ) ∣ x ≤ ⋁ I } . By Proposition 2.6, ⋂_j∈J I _j = ↡ x. So we only need to prove that x ∈ ↓(⋁⋂_j∈J I _j). It is clear that x ∈ ⋂_j∈J ↓(⋁I _j) by the definition of { I j } j ∈ J . Since { ⋁ I j ∣ j ∈ J } ∈ A , we have

↓ ⋁ ⋂ j ∈ J I j = ⋂ j ∈ J ↓ ⋁ I j .

Then the conclusion is proved.

4 Some examples

In this section, we will give some examples. First, we illustrate that there is a poset which belongs to the subclass of continuous semilattice discussed in Corollary 3.5. Before this, we give the following concept of an A-domain.

Definition 4.1.

A poset L is called an A-domain if ↑ x is a finite lattice for each x ∈ L under the hereditary order.

Lemma 4.2.

If L is an A-domain, then L is an algebraic domain.

Proof.

Assume D is a directed set of L. Fix d ∈ D, then ↑d ∩ D is a finite directed set. Hence, ⋁(↑d ∩ D) exists in D. Moreover, all elements in L are compact elements because ⋁D exists in D for each directed set D. It follows that L is algebraic.

Example 4.3.

Let L = { F ⊆ N ∣ F i s fi n i t e } be ordered by reverse inclusion and A be all of compact subsets of L with respect to Alexandrov topology. Then L is an A D M dcpo and an A-domain.

Lemma 4.4.

Let L be an A-domain, and A be the family of all the principal filters and singleton sets. Then L is an A D M dcpo.

Proof.

Clearly, L is an A M dcpo. Assume { D j } j ∈ J = { { x j , k ∣ k ∈ K ( j ) } } j ∈ J is a family of directed sets with { ⋁ D j ∣ j ∈ J } ∈ A . We can choose x j , k j = ⋁ D j for each j ∈ J. And there is a choice function f : J → ⋃ _j∈J K(j) given by f(j) = k _j for each j ∈ J. Then

{ x j , f ( j ) ∣ j ∈ J } = ⋁ D j ∣ j ∈ J ∈ A .

Thus, L is an A D M dcpo.

Thus each A-domain L can seen be as an A D M dcpo where A = P F S ( L ) . The following example indicates that there is an A D M dcpo, which may be not an L-domain or a continuous semilattice.

Example 4.5.

Let L = {a, b, c} be a set. Define a relation ≤ on L by

x ≤ y i ff x = y o r y = c .

Obviously, ≤ is a partial order on L and (L, ≤) is an A-domain. But (L, ≤) is neither a continuous semilattice nor an L-domain.

Finally, we propose the following example to reveal that the method of using ideals can not be used to characterize all of domains.

Example 4.6.

Let F = { n n + 1 ∣ n ∈ N } ⋃ { 1 } and L = {{0, 1} × F}⋃{F × {0}}. We define a partial order ≤* on L below:

( a 1 , b 1 ) ≤ * ( a 2 , b 2 ) iff a 2 ∉ F , b 1 ≤ b 2 , a 1 = 0 ⋅ ⋅ ⋅ ⋅ ≤ 1 , , a 2 = 0 , a 1 ≤ b 2 , a 1 ∈ F ⋅ ⋅ ⋅ ⋅ ≤ 2 , , a 2 ∈ F , a 1 ≤ a 2 , a 1 ∈ F ⋅ ⋅ ⋅ ⋅ ≤ 3 , , a 2 = 1 , a 1 ∈ F ⋅ ⋅ ⋅ ⋅ ≤ 4 , , a 2 = 1 , b 1 ≤ b 2 , a 1 = 1 ⋅ ⋅ ⋅ ⋅ ≤ 5 , ,

where “ ≤ ” is the usual order on R .

We now prove that the relation ≤* forms a partial order.

We first prove that ≤* is reflexive. Given (a, b) ∈ L.

a = 0, b ∈ F. Then (a, b) ≤*(a, b) by ≤ 1 * .
a = 1, b ∈ F. Then (a, b) ≤*(a, b) by ≤ 5 * .
a ∈ F, b = 0. Then (a, b) ≤*(a, b) by ≤ 3 * .

Secondly, we prove that ≤* is antisymmetric. Given (a, b), (c, d) ∈ L with (a, b) ≤*(c, d) and (c, d) ≤*(a, b).

( a , b ) ≤ 1 * ( c , d ) . Then c = 0 or 1. By ≤ 4 * , ≤ 5 * , we have c ≠ 1. Hence c = 0 and it follows that ( a , b ) ≤ 1 * ( c , d ) and ( c , d ) ≤ 1 * ( a , b ) . It implies that a = c = 0, b = d.
( a , b ) ≤ 2 * ( c , d ) . Then c = 0. Because (c, d) ≤*(a, b), we have ( c , d ) ≤ 1 * ( a , b ) and a∉F, a contradiction.
( a , b ) ≤ 3 * ( c , d ) . Then b = d = 0. And c = d, by ( c , d ) ≤ 3 * ( a , b ) . That is a = c.
( a , b ) ≤ 4 * ( c , d ) . Then c = 1. By ( c , d ) ≤ 5 * ( a , b ) , we have a = 1, a contradiction.
( a , b ) ≤ 5 * ( c , d ) . Then a = c = 1. Since ( c , d ) ≤ 5 * ( a , b ) , it holds that b = d.

Finally, we prove the transitivity of ≤*. Given (a, b) ≤*(c, d) ≤*(e, f).

( a , b ) ≤ 1 * ( c , d ) , ( c , d ) ≤ 1 * ( e , f ) .

Then a = c = 0, e = 0 or 1, b ≤ d ≤ f. And (a, b) ≤*(e, f), by ≤ 1 * .

( a , b ) ≤ 1 * ( c , d ) , ( c , d ) ≤ 5 * ( e , f ) .

Then a = 0, c = e = 1, b ≤ d ≤ f. And (a, b) ≤*(e, f), by ≤ 1 * .

( a , b ) ≤ 2 * ( c , d ) , ( c , d ) ≤ 1 * ( e , f ) .

Then a ≤ d, c = 0, e = 0 or 1, d ≤ f.

If e = 0, then ( a , b ) ≤ 2 * ( e , f ) .

If e = 1, then ( a , b ) ≤ 4 * ( e , f ) .

( a , b ) ≤ 3 * ( c , d ) .

If ( c , d ) ≤ 2 * ( e , f ) , then e = 0 and a ≤ c ≤ f, i.e., ( a , b ) ≤ 2 * ( e , f ) .

If ( c , d ) ≤ 3 * ( e , f ) , then a ≤ c ≤ e and ( a , b ) ≤ 3 * ( e , f ) .

If ( c , d ) ≤ 4 * ( e , f ) , then e = 1 and ( a , b ) ≤ 4 * ( e , f ) .

( a , b ) ≤ 4 * ( c , d ) , ( c , d ) ≤ 5 * ( e , f ) .

Then c = e = 1 and ( a , b ) ≤ 4 * ( e , f ) .

( a , b ) ≤ 5 * ( c , d ) , ( c , d ) ≤ 5 * ( e , f ) .

Then a = c = e = 1 and b ≤ d ≤ f. That is ( a , b ) ≤ 5 * ( e , f ) .

Hence ≤* is a partial order.

To understand the above dcpo L more intuitively, we can see Figure 1.

For r ∈ F and (a, b) ∈ L, we set ↓ r = { x ∈ R : x ≤ r in R } , ↓*(a, b) = {(c, d) ∈ L : (c, d) ≤*(a, b)} and ↑*(a, b) = {(c, d) ∈ L : (a, b) ≤*(c, d)}.

There is an approximate identity { h n } n ∈ N consisting of finitely separating functions defined below.

h n ( ( a , b ) ) = 0 , max ↓ b ∩ ↓ n n + 1 , a = 0 , max ↓ a ∩ ↓ n n + 1 , 0 , b = 0 , 1 , max ↓ b ∩ ↓ n n + 1 , e l s e .

Thus L is an FS-domain, and also a domain.

Let A be all of principal filters of L. Then L \{(1, 1), (0, 1)} and ↓*(0, 1) are two ideals, and

↑ * ( 0,1 ) = { ⋁ ( L \ { ( 1,1 ) , ( 0,1 ) } ) , ⋁ ↓ * ( 0,1 ) } ∈ A .

However, L \{(1, 1), (0, 1)}∩↓(0, 1) is not an ideal. It shows that L is not an A I M dcpo. Therefore, L is not an A D M dcpo, even though L is an FS-domain.

$Figure 1: The poset L in Example 4.6.$

Figure 1:

The poset L in Example 4.6.

Example 4.6 also tells us that an equational characterization for FS-domains has not been established. So it is a good follow-up effort to resolve the above question.

Corresponding author: Xiangrui Li, School of Mathematics, Hunan University, Changsha, Hunan, 410082, China, E-mail: 17731912783@163.com

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. All authors contributed equally in this work.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: Authors state no conflicts of interest.
Research funding: None declared.
Data availability: Data sharing is not applicable for this article since no dataset was created or analyzed during the study.

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Received: 2025-02-20

Accepted: 2025-11-07

Published Online: 2025-12-19

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https://doi.org/10.1515/math-2025-0219

Schlagwörter für diesen Artikel

domains; continuous semilattices; equational characterization; ideals; family of dcpos

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