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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Directed colimits of some flatness properties and purity of epimorphisms in S-posets

Xingliang Liang
• Corresponding author
• Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, Shaanxi, China
• Email
• Other articles by this author:
/ Roghaieh Khosravi
Published Online: 2018-07-04 | DOI: https://doi.org/10.1515/math-2018-0061

## Abstract

Let S be a pomonoid. In this paper, we introduce some new types of epimorphisms with certain purity conditions, and obtain equivalent descriptions of various flatness properties of S-posets, such as strong flatness, Conditions (E), (E′), (P), (Pw), (WP), (WP)w, (PWP) and (PWP)w. Thereby, we present other equivalent conditions in the Stenström-Govorov-Lazard theorem for S-posets. Furthermore, we prove that these new epimorphisms are closed under directed colimits. Meantime, this implies that by a new approach we can show that most of flatness properties of S-posets can be transferred to their directed colimit. Finally, we prove that every class of S-posets having a flatness property is closed under directed colimits.

MSC 2010: 06F05; 20M50

## 1 Introduction and preliminaries

Motivated by the work of Lazard and Govorov for modules over a ring, Stenström in 1971 introduced the concept of pure epimorphisms and established the Stenström-Govorov-Lazard theorem in the context of S-acts (see [1]). Based on the method of this theorem, Normak then in [2], used 1-pure epimorphisms to obtain some equivalent descriptions of Condition (E). Recently, Bailey and Renshaw in [3] continued the investigation of purity of epimorphisms. They found that there are some connections between some flatness properties of S-acts and purity of epimorphisms (for example, [3, Propositions 3.11, 3.12]). They also proved that every surjective S-act morphism is pure if and only if it is a directed colimit of split epimorphisms. Moreover, in [4] the authors proved tha every class of S-acts having a flatness property is closed under directed colimits.

In 2005, Bulman-Fleming and Laan [5] introduced the ordered analogues of pure epimorphisms and directed colimits, and extended the Stenström-Govorov-Lazard theorem to S-posets. From the Stenström-Govorov-Lazard theorem for S-posets, we see that a right S-poset AS is strongly flat, which means that AS has both Condition (P) and Condition (E), if and only if every surjective S-poset morphism BSAS is pure. During recent years, a number of papers on flatness properties of S-posets have appeared, but vast majority of them have focused on the homological classification of pomonoids (see, for example, [6-9]). Up to now, the research on purity conditions of epimorphisms which are used to characterize flatness properties of S-posets (with the exception of strong flatness), has not been completed. The present paper addresses some versions of this matter. As applications of the results in this paper, some results on S-acts can be also obtained.

Preliminary work on flatness properties of S-posets was done by Fakhruddin in the 1980s (see [10, 11]). For background material on S-posets, the reader may consult [6, 12, 13] and the references cited therein.

Let S be a pomonoid. A nonempty poset A is called a right S-poset if there exists a right action A × SA, (a, s) ↦ as, which satisfies that (1) the action is monotonic in each of the variables, and (2) a(st) = (as)t and a1 = a for all aA and s, tS. Left S-posets are defined analogously. The notations AS and SB will respectively denote a right and left S-poset, and ΘS = {θ} is the one-element right S-poset. By an S-poset morphism, we mean a monotone map between S-posets which preserves the S-action. We denote the category of all right (resp., left) S-posets, with S-poset morphisms between them, by Pos-S (resp., S-Pos).

Let AS be a right S-poset. An S-poset congruence θ on AS is an S-act congruence that has the further property that the factor act A/θ can be equipped with a compatible order so that the natural map AA/θ is an S-poset morphism. Because for a given congruence θ the factor S-act A/θ may support several different compatible orders, Bulman-Fleming and Laan in [5] provided a detailed treatment to the factor S-posets A/θ, the essential part was repeated as follows:

Suppose that α is any binary relation on AS that is reflexive, transitive, and compatible with the S-action. For a, a′ ∈ A, we write $\begin{array}{}a\underset{\alpha }{\le }{a}^{\prime }\end{array}$ if a so-called α-chain

$a≤a1 α a1′≤a2 α a2′≤⋯≤an α an′≤a′$

from a to a′ exists in AS, where each ai, $\begin{array}{}{a}_{i}^{\prime }\end{array}$A for 1 ≤ in. It can be shown that an S-act congruence θ on an S-poset AS is an S-poset congruence if and only if a θ a′ whenever $\begin{array}{}a\underset{\theta }{\le }{a}^{\prime }\underset{\theta }{\le }a.\end{array}$

The next concept will be used frequently in this paper. Let AS be a right S-poset and let HA × A. The S-poset congruence on AS induced by H is described as follows:

$a ν(H) a′ if and only if a≤α(H)a′≤α(H)a,$

where a α(H) a′ if and only if a = a′ or

$a=x1s1,y1s1=x2s2,⋯,yn−1sn−1=xnsn,ynsn=a′$

for some (xi, yi) ∈ H and siS, i = 1, ⋯, n. The order relation on A/ν(H) is given by

$[a]ν(H)≤[a′]ν(H) if and only if a≤α(H)a′.$

One can show that ν(H) is the smallest congruence on AS such that [x]ν(H) ≤ [x′]ν(H) whenever (x, x′) ∈ H. The S-poset congruence on AS generated by H is θ(H) = ν(HHop), and as usual is the smallest congruence on AS that contains H.

A right S-poset AS is called cyclic if A = aS for some aA. It is clear that AS is cyclic if and only if AS is isomorphic to S/ρ for some right S-poset congruence ρ on SS. A congruence ρ on an S-poset AS is called finitely generated if ρ = ν(H) for some finite subset H of A × A. An S-poset AS is called finitely presented if it is isomorphic to F/ν(H) for some finitely generated free S-poset FS and some finite subset HF × F.

Conditions (E), (P) and (Pw) are formulated below (see for example [5] and [9]).

$(E):(∀a∈A)(∀s,s′∈S) (as≤as′⟹(∃a′∈A)(∃u∈S)(a=a′u∧us≤us′)).(P):(∀a,a′∈A)(∀s,s′∈S) (as≤a′s′⟹(∃a″∈A)(∃u,v∈S)(a=a″u∧a′=a″v∧us≤vs′)).(Pw):(∀a,a′∈A)(∀s,s′∈S) (as≤a′s′⟹(∃a″∈A)(∃u,v∈S)(a≤a″u∧a″v≤a′∧us≤vs′)).$

It is clear that Condition (P) implies Condition (Pw). According to [9], an S-poset AS is said to be strongly flat if it satisfies Conditions (E) and (P). Equivalently, an S-poset AS is strongly flat if and only if it is subpullback flat and subequalizer flat [that is, the functor AS ⊗ – (from the category of left S-posets to the category of posets) preserves subpullbacks and subequalizers ].

Weak and principal weak versions of Conditions (P) and (Pw) are defined in [7] as follows. An S-poset AS is said to satisfy Condition (WP) if the corresponding ϕ is surjective for every subpullback diagram P(I, I, f, f, S), where I is a left ideal of S. An S-poset AS is said to satisfy Condition (WP)w if, for all elements s, tS, all homomorphisms f: S(SsSt) → SS, and all a, a′ ∈ A, af(s) ≤ af(t) implies asa″⊗us′ and a″⊗vt′ ≤ a′⊗t in AS (SsSt) for some a″ ∈ A, u, vS and s′, t′ ∈ {s, t} with f(us′) ≤ f(vt′). An S-poset AS is said to satisfy Condition (PWP) if the corresponding ϕ is surjective for every subpullback diagram P(Ss, Ss, f, f, S), sS. An S-poset AS is said to satisfy Condition (PWP)w if, for all a, a′ ∈ A, sS, asas implies aau, ava′ for some a″ ∈ A, u, vS with usvs.

The definitions of free and projective S-posets can be found, for example, in [9, 14].

By [7] and [9], the relations of the above-mentioned properties are as follows.

The rest of this paper is organized as follows. In Section 2, we first introduce the concept of 1-pure epimorphisms for S-posets, and obtain an alternative description of Condition (E) which is the ordered version of [2, Proposition 2.2]. In particular, we can give a more brief characterization for Condition (E) by using a new purity epimorphism. Further, characterizations for Condition (E) may serve as a template for Condition (E′), as we show that Condition (E′) is equivalent to certain purity conditions of epimorphisms. We also consider similar questions to Conditions (P), (Pw), (WP), (WP)w, (PWP) and (PWP)w. Thereby, we obtain other equivalent conditions in the Stenström-Govorov-Lazard theorem for S-posets. In Section 3, we initiate a study of directed colimits of purity epimorphisms for S-posets, and prove that these new epimorphisms introduced in Section 2 are closed under directed colimits. Then, we deduce that an S-poset epimorphism is pure if and only if it is a directed colimit of split epimorphisms. In Section 4, we consider the behavior of subequlizers and subpullback diagrams under directed colimits. Finally, we show that flatness properties transferred under directed colimits.

## 2 Purity of epimorphisms for S-posets

In this section, we give equivalent descriptions of some flatness properties of S-posets in accordance with certain purity conditions of epimorphisms.

We first recall the concept of pure epimorphisms for S-posets.

Let ψ : BSAS be a surjective S-poset morphism. We say that ψ is a pure (m-pure) epimorphism [5] if, for every m ∈ ℕ, a1, ⋯, amA and relations

$aαisi≤aβiti(i=1,2,⋯,l),$

where αi, βi ∈ {1, ⋯, m} and si, tiS, there exist b1,⋯, bmB such that ψ(br) = ar for all 1 ≤ rm and bαisibβiti for all i.

We are interested here in the cases m = 1 and m = 2. In both cases, we shall call ψ 1-pure and 2-pure, respectively. It is easy to check that every 2-pure epimorphism is 1-pure, but the following example shows that the converse is not true in general.

#### Example 2.1

Let S = {0, 1} with the usual order. Let

$A={x,y,z | x⋅0=y⋅0=z⋅0=z, x⋅1=x, y⋅1=y, z⋅1=z}$

with the only nontrivial order relation is z < x and z < y, and

$B={a,b,c| a⋅0=a⋅1=a, b⋅0=b⋅1=b, c⋅0=c⋅1=c}$

with the discrete order. Define an S-poset epimorphism ψ : BSAS by

$ψ(a)=x, ψ(b)=y, ψ(c)=z.$

It is not hard to verify that ψ is 1-pure. However, it is not 2-pure, since x · 0 ≤ y · 1 but a · 0 ≰ b · 1.

We begin our investigation with Condition (E). The following proposition is an ordered analogue of [2, Proposition 2.2]. The technique of the proof is taken from the unordered case.

#### Proposition 2.2

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS has Condition (E).

2. Every surjective S-poset morphism BSAS is 1-pure.

3. There exists a 1-pure epimorphism BSAS where BS is subequalizer flat.

4. Every S-poset morphism BSAS where BS is a finitely presented cyclic S-poset may be factorized through a free S-poset.

5. Every S-poset morphism BSAS where BS is a finitely presented cyclic S-poset may be factorized through a subequalizer flat S-poset.

The next proposition gives another description of Condition (E), whose formulation is quite brief. To do this, we require the following.

#### Definition 2.3

Let ψ : BSAS be a surjective S-poset morphism. We say that φ is quasi-1-pure if, for every aA and asat, there exists bB such that ψ(b) = a and bsbt.

#### Proposition 2.4

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS has Condition (E).

2. Every surjective S-poset morphism BSAS is quasi-1-pure.

3. There exists a quasi-1-pure epimorphism BSAS where BS has Condition (E).

#### Proof

(1) ⇒ (2). Suppose that ψ : BSAS is an S-poset epimorphism. Let asat in AS for aA and s, tS. Since AS satisfies Condition (E), there exist uS and a′ ∈ A such that a = au and usut. Applying surjectivity of ψ, obtains bB with ψ(b) = a′. Further, we have busbut in BS, and a = au = ψ(bu), as required.

(2) ⇒ (3). From [5, Proposition 2.4] we know that every S-poset is isomorphic to the quotient of a free S-poset. So, without loss of generality, we can always choose an S-poset BS such that ψ : BSAS is an S-poset epimorphism and BS satisfies Condition (E). Then by (2), ψ is quasi-1-pure.

(3) ⇒ (1). Let AS be a right S-poset. Then by (3) there exists a quasi-1-pure epimorphism ψ : BSAS with BS satisfies Condition (E). Now assume asat in AS for aA, s, tS. Then there exists b′ ∈ B such that bsbt in BS, and ψ(b′) = a. Further, because BS satisfies Condition (E), from the inequality bsbt, yields bB and uS such that b′ = bu and usut. Consequently, a = ψ(b′) = ψ(bu) = ψ(b)u in AS, and the proof is complete. □

Similar considerations apply to a generalization of Condition (E) for S-posets.

#### Definition 2.5

A right S-poset AS satisfies Condition (E′) if, for all aA, s, s′, zS with asasand sz = sz, there exist a′ ∈ A, uS such that a = au and usus′.

It is immediate from the above definition that Condition (E) implies Condition (E′), but we will see in the sequel that the converse is not true in general.

#### Definition 2.6

Let ψ : BSAS be a surjective S-poset morphism. We say ψ is

1. 1′-pure if, for every aA and relations

$asi≤atiandsizi=tizi(i=1,⋯,n),$

there exists bB such that ψ(b) = a and bsibti for all i;

2. quasi-1′-pure if, for every aA and relations

$as≤atandsz=tz,$

there exists bB such that ψ(b) = a and bsbt.

The following lemma is useful to characterize Condition (E′).

#### Lemma 2.7

A surjective S-poset morphism ψ : BSAS is 1′-pure if and only if for every cyclic right S-poset S/ρ and every S-poset morphism α : S/ρAS, where ρ = ν(H) for some set H of the form

$H={(si,ti)∈S×S | sizi=tizi for some zi∈S, i=1,2,⋯,n},$

there exists an S-poset morphism β : S/ρBS such that the diagram

commutes.

#### Proof

Necessity. Suppose that ψ : BSAS is a 1′-pure epimorphism and suppose that α : S/ρAS is an S-poset morphism, where ρ is an S-poset congruence on SS induced by a set of the form {(si, ti) ∈ S × S | sizi = tizi for some ziS, i = 1, 2, ⋯, n}. Then we have α([1])siα([1])ti in AS and sizi = tizi for all i. By the assumption, there is an element bB such that α([1]) = ψ(b) and bsibti for all i. Now we define a mapping β : S/ρBS by β([s]) = bs. It is easy to see that β is a well-defined S-poset morphism such that α = ψβ.

Sufficiency. Let ψ : BSAS be an S-poset epimorphism and let asiati and sizi = tizi for aA, si, ti, ziS, i = 1, 2, ⋯, n. Consider the S-poset congruence ρ on SS induced by the set {(si, ti)| i = 1, 2, ⋯, n}. Then α : S/ρAS, given by α([s]) = as for sS, is a well-defined S-poset morphism. By the assumption, there exists an S-poset morphism β : S/ρBS such that α = ψβ. Thus, we can deduce a = α([1]) = ψ(β([1])), and β([1])si = β([si]) ≤ β([ti]) = β([1])ti in BS for all i, exactly as needed. □

For the sake of completeness we now prove the following

#### Proposition 2.8

For any right S-poset AS, the following statements are equivalent.

1. AS has Condition (E′).

2. Every surjective S-poset morphism BSAS is 1′-pure.

3. There exists a 1′-pure epimorphism BSAS, where BS has Condition (E′).

4. Every S-poset morphism S/ρAS may be factorized through a free right S-poset, where ρ is as in Lemma 2.7.

5. Every surjective S-poset morphism BSAS is quasi-1′-pure.

6. There exists a quasi-1′-pure epimorphism BSAS, where BS has Condition (E′).

#### Proof

(1) ⇒ (2). Let, first, ψ : BSAS be a surjective S-poset morphism, and let asiati and sizi = tizi for some aA and si, ti, ziS, i = 1, 2, ⋯, n. We start by applying Condition (E′) for i = 1, and obtain a1A and u1S with a = a1u1 and u1s1u1t1. We substitute the expression a = a1u1 into the relation as2at2 (for i = 2), and get a1u1s2a1u1t2 and u1s2z2 = u1t2z2. Again using Condition (E′), yields a2A and u2S such that a1 = a2u2 and u2u1s2u2u1t2, and so we have a = a2u2u1. Continuing in this way, we end up with expressions a = anunu1 and (unu1)sn ≤ (unu1)tn (for i = n). Denoting u = unu1, we have a = anu and usiuti for all i. Applying surjectivity of ψ, there exists bB with ψ(b) = an. Therefore, a = anu = ψ(bu), and busibuti in BS for all i, as desired.

The implications (2) ⇒ (3) and (5) ⇒ (6) follow from [5, Proposition 2.4].

The implications (2) ⇒ (5) and (3) ⇒ (6) are all clear.

(3) ⇒ (4). Suppose that α : S/ρAS is an S-poset morphism, where ρ = ν(H), for some set H = {(si, ti) ∈ S × S | sizi = tizi for some ziS, i = 1, 2, ⋯, n}. By (3) there exists a 1′-pure epimorphism ψ : BSAS, where BS has Condition (E′). In view of Lemma 2.7, there exists an S-poset morphism β : S/ρBS such that α = ψβ. Then we see β([1])siβ([1])ti in BS and sizi = tizi for all i. Also, as BS satisfies Condition (E′), there exist bB and uS with β([1]) = bu and usiuti for all i. So the mappings ϕ : S/ρSS and φ : SSAS, defined by ϕ([s]) = us and φ(s) = ψ(bs), respectively, are S-poset morphisms such that α = φϕ, as required.

(4) ⇒ (1). Let asat and sz = tz for aA and s, t, zS. We consider the S-poset congruence ρ on SS induced by the pair (s, t). Then the mapping α : S/ρAS, defined by α([u]) = au, is a well-defined S-poset morphism. By assumption there are a free S-poset BS and two S-poset morphisms β : S/ρBS and ψ : BSAS such that α = ψβ. Then [s] ≤ [t] implies β([1])sβ([1])t in BS. Since BS is free, but it must also be Condition (E′), from the last inequality and sz = tz we obtain bB and vS with β([1]) = bv and vsvt. Therefore, we can calculate ψ(b)v = ψ(bv) = ψβ([1]) = α([1]) = a, as required.

(6) ⇒ (1). It is similar to the implication (3) ⇒ (1) of Proposition 2.4. □

As mentioned before, Condition (E′) does not imply Condition (E). The example below is illustrative of the fact.

#### Example 2.9

Let S = {1, x|x2 = 1} with the order of S to be discrete. Clearly, S is a pomonoid. Now consider an S-poset epimorphism ψ : SSΘS. It is not hard to verify that ψ is quasi-1′-pure, and so ΘS satisfies Condition (E′) by Proposition 2.8. However, ψ is not quasi-1-pure, because θ · 1 ≤ θ · x, but there cannot exist sS such that ψ(s) = θ and s · 1 ≤ s · x. It follows from Proposition 2.4 that ΘS does not satisfy Condition (E).

As we known that the situations of Conditions (P) and (Pw) are unknown since now, here we wish to consider them.

#### Definition 2.10

Let ψ : BSAS be a surjective S-poset morphism. We say ψ is (weakly) quasi-2-pure, if for any a, a′ ∈ A and asat, there exist b, b′ ∈ B such that bsbt, a = (≤) ψ(b) and ψ(b′) = (≤) a′.

Note that, quasi-2-purity and quasi-1-purity are two unrelated notions. Indeed, on the one hand, it follows easily from Example 2.1 that quasi-1-purity does not imply quasi-2-purity. On the other hand, if S = x ∪ {0}, where x is the monogenic free monoid generated by x, equipped with the order in which

$1

and 0 is isolated. Then S is a pomonoid. Consider the natural epimorphism f : SSS/ρ(1, x), where ρ(1, x) is an S-poset congruence on SS generated by the pair (x, 1). Since x ρ(x, 1) 1, this implies [1]x ≤ [1]1, but there cannot exist uS\{0} with uxu, and so f is not quasi-1-pure. But, it is easy to check that f is quasi-2-pure.

Now, we provide equivalent descriptions of Conditions (P) and (Pw).

#### Proposition 2.11

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS has Condition (P) (Condition (Pw)).

2. Every surjective S-poset morphism BSAS is (weakly) quasi-2-pure.

3. There exists a (weakly) quasi-2-pure epimorphism BSAS where BS has Condition (P).

#### Proof

(1) ⇒ (2). We deal only with Condition (P), the proof for Condition (Pw) being similar. Now suppose that ψ : BSAS is an S-poset epimorphism and suppose that a, a′ ∈ A, s, tS are such that asat in AS. Since AS satisfies Condition (P), there exist u, vS and a″ ∈ A such that a = au, a′ = av and usvt. Applying surjectivity of ψ, we obtain bB with ψ(b) = a″. Thus, we have a = au = ψ(bu), a′ = av = ψ(bv), and busbvt in BS, as required.

(2) ⇒ (3). This is given by [5, Proposition 2.4].

(3) ⇒ (1). Let AS be a right S-poset. By (3) there exists a quasi-2-pure epimorphism ψ : BSAS, where BS satisfies Condition (P). Now suppose that asat in AS for a, a′ ∈ A, s, tS. Then we see that bsbt in BS, ψ(b′) = a′ and ψ(b) = a for some b, b′ ∈ B. Also, because BS satisfies Condition (P), from the inequality bsbt, we obtain b″ ∈ B and u, vS with b = bu, b′ = bv and usvt. Consequently, a = ψ(b) = ψ(bu) = ψ(b″)u and a′ = ψ(b′) = ψ(bv) = ψ(b″)v in AS, and the proof is complete. □

Since 2-pure epimorphisms are quasi-2-pure, we have the following corollary, which is the S-poset version of [3, Proposition 3.12]

#### Corollary 2.12

Let S be a pomonoid and let ψ : BSAS be an S-poset epimorphism in which BS has Condition (P). If ψ is 2-pure then AS has Condition (P).

From [9] we remark that Condition (P) implies Condition (Pw). But we can show that this implication is strict by using an example of a purity epimorphism.

#### Example 2.13

Let S = {1, x|x2 = x} with the order of S to be discrete. Let A = {a, b|a · 1 = a, a · x = b · x = b · 1 = b with the order a < b. Define an S-poset epimorphism f : SSAS by f(1) = a and f(x) = b. Since a · 1 ≤ b · x, and we want to reach 1 · 1 ≤ x · x, but this is impossible. Hence f is not quasi-2-pure, and so by Proposition 2.11, AS fails to satisfy Condition (P). However, it is easy to see that f is weakly quasi-2-pure, and so AS satisfies Condition (Pw) by Proposition 2.11.

Bulman-Fleming and Laan in [5] established the Stenström-Govorov-Lazard Theorem in the context of S-posets. Further, synthesizing Propositions 2.2, 2.4 and 2.11, we can deduce the following

#### Corollary 2.14

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS is strongly flat.

2. AS has Conditions (P) and (E).

3. Every surjective S-poset morphism BSAS is pure.

4. Every S-poset morphism BSAS where BS is finitely presented factors through a finitely generated, free S-poset.

5. AS is isomorphic to a directed colimit of a family of finitely generated, free S-posets.

6. AS is subpullback flat and subequalizer flat.

7. Every surjective S-poset morphism BSAS is 2-pure.

8. There exists a 2-pure epimorphism BSAS where BS is strongly flat.

9. Every surjective S-poset morphism BSAS is quasi-2-pure and (quasi-)1-pure.

10. There exists a quasi-2-pure and (quasi-)1-pure epimorphism BSAS where BS is strongly flat.

We make immediate use of the above ideas to the next propositions, which describe Conditions (WP), (WP)w, (PWP) and (PWP)w by certain purity epimorphisms.

#### Definition 2.15

Let ψ : BSAS be a surjective S-poset morphism. We say ψ is

• (weakly) quasi-w-2-pure if, for all elements a, a′ ∈ A, s, tS, all S-poset morphisms f : S(SsSt) → SS, and af(s) ≤ af(t), there exist b, b′ ∈ B and s′, t′ ∈ {s, t} such that bf(s′) ≤ bf(t′), as = (≤) ψ(b) ⊗ sand ψ(b′) ⊗ t′ = (≤) a′ ⊗ t in AS (SsSt).

• (weakly) quasi-pw-2-pure if, for any a, a′ ∈ A and asas, there exist b, b′ ∈ B such that bsbs, a = (≤) ψ(b) and ψ(b′) = (≤) a′.

Clearly, quasi-w-2-purity implies weakly quasi-w-2-purity, and quasi-pw-2-purity implies weakly quasi-pw-2-purity. But, it follows easily from Example 2.13 that these two implications are strict.

Notice from the proof of Proposition 2.11 that we can deduce the following results.

#### Proposition 2.16

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS has Condition (WP) (Condition (WP)w).

2. Every surjective S-poset morphism BSAS is (weakly) quasi-w-2-pure.

3. There exists a (weakly) quasi-w-2-pure epimorphism BSAS where BS has Condition (WP).

#### Proposition 2.17

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS has Condition (PWP) (Condition (PWP)w).

2. Every surjective S-poset morphism BSAS is (weakly) quasi-pw-2-pure.

3. There exists a (weakly) quasi-pw-2-pure epimorphism BSAS where BS has Condition (PWP).

In [15], the authors introduced a generalization of Condition (P) called Condition (P′) for S-acts. In this way, we introduce Conditions (P′) and ( $\begin{array}{}{\mathrm{P}}_{w}^{\prime }\end{array}$) for S-posets.

#### Definition 2.18

Let AS be a right S-poset. We say that

1. AS satisfies Condition (P′) if, for all aA, s, s′, zS with asasand sz = sz, there exist a″ ∈ A, u, vS such that a = au, a′ = av and usvs′.

2. AS satisfies Condition ( $\begin{array}{}{\mathrm{P}}_{w}^{\prime }\end{array}$) if, for all aA, s, s′, zS with asasand sz = sz, there exist a″ ∈ A, u, vS such that aau, avaand usvs′.

Now, we define two kinds of purity conditions of epimorphisms in order to consider Conditions (P′) and ( $\begin{array}{}{\mathrm{P}}_{w}^{\prime }\end{array}$).

#### Definition 2.19

Let ψ : BSAS be a surjective S-poset morphism. We say ψ is

1. quasi-2′-pure if, for every a, a′ ∈ A and relations

$as≤a′tandsz=tz,$

there exist b, b′ ∈ B such that a = ψ(b), ψ(b′) = a′, and bsbt.

2. weakly quasi-2′-pure if, for every a, a′ ∈ A and relations

$as≤a′tandsz=tz,$

there exist b, b′ ∈ B such that aψ(b), ψ(b′) ≤ a′, and bsbt.

Similarly to the argument of Proposition 2.11 one could prove the following result.

#### Proposition 2.20

Let AS be a right S-poset. Then the following assertions are equivalent.

1. AS has Condition (P′) (Condition (P′w)).

2. Every surjective S-poset morphism BSAS is (weakly) quasi-2′-pure.

3. There exists a (weakly) quasi-2′-pure epimorphism BSAS where BS has Condition (P′).

Concluding this section, we summarize in the following Table 1 the results concerning equivalent descriptions of some flatness properties for S-posets (starting with strong flatness and ending with Condition (PWP)w) by certain purity conditions of epimorphisms.

Table 1

Equivalent descriptions of some flatness properties

## 3 Directed colimits for S-posets

In this section, we show that these purity epimorphisms of S-posets introduced in Section 2 are preserved under directed colimits.

Directed colimits (also called direct limits) of families of right S-posets are introduced in [5], but note that the definition of directed colimits is omitted in that paper. For completeness, we will provide a full definition of directed colimits for S-posets.

Let I be a quasi-ordered (that is, a reflexive and transitive relation) set. A direct system in Pos-S is a collection of right S-posets (Ai)iI and a collection of right S-poset morphisms ϕi,j : AiAj (ij) with the following properties:

1. ϕi,i = 1Ai for all iI;

2. ϕj,kϕi,j = ϕi,k, whenever ijk.

The colimit of the system (Ai, ϕi,j) is a right S-poset AS together with right S-poset morphisms ϕi : AiA such that

1. ϕjϕi,j = ϕi, whenever ij;

2. If BS is a right S-poset and αi : AiB are right S-poset morphisms such that αjϕi,j = αi whenever ij, then there exists a unique S-poset morphism ψ : AB such that the diagram

commutes for all iI.

Further, if the indexing set I satisfies the property that for all i, jI there exists kI such that ki, j, then we say that I is directed. In this case, we call the colimit (A, ϕi) is a directed colimit.

From the following lemma, we remark that in Pos-S, directed colimits of directed systems of S-posets exist.

#### Lemma 3.1

([5, Proposition 2.5]). The directed colimit of any directed system ((Ai)iI, (ϕi,j)ij) of right S-posets exists, and may be represented as (A/θ, (ϕi)iI), where

1. $\begin{array}{}A=\coprod _{i\in I}{A}_{i}\mathit{;}\end{array}$

2. a θ a′ (aAi, a′ ∈ Aj) if and only if ϕi,k(a) = ϕj,k(a′) in Ak for some ki, j;

3. [a]θ ≤ [a′]θ (aAi, a′ ∈ Aj) if and only if ϕi,k(a) ≤ ϕj,k(a′) for some ki, j;

4. for each iI and aAi, ϕi(a) = [a]θ.

The next result is often useful in dealing with directed colimits.

#### Lemma 3.2

([5, Proposition 2.6]). Let (Ai)iI, (ϕi,j)ij be a direct system of right S-posets. Then the directed colimit (A, (ϕi)iI) is characterized up to isomorphism by the conditions

1. each ϕi : AiA is an S-poset morphism;

2. ϕjϕi,j = ϕi, whenever ij;

3. $\begin{array}{}A=\bigcup _{i\in I}{\varphi }_{i}\left[{A}_{i}\right];\end{array}$

4. $\begin{array}{}{\varphi }_{i}^{-1}\left[R\right]\subseteq \bigcup _{j\ge i}{\varphi }_{i,j}^{-1}\left[{R}_{j}\right],\end{array}$ where R (resp. Rj) denotes the graph of the relationin the S-poset A (resp. Aj).

Actually, the property (4) of Lemma 3.2 states the order relation on AS, that is, ϕi (a) ≤ ϕj(a′) if and only if ϕi,k(a) ≤ ϕj,k(a′) for some ki, j. Thereby, we obtain the fact that ϕi is an order-embedding if and only if ϕi,j is an order-embedding for all ij.

Using these properties of directed colimits, the following result is obtained.

#### Lemma 3.3

Let (Ai, ϕi,j) be a direct system of right S-posets with directed index set I and directed colimit (A, ϕi). For every family a1, ⋯, amA and relations

$aαisi≤aβiti(i=1,⋯,n),$

there exist some lI and $\begin{array}{}{a}_{1}^{\prime },\cdots ,{a}_{m}^{\prime }\end{array}$Al such that ϕl( $\begin{array}{}{a}_{j}^{\prime }\end{array}$ ) = aj for all 1 ≤ jm, and $\begin{array}{}{a}_{{\alpha }_{i}}^{\prime }{s}_{i}\le {a}_{{\beta }_{i}}^{\prime }{t}_{i}\end{array}$ for all 1 ≤ in.

#### Proof

The technique used in [3, Lemma 2.3] will be employed. □

In order to prepare for our main results, we will introduce the concept of directed colimits of S-poset morphisms.

Suppose that (Ai, ϕi,j) and (Bi, φi,j) are direct systems of S-posets and S-poset morphisms. Suppose that for each iI there exists an S-poset morphism ψi: AiBi and suppose that (A, ϕi) and (B, φi), the directed colimits of these systems, are such that the diagrams

commute for all ijI. Then we shall refer to ψ as the directed colimit of the ψi.

In light of Lemma 3.2, the following easily proved result is probably well-known.

#### Proposition 3.4

Let S be a pomonoid. Directed colimits of (order-embeddings) epimorphisms of S-posets are (order-embeddings) epimorphisms.

We now establish one of our main results.

#### Proposition 3.5

Let S be a pomonoid. Directed colimits of pure epimorphisms of right S-posets are pure.

#### Proof

Suppose that (Ai, ϕi,j) and (Bi, φi,j) are direct systems of right S-posets and S-poset morphisms. Suppose that for each iI there exists a pure epimorphism ψi: AiBi and suppose that (A, ϕi) and (B, φi), the directed colimits of these systems, are such that the diagrams

commute for all ijI.

Suppose that there are b1, ⋯, bmB, s1, ⋯, sn, t1, ⋯, tnS and relations

$bαisi≤bβiti(i=1,⋯,n).$

In view of Lemma 3.3, there exist lI and $\begin{array}{}{b}_{1}^{\prime },\cdots ,{b}_{m}^{\prime }\end{array}$Bl such that φl( $\begin{array}{}{b}_{j}^{\prime }\end{array}$ ) = bj for all 1 ≤ jm, and

$bαi′si≤bβi′ti(i=1,⋯,n).$

Since ψl is a pure epimorphism, there exist a1, ⋯, amAl with ψl(aj) = $\begin{array}{}{b}_{j}^{\prime }\end{array}$ for all 1 ≤ jm, and aαisiaβiti for all 1 ≤ in. Then we can calculate that

$ψ(ϕl(aj))=(ψϕl)(aj)=(φlψl)(aj)=φl(ψl(aj))=φl(bj′)=bj$

for all j, and

$ϕl(aαi)si≤ϕl(aβi)ti$

for all i. Hence ψ is pure. □

By the same approach, one can get

#### Proposition 3.6

Let S be a pomonoid. Directed colimits of 1-pure (1′-pure) epimorphisms of right S-posets are 1-pure (1′-pure).

In what follows, we consider the situation for quasi-2-pure epimorphisms.

#### Proposition 3.7

Let S be a pomonoid. Directed colimits of quasi-2-pure epimorphisms of right S-posets are quasi-2-pure.

#### Proof

Suppose that (Ai, ϕi,j) and (Bi, φi,j) are direct systems of right S-posets and S-poset morphisms. Suppose that for each iI there exists a quasi-2-pure epimorphism ψi: AiBi and suppose that (A, ϕi) and (B, φi), the directed colimits of these systems, are such that the diagrams

commute for all ijI.

Suppose that b1, b2B, s, tS are such that b1sb2t. By Lemma 3.2(3), there exist i, jI, biBi and bjBj with b1 = φi(bi) and b2 = φj(bj). According to the order relation on BS, we obtain ki, j with φi,k(bi)sφj,k(bj)t in Bk. Since ψk is quasi-2-pure, there exist a1, a2Ak such that ψk(a1) = φi,k(bi), ψk(a2) = φj,k(bj) and a1sa2t. Now we can compute that ϕk(a1)sϕk(a2)t and

$b1=φi(bi)=φk(φi,k(bi))=φk(ψk(a1))=(φkψk)(a1)=(ψϕk)(a1)=ψ(ϕk(a1)).$

In the same way, b2 = ψ(ϕk(a2)), and so ψ is quasi-2-pure. □

Note that, the situation for quasi-1-pure (resp., quasi-1′-pure, quasi-2′-pure, weakly quasi-2-pure, weakly quasi-2′-pure, quasi-w-2-pure, weakly quasi-w-2-pure, quasi-pw-2-pure and weakly quasi-pw-2-pure) epimorphisms, is similar in nature to Proposition 3.7, and so here will be omitted.

We have already seen in Section 2 that some flatness properties are equivalent to certain purity conditions of epimorphisms. And then, using Propositions 3.5, 3.6, 3.7 and the above note, the following result is immediately established.

#### Corollary 3.8

Let S be a pomonoid. Every directed colimit of a direct system of right S-posets that are strongly flat (resp., have Conditions (E), (E′), (P), (P′), (Pw), (P′ w), (WP), (WP)w, (PWP) and (PWP)w) has again these properties.

Observing the result above, by a new way we have obtained that many flatness properties of S-posets are preserved under directed colimits.

However, as for acts, the situation for projective S-posets is slightly different.

#### Proposition 3.9

Let S be a pomonoid. Every directed colimit of a direct system of projective right S-posets is projective if and only if S is right (po-)perfect.

#### Proof

Necessity. In light of [14, Theorem 6.3], it suffices to show that every strongly flat right S-poset is projective, so now suppose that AS is a strongly flat S-poset. It then follows from [5, Proposition 4.4] that AS is isomorphic to a directed colimit of a family of finitely generated, free S-posets. So by assumption, AS is projective since free S-posets are projective.

Sufficiency. Let (A, ϕi) be a directed colimit of a direct system of projective right S-posets. In view of Corollary 3.8, AS is strongly flat because projective S-posets are strongly flat. Also, since S is right (po-)perfect, by [14, Theorem 6.3], AS is projective, and the proof is complete. □

From [5, Proposition 4.6], it follows that an S-poset epimorphism φ: BSAS is pure if and only if, for every finitely presented S-poset CS and morphism η : CSAS there exists a morphism μ : CSBS such that φμ = η. It is easy to check that split epimorphisms are pure. But the following example from [3] shows that the converse is false. Let S = (ℕ, max) with the discrete order. Consider the one-element S-poset ΘS and note that SSΘS is a pure epimorphism. On the other hand, since S does not contain a fixed point then it does not split.

But, the following result is a small improvement. Its straightforward proof is omitted.

#### Proposition 3.10

Let ψ: ASBS be a surjective S-poset morphism with BS is finitely presented. Then ψ is pure if and only if it is split.

For future use, we record

#### Lemma 3.11

Let S be a pomonoid, let

be a pullback diagram of S-posets and suppose that ψ is a pure epimorphism. Then ϕ is also a pure epimorphism.

#### Proof

It is routine. □

Note that in the above result, if we replace “pullback diagram” by “subpullback diagram”, then this result is also valid.

As previously discussed, not every pure epimorphism splits, but every pure epimorphism is a directed colimit of split epimorphisms. The strategy for the proof the following proposition is taken from the unordered case in [3].

#### Proposition 3.12

Let S be a pomonoid and let ψ: ASBS be a surjective S-poset morphism. Then ψ is pure if and only if it is a directed colimit of split epimorphisms.

#### Proof

Suppose that ψ: ASBS is a pure epimorphism. From [5, Proposition 4.4] it follows that BS is a directed colimit of finitely presented S-posets (Bi, ϕi,j) and so let ϕi: BiB be the canonical morphisms. For each Bi let

be a pullback diagram so that by Lemma 3.11 ψi is pure. Because Bi is finitely presented, it follows from Proposition 3.10 that ψi splits. Notice that

$Ai={(bi,a)∈Bi×A| ϕi(bi)=ψ(a)},$

ψi(bi, a) = bi and φi(bi, a) = a, and that since ψ is onto then Ai ≠ ∅.

For ij define φi,j: AiAj by φi,j(bi, a) = (ϕi,j(bi), a) and notice that φjφi,j = φi and ψjφi,j = ϕi,jψi. Next we show that (A, φi) is the directed colimit of (Ai, φi,j). So assume there exist an S-poset CS and S-poset morphisms αi: AiC with αjφi,j = αi for all ij. We now define α : AC by α(a) = αi(bi, a) where i and bi are chosen so that φi(bi) = ψ(a). Then α is well-defined since if ψ(a) = ϕj(bj) then there exists ki, j with ϕi,k(bi) = ϕj,k(bj) and

$αi(bi,a)=αkφi,k(bi,a)=αk(ϕi,k(bi,a))=αk(ϕj,k(bj,a))=αkφj,k(bj,a)=αj(bj,a).$

Then α is an S-poset morphism and clearly αφi = αi. Finally, if β: AC is such that βφi = αi for all i, then β(a) = βφi(bi, a) = αi(bi, a) = α(a) and so α is unique. We therefore see that we have reached the desired conclusion.

The converse holds by Proposition 3.5. □

## 4 Directed colimits of flatness properties

From previous section, we find some flatness properties that are closed under directed colimits. But for the other flatness properties, we need to characterize the behavior of subpullbacks diagrams under directed colimits. In this section, we first prove that every directed colimit of a direct system of subequalizer flat S-posets is subequalizer flat. Then, we consider subpullbacks in order to prove that every class of S-posets having a flatness property is closed under directed colimits. First let us to recall the concept of subpullbacks and subequalizers in the category S-Pos which are defined in [5] as follows.

The categories S-Pos and Pos are poset-enriched concrete categories, where the order relation on morphism sets is defined pointwise (i.e. fg for f, g: AB if and only if f(a) ≤ g(a) for every aA). In such categories, a diagram

is called a subpullback diagram for α and β if

1. α p1βp2 and

2. for any diagram

with $\begin{array}{}\alpha {p}_{1}^{\prime }\le \beta {p}_{2}^{\prime },\end{array}$ there exists a unique morphism φ: P′ → P such that piφ = $\begin{array}{}{p}_{i}^{\prime }\end{array}$ for i = 1, 2. In S-Pos or Pos, P may in fact be realized as

$P={(m,n)∈M×N| α(m)≤β(n)}.$

The first subpullback diagram is denoted by P(M, N, α, β, Q) and tensoring it by any right S-poset AS one gets the diagram

in Pos. Take

$P′={(a⊗m,a⊗n)∈(AS⊗SM)⊗(AS⊗SN)| a⊗α(m)≤a⊗β(n)}$

with $\begin{array}{}{p}_{1}^{\prime },{p}_{2}^{\prime }\end{array}$ being the restrictions of the projections. From the definition of subpullbacks it follows the existence of a unique monotonic mapping ϕ: ASSPP′ such that $\begin{array}{}{p}_{i}^{\prime }\varphi \end{array}$ = 1Api for i = 1, 2. This mapping is called the ϕ corresponding to the subpullback diagram P(M, N, α, β, Q) for AS. It can be checked that ϕ(a ⊗ (m, n)) = (am, an).

A subequalizer diagram for α and β

is defined similarly, where E = {mM |α(m) ≤ β(m)}. As we mentioned earlier, an S-poset AS is called subpullback flat (subequalizer flat) if the functor AS ⊗ − takes subpullbacks (subequalizers) in S-Pos to subpullbacks (subequalizers) in Pos. We begin by proving the result for subequalizer flat. Before that we need some preparations.

If AS is an S-subposet of BS and x, y are different elements not belonging to BS, then

$BS∐ABS={(a,x)| a∉AS}∪˙AS∪˙{(a,y)| a∉AS}.$

Define an S-action

$(a,w)s=(as,w) ifas∉ASas otherwise,$

for every aB SAS, sS and w ∈ {x, y}.

The order on BSA BS is given by

$(a,w1)≤(b,w2)⇔(w1=w2, a≤b) or (w1≠w2, a≤a′≤b for some a′∈AS),$

where {w1, w2} = {x, y} and a, bAS. For w ∈ {x, y}, bAS and aA,

$(b,w)≤a⇔b

It is easily checked BSA BS is a right S-poset.

#### Lemma 4.1

Every regular monomorphism h : ASBS in Pos-S can be consider as the subequalizer of the diagram

#### Proof

Let h : ASBS be a regular monomorphism. Define α, β: BSBSh(A) BS as

$α(b)=(b,x) ifb∉h(A)b otherwise$

and

$β(b)=(b,y) ifb∉h(A)b otherwise.$

To show that (AS, h) is a subequalizer diagram for α and β, clearly αhβ h, and consider the diagram

where CS and f are such that αfβf. By the construction of BSh(A) BS we see that f(CS) ⊆ h(AS). Now, since h is a regular monomorphism, one can define a mapping f : CSAS by f(c) = h−1(f(c)) for cCS. Then f is a well-defined S-morphism and unique with the property f = hf. □

From the previous lemma, it follows that every subequalizer flat S-poset is po-flat. Now we are ready to consider directed colimit of a direct system of subequalizer flat S-posets.

#### Proposition 4.2

Let S be a pomonoid. Every directed colimit of a direct system of subequalizer flat S-posets is subequalizer flat.

#### Proof

Let (Ai, ϕi,j) be a direct system of S-psets and S-morphisms with directed index set I and directed colimit (A, ϕi). If every Ai is subequalizer flat, we will prove that A is subequalizer flat. Let the pair (E, l) be a subequalizer in the following diagram

Since every subequalizer flat S-poset is po-flat, from po-flatness of AS we have the following diagram

and 1Al is a regular monomorphism. By the definition, the subequalizer of 1Aα and 1Aβ is

$E′={a⊗m∈A⊗M| (1A⊗α)(a⊗m)≤(1A⊗β)(a⊗m),a∈A,m∈M}.$

By the definition of the subequalizer, it is easily checked that AEE′, next we want to show that E′ ⊆ AE. Let aA, mM be such that amE′. So aα(m) ≤ aβ(m) in AN implies that the existence of u1, v1, …, un, vnS, a1, …, anAS, y2, …, ynS N, n ∈ ℕ such that

$a≤a1u1a1v1≤a2u2 u1α(m)≤v1y2 ⋮ ⋮anvn≤a unyn≤vnβ(m).$

Denote a by a0, then there exist aijAij such that aj = ϕij(aij), j = 0, 1, .., n. Then we get

$ϕi0(ai0)≤ϕi1(ai1)u1ϕi1(ai1)v1≤ϕi2(ai2)u2 u1α(m)≤v1y2 ⋮ ⋮ϕin(ain)vn≤ϕi0(ai0) unyn≤vnβ(m).$

Since I is directed, there exists li1, i2, …, in such that

$ϕi0,l(ai0)≤ϕi1,l(ai1)u1ϕi1,l(ai1)v1≤ϕi2,l(ai2)u2 u1α(m)≤v1y2 ⋮ ⋮ϕin,l(ain)vn≤ϕi0,l(ai0) unyn≤vnβ(m).$

This means that ϕi0,l(ai0) ⊗ α(m) ≤ ϕi0,l(ai0) ⊗ β(m) in AlN. Now, from the fact that Al is subequalizer flat, and AlE is the subequalizer of 1Alα and 1Alβ, it follows that ϕi0,l(ai0) ⊗ mAlE. So mE, amAE and we are done.□

In the rest of this section we concentrate on subpullback diagrams. We now present two fundamental propositions that yield the main result of this section.

#### Lemma 4.3

Let S be a pomonoid, and (Ai, ϕi,j) be a direct system of S-posets with directed index set I and let (A, ϕi) be the directed colimit. Suppose that for each Ai the mapping ϕ corresponding to the subpullback diagram P(M, N, α, β, Q) is surjective, then for A the mapping ϕ corresponding to the subpullback diagram P(M, N, α, β, Q) is surjective.

#### Proof

Suppose that (xm, yn) belongs to the subpullback of P(AM, AN, 1Aα, 1Aβ, AQ), where x, yA, mM, nN. Then xα(m) ≤ yβ(n) in AQ. We should find aA, (m, n) ∈ P(m, N, α, β, Q) such that ϕ(A ⊗ (m, n)) = (xm, yn). Since xα(m) ≤ yβ(n), there exist k ∈ ℕ and elements a1, …, akAS, q2, …, qkSQ, u1, v1, …, uk, vkS such that

$x≤a1u1a1v1≤a2u2 u1α(m)≤v1q2 ⋮ ⋮akvk≤y ukqk≤vkβ(n).$

Denote x by a0 and y by ak+1, so there exist aijAij such that aj = ϕij ( $\begin{array}{}{a}_{{i}_{j}}^{\prime }\end{array}$ ), where ijI and j = 0, 1, …, k, k + 1. Hence we have

$ϕi0(ai0′)≤ϕi1(ai1′)u1ϕi1(ai1′)v1≤ϕi2(ai2′)u2 u1α(m)≤v1q2 ⋮ ⋮ϕik(aik′)vk≤ϕik+1(aik+1′) ukqk≤vkβ(n).$

Since I is directed, there exists li0, i1, …, ik+1 such that

$ϕi0(ai0′)≤ϕi1(ai1′)u1ϕi1(ai1′)v1≤ϕi2(ai2′)u2 u1α(m)≤v1q2 ⋮ ⋮ϕik(aik′)vk≤ϕik+1(aik+1′) ukqk≤vkβ(n).$

Then ϕi0,l( $\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ) ⊗ α(m) ≤ ϕik+1,l( $\begin{array}{}{a}_{{i}_{k+1}}^{\prime }\end{array}$ ) ⊗ β(n) in AlQ. That is (ϕi0,l ($\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ) ⊗ m, ϕik+1,l( $\begin{array}{}{a}_{{i}_{k+1}}^{\prime }\end{array}$ ) belongs to the subpullback of P(AlM,AlN, 1Alα, 1Alβ, AlQ). By the assumption, for Al the mapping ϕ corresponding to the subpullback diagram P(M, N, α, β, Q) is surjective, so there exist a″ ∈ Al, m′ ∈ M and n′ ∈ N such that ϕ(a″ ⊗ (m′, n′)) = (ϕi0,l( $\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ) ⊗ m, ϕik+1,l( $\begin{array}{}{a}_{{i}_{k+1}}^{\prime }\end{array}$ ) ⊗ n), where α(m′) ≤ β(n′). That is ϕi0,l( $\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ) ⊗ m = a″ ⊗ m′ and ϕik+1,l( $\begin{array}{}{a}_{{i}_{k+1}}^{\prime }\end{array}$ ) ⊗ n = a″ ⊗ n′. Since ϕi0,l( $\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ) ⊗ m = a″ ⊗ m′, there exist elements ci, $\begin{array}{}{c}_{j}^{\prime }\end{array}$Al, m2, …, mp, $\begin{array}{}{m}_{2}^{\prime }\end{array}$ , …, $\begin{array}{}{m}_{{p}^{\prime }}^{\prime }\end{array}$SM, si, ti, $\begin{array}{}{s}_{j}^{\prime },{t}_{j}^{\prime }\end{array}$S, 1 ≤ ip, 1 ≤ jp′ for p, p′ ∈ ℕ such that

$ϕi0,l(ai0′)≤c1s1$

$c1t1≤c2s2 s1m≤t1m2 ⋮ ⋮cptp≤a″ tpmp≤tpm′, a″≤c1′s1′c1′t1′≤c2′s2′ s1′m′≤t1′m2′ ⋮ ⋮cp′′tp′′≤ϕi0,l(ai0′) tp′′mp′′≤tp′′m.$

Acting ϕl on left column inequations and since x = ϕi0 ( $\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ) = ϕlϕi0,l( $\begin{array}{}{a}_{{i}_{0}}^{\prime }\end{array}$ ), we get

$x≤ϕl(c1)s1ϕl(c1)t1≤ϕl(c2)s2 s1m≤t1m2 ⋮ ⋮ϕl(cp)tp≤ϕl(a″) tpmp≤tpm′,ϕl(a″)≤ϕl(c1′)s1′ϕl(c1′)t1′≤ϕl(c2′)s2′ s1′m′≤t1′m2′ ⋮ ⋮ϕl(cp′′)tp′′≤x tp′′mp′′≤tp′′m.$

Thus xm = ϕl(a″) ⊗ m′ in AM. Since ϕik+1,l( $\begin{array}{}{a}_{{i}_{k+1}}^{\prime }\end{array}$ ) ⊗ n = a″ ⊗ n′, by a similar way we can prove yn = ϕl(a″) ⊗ n′ in AN. Therefore, ϕ(ϕl(a″) ⊗ (m′, n′)) = (ϕl(a″) ⊗ m′, ϕl(a″) ⊗ n′) = (xm, yn), as desired. □

#### Lemma 4.4

Let S be a pomonoid, and (Ai, ϕi,j) be a direct system of S-posets with directed index set I and let (A, ϕi) be the directed colimit. Suppose that for each Ai the mapping ϕ corresponding to the subpullback diagram P(m, N, α, β, Q) is an order-embedding, then for A the mapping ϕ corresponding to the subpullback diagram P(m, N, α, β, Q) is an order-embedding.

#### Proof

Suppose that ϕ(a ⊗ (m, n)) ≤ ϕ(a′ ⊗ (m′, n′)) belongs to the subpullback of P(AM, AN, 1Aα, 1Aβ, AQ), where a, a′ ∈ A and (m, n), (m′, n′) ∈ SP. So we have α(m) ≤ β(n), α(m′) ≤ β(n′), and

$a⊗m≤a′⊗m′ in AS⊗SM, a⊗n≤a′⊗n′ in AS⊗SN.$

We will show that a ⊗ (m, n) ≤ a′ ⊗ (m′, n′) in ASSP. Since ama′ ⊗ m′ in ASS M and ana′ ⊗ n′ in ASSN, there exist p, p′ ∈ ℕ and elements ai, cjAS, m2, …, mpSM, n2, …, npSN, ui, vi, $\begin{array}{}{u}_{j}^{\prime },{v}_{j}^{\prime }\end{array}$S, 1 ≤ ip, 1 ≤ jp′ such that

$a≤a1u1a1v1≤a2u2 u1m≤v1m2 ⋮⋮apvp≤a′ upmp≤vmm′,a≤c1u1′c1v1′≤c2u2′ u1′n≤v1′n2 ⋮⋮cp′vp′′≤a′ up′′np′≤vp′′n′.$

By Lemma 3.3, there exist lI and $\begin{array}{}{a}_{0}^{\prime },{a}_{1}^{\prime },\cdots ,{a}_{p}^{\prime },{c}_{1}^{\prime },...,{c}_{{p}^{\prime }}^{\prime },{c}_{0}^{\prime }\end{array}$Al such that $\begin{array}{}{\varphi }_{l}\left({a}_{0}^{\prime }\right)=a,{\varphi }_{l}\left({a}_{i}^{\prime }\right)\end{array}$ = a, $\begin{array}{}{\varphi }_{l}\left({c}_{j}^{\prime }\right)={c}_{j},{\varphi }_{l}\left({c}_{0}^{\prime }\right)={a}^{\prime }\end{array}$ for all 1 ≤ ip, 1 ≤ jp′, and

$a0′≤a1′u1a1′v1≤a2′u2 u1m≤v1m2 ⋮ ⋮apvp≤c0′ upmp≤vmm′,a0′≤c1′u1′c1′v1′≤c2′u2′ u1′n≤v1′n2 ⋮ ⋮cp′′vp′′≤c0′ up′′np′≤vp′′n′.$

This means that

$a0′⊗m≤c0′⊗m′ in Al⊗SM, α(m)≤β(n),a0′⊗n≤c0′⊗n′ in Al⊗SN, α(m′)≤β(n′).$

Hence $\begin{array}{}\varphi \left({a}_{0}^{\prime }\otimes \left(m,n\right)\right)\le \varphi \left({c}_{0}^{\prime }\otimes \left({m}^{\prime },{n}^{\prime }\right)\right).\end{array}$ Since for Al the mapping ϕ corresponding to the subpullback diagram P(m, N, α, β, Q) is an order-embedding, we imply $\begin{array}{}{a}_{0}^{\prime }\otimes \left(m,n\right)\le {c}_{0}^{\prime }\otimes \left({m}^{\prime },{n}^{\prime }\right)\end{array}$ in AlS P. So there exist d1, …, drAl, (m2, n2), …, (mr,nr) ∈ SP, x1, y1, …, xr, yrS, r ∈ ℕ such that

$a0′≤d1x1d1y1≤d2x2 x1(m,n)≤y1(m2,n2) ⋮⋮dryr≤c0′ xr(mr,nr)≤yr(m′,n′).$

Acting ϕl on left column inequations and since $\begin{array}{}a={\varphi }_{l}\left({a}_{0}^{\prime }\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{a}^{\prime }={\varphi }_{l}\left({c}_{0}^{\prime }\right),\end{array}$ we get

$a≤ϕl(d1)x1ϕl(d1)y1≤ϕl(d2)x2 x1(m,n)≤y1(m2,n2) ⋮ ⋮ϕl(dr)yr≤a′ xr(mr,nr)≤yr(m′,n′).$

Therefore, a ⊗ (m, n) ≤ a′ ⊗ (m′, n′) in ASSP and the result follows. □

It is shown in [7, Theorems 2.1-2.4, 3.2, 4.1, 5.3] and [16, Definition 2.2] that most of flatness properties of S-posets over a pomonoid S are equivalent to the surjectivity or bijectivity of mappings corresponding to the subpullback diagrams in special cases. Using that results and two previous propositions we conclude the main result of this section.

#### Theorem 4.5

Every class of S-posets having a flatness property such as torsion freeness, principal weak flatness, weak flatness, flatnes, pullback flatness, subpullback flatness, principal weak kernel po-flatness, weak kernel po-flatness, translation kernel po-flatness, and satisfying Conditions (P), (WP) or (PWP) is closed under directed colimits.

## Acknowledgement

This research was partially supported by the Natural Science Foundation of Shaanxi Province(No.2017JQ1026) and Special Research Project of Shaanxi Department of Education (No.17JK0102).

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Accepted: 2018-04-11

Published Online: 2018-07-04

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 669–687, ISSN (Online) 2391-5455,

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