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

Xingliang Liang; Roghaieh Khosravi

doi:10.1515/math-2018-0061

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

Xingliang Liang and Roghaieh Khosravi

Published/Copyright: July 4, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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), (P_w), (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.

Keywords: S-poset; Flatness property; Pure epimorphism; Directed colimit

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 A_S is strongly flat, which means that A_S has both Condition (P) and Condition (E), if and only if every surjective S-poset morphism B_S → A_S 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 × S → A, (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 a ∈ A and s, t ∈ S. Left S-posets are defined analogously. The notations A_S and _SB will respectively denote a right and left S-poset, and Θ_S = {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 A_S be a right S-poset. An S-poset congruenceθ onA_S 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 A → A/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 A_S that is reflexive, transitive, and compatible with the S-action. For a, a′ ∈ A, we write a≤αa′ if a so-called α-chain

a≤a1 α a1′≤a2 α a2′≤⋯≤an α an′≤a′

from a to a′ exists in A_S, where each a_i, ai′ ∈ A for 1 ≤ i ≤ n. It can be shown that an S-act congruence θ on an S-poset A_S is an S-poset congruence if and only if aθa′ whenever a≤θa′≤θa.

The next concept will be used frequently in this paper. Let A_S be a right S-poset and let H ⊆ A × A. The S-poset congruence onA_Sinduced byH is described as follows:

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

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

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

for some (x_i, y_i) ∈ H and s_i ∈ S, 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 A_S such that [x]_ν(H) ≤ [x′]_ν(H) whenever (x, x′) ∈ H. The S-poset congruence onA_Sgenerated byH is θ(H) = ν(H ∪ H^op), and as usual is the smallest congruence on A_S that contains H.

A right S-poset A_S is called cyclic if A = aS for some a ∈ A. It is clear that A_S is cyclic if and only if A_S is isomorphic to S/ρ for some right S-poset congruence ρ on S_S. A congruence ρ on an S-poset A_S is called finitely generated if ρ = ν(H) for some finite subset H of A × A. An S-poset A_S is called finitely presented if it is isomorphic to F/ν(H) for some finitely generated free S-poset F_S and some finite subset H ⊆ F × F.

Conditions (E), (P) and (P_w) 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 (P_w). According to [9], an S-poset A_S is said to be strongly flat if it satisfies Conditions (E) and (P). Equivalently, an S-poset A_S is strongly flat if and only if it is subpullback flat and subequalizer flat [that is, the functor A_S ⊗ – (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 (P_w) are defined in [7] as follows. An S-poset A_S 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 A_S is said to satisfy Condition (WP)_w if, for all elements s, t ∈ S, all homomorphisms f: _S(Ss∪St) → _SS, and all a, a′ ∈ A, af(s) ≤ a′f(t) implies a⊗s ≤ a″⊗us′ and a″⊗vt′ ≤ a′⊗t in A ⊗_S (Ss ∪ St) for some a″ ∈ A, u, v ∈ S and s′, t′ ∈ {s, t} with f(us′) ≤ f(vt′). An S-poset A_S is said to satisfy Condition (PWP) if the corresponding ϕ is surjective for every subpullback diagram P(Ss, Ss, f, f, S), s ∈ S. An S-poset A_S is said to satisfy Condition (PWP)_w if, for all a, a′ ∈ A, s ∈ S, as ≤ a′s implies a ≤ a″u, a″v ≤ a′ for some a″ ∈ A, u, v ∈ S with us ≤ vs.

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), (P_w), (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 : B_S → A_S be a surjective S-poset morphism. We say that ψ is a pure (m-pure) epimorphism [5] if, for every m ∈ ℕ, a₁, ⋯, a_m ∈ A and relations

aαisi≤aβiti(i=1,2,⋯,l),

where α_i, β_i ∈ {1, ⋯, m} and s_i, t_i ∈ S, there exist b₁,⋯, b_m ∈ B such that ψ(b_r) = a_r for all 1 ≤ r ≤ m and b_{α_i}s_i ≤ b_{β_i}t_i for all i.

We are interested here in the cases m = 1 and m = 2. In both cases, we shall call ψ1-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 epimorphismψ : B_S → A_S by

ψ(a)=x, ψ(b)=y, ψ(c)=z.

It is not hard to verify thatψis 1-pure. However, it is not 2-pure, sincex · 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 A_S be a right S-poset. Then the following assertions are equivalent.

A_S has Condition (E).
Every surjectiveS-poset morphismB_S → A_S is 1-pure.
There exists a 1-pure epimorphismB_S → A_S whereB_S is subequalizer flat.
Every S-poset morphismB_S → A_S where B_S is a finitely presented cyclic S-poset may be factorized through a free S-poset.
Every S-poset morphismB_S → A_SwhereB_S 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 : B_S → A_S be a surjectiveS-poset morphism. We say that φ is quasi-1-pure if, for every a ∈ A and as ≤ at, there exists b ∈ B such that ψ(b) = a and bs ≤ bt.

Proposition 2.4

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

A_S has Condition (E).
Every surjectiveS-poset morphismB_S → A_S is quasi-1-pure.
There exists a quasi-1-pure epimorphismB_S → A_SwhereB_S has Condition (E).

Proof

(1) ⇒ (2). Suppose that ψ : B_S → A_S is an S-poset epimorphism. Let as ≤ at in A_S for a ∈ A and s, t ∈ S. Since A_S satisfies Condition (E), there exist u ∈ S and a′ ∈ A such that a = a′u and us ≤ ut. Applying surjectivity of ψ, obtains b ∈ B with ψ(b) = a′. Further, we have bus ≤ but in B_S, and a = a′u = 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 B_S such that ψ : B_S → A_S is an S-poset epimorphism and B_S satisfies Condition (E). Then by (2), ψ is quasi-1-pure.

(3) ⇒ (1). Let A_S be a right S-poset. Then by (3) there exists a quasi-1-pure epimorphism ψ : B_S → A_S with B_S satisfies Condition (E). Now assume as ≤ at in A_S for a ∈ A, s, t ∈ S. Then there exists b′ ∈ B such that b′s ≤ b′t in B_S, and ψ(b′) = a. Further, because B_S satisfies Condition (E), from the inequality b′s ≤ b′t, yields b ∈ B and u ∈ S such that b′ = bu and us ≤ ut. Consequently, a = ψ(b′) = ψ(bu) = ψ(b)u in A_S, and the proof is complete. □

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

Definition 2.5

A rightS-posetA_S satisfies Condition (E′) if, for alla ∈ A, s, s′, z ∈ S with as ≤ as′ and sz = s′z, there exista′ ∈ A, u ∈ S such thata = a′u and us ≤ us′.

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

Letψ : B_S → A_S be a surjective S-poset morphism. We sayψ is

1′-pure if, for everya ∈ Aand relations
asi≤atiandsizi=tizi(i=1,⋯,n),
there existsb ∈ B such thatψ(b) = aandbs_i ≤ bt_ifor alli;
quasi-1′-pure if, for everya ∈ Aand relations
as≤atandsz=tz,
there existsb ∈ Bsuch thatψ(b) = aand bs ≤ bt.

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

Lemma 2.7

A surjectiveS-poset morphismψ : B_S → A_Sis 1′-pure if and only if for every cyclic rightS-posetS/ρand everyS-poset morphismα : S/ρ → A_S, whereρ = ν(H) for some setHof the form

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

there exists anS-poset morphismβ : S/ρ → B_Ssuch that the diagram

commutes.

Proof

Necessity. Suppose that ψ : B_S → A_S is a 1′-pure epimorphism and suppose that α : S/ρ → A_S is an S-poset morphism, where ρ is an S-poset congruence on S_S induced by a set of the form {(s_i, t_i) ∈ S × S | s_iz_i = t_iz_i for some z_i ∈ S, i = 1, 2, ⋯, n}. Then we have α([1])s_i ≤ α([1])t_i in A_S and s_iz_i = t_iz_i for all i. By the assumption, there is an element b ∈ B such that α([1]) = ψ(b) and bs_i ≤ bt_i for all i. Now we define a mapping β : S/ρ → B_S 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 : B_S → A_S be an S-poset epimorphism and let as_i ≤ at_i and s_iz_i = t_iz_i for a ∈ A, s_i, t_i, z_i ∈ S, i = 1, 2, ⋯, n. Consider the S-poset congruence ρ on S_S induced by the set {(s_i, t_i)| i = 1, 2, ⋯, n}. Then α : S/ρ → A_S, given by α([s]) = as for s ∈ S, is a well-defined S-poset morphism. By the assumption, there exists an S-poset morphism β : S/ρ → B_S such that α = ψβ. Thus, we can deduce a = α([1]) = ψ(β([1])), and β([1])s_i = β([s_i]) ≤ β([t_i]) = β([1])t_i in B_S for all i, exactly as needed. □

For the sake of completeness we now prove the following

Proposition 2.8

For any rightS-posetA_S, the following statements are equivalent.

A_Shas Condition (E′).
Every surjectiveS-poset morphismB_S → A_Sis 1′-pure.
There exists a 1′-pure epimorphismB_S → A_S, whereB_Shas Condition (E′).
EveryS-poset morphismS/ρ → A_Smay be factorized through a free rightS-poset, whereρis as inLemma 2.7.
Every surjectiveS-poset morphismB_S → A_Sis quasi-1′-pure.
There exists a quasi-1′-pure epimorphismB_S → A_S, whereB_Shas Condition (E′).

Proof

(1) ⇒ (2). Let, first, ψ : B_S → A_S be a surjective S-poset morphism, and let as_i ≤ at_i and s_iz_i = t_iz_i for some a ∈ A and s_i, t_i, z_i ∈ S, i = 1, 2, ⋯, n. We start by applying Condition (E′) for i = 1, and obtain a₁ ∈ A and u₁ ∈ S with a = a₁u₁ and u₁s₁ ≤ u₁t₁. We substitute the expression a = a₁u₁ into the relation as₂ ≤ at₂ (for i = 2), and get a₁u₁s₂ ≤ a₁u₁t₂ and u₁s₂z₂ = u₁t₂z₂. Again using Condition (E′), yields a₂ ∈ A and u₂ ∈ S such that a₁ = a₂u₂ and u₂u₁s₂ ≤ u₂u₁t₂, and so we have a = a₂u₂u₁. Continuing in this way, we end up with expressions a = a_nu_n ⋯ u₁ and (u_n ⋯ u₁)s_n ≤ (u_n ⋯ u₁)t_n (for i = n). Denoting u = u_n ⋯ u₁, we have a = a_nu and us_i ≤ ut_i for all i. Applying surjectivity of ψ, there exists b ∈ B with ψ(b) = a_n. Therefore, a = a_nu = ψ(bu), and bus_i ≤ but_i in B_S 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/ρ → A_S is an S-poset morphism, where ρ = ν(H), for some set H = {(s_i, t_i) ∈ S × S | s_iz_i = t_iz_i for some z_i ∈ S, i = 1, 2, ⋯, n}. By (3) there exists a 1′-pure epimorphism ψ : B_S → A_S, where B_S has Condition (E′). In view of Lemma 2.7, there exists an S-poset morphism β : S/ρ → B_S such that α = ψβ. Then we see β([1])s_i ≤ β([1])t_i in B_S and s_iz_i = t_iz_i for all i. Also, as B_S satisfies Condition (E′), there exist b ∈ B and u ∈ S with β([1]) = bu and us_i ≤ ut_i for all i. So the mappings ϕ : S/ρ → S_S and φ : S_S → A_S, defined by ϕ([s]) = us and φ(s) = ψ(bs), respectively, are S-poset morphisms such that α = φϕ, as required.

(4) ⇒ (1). Let as ≤ at and sz = tz for a ∈ A and s, t, z ∈ S. We consider the S-poset congruence ρ on S_S induced by the pair (s, t). Then the mapping α : S/ρ → A_S, defined by α([u]) = au, is a well-defined S-poset morphism. By assumption there are a free S-poset B_S and two S-poset morphisms β : S/ρ → B_S and ψ : B_S → A_S such that α = ψβ. Then [s] ≤ [t] implies β([1])s ≤ β([1])t in B_S. Since B_S is free, but it must also be Condition (E′), from the last inequality and sz = tz we obtain b ∈ B and v ∈ S with β([1]) = bv and vs ≤ vt. 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

LetS = {1, x|x² = 1} with the order ofSto be discrete. Clearly, Sis a pomonoid. Now consider anS-poset epimorphismψ : S_S → Θ_S. It is not hard to verify thatψis quasi-1′-pure, and soΘ_Ssatisfies Condition (E′) by Proposition 2.8. However, ψis not quasi-1-pure, becauseθ · 1 ≤ θ · x, but there cannot exists ∈ Ssuch thatψ(s) = θand s · 1 ≤ s · x. It follows from Proposition 2.4 thatΘ_Sdoes not satisfy Condition (E).

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

Definition 2.10

Letψ : B_S → A_S be a surjective S-poset morphism. We say ψ is (weakly) quasi-2-pure, if for any a, a′ ∈ A and as ≤ a′t, there exist b, b′ ∈ B such that bs ≤ b′t, 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<x<x2<⋯

and 0 is isolated. Then S is a pomonoid. Consider the natural epimorphism f : S_S → S/ρ(1, x), where ρ(1, x) is an S-poset congruence on S_S generated by the pair (x, 1). Since xρ(x, 1) 1, this implies [1]x ≤ [1]1, but there cannot exist u ∈ S\{0} with ux ≤ u, 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 (P_w).

Proposition 2.11

LetA_Sbe a rightS-poset. Then the following assertions are equivalent.

A_Shas Condition (P) (Condition (P_w)).
Every surjectiveS-poset morphismB_S → A_Sis (weakly) quasi-2-pure.
There exists a (weakly) quasi-2-pure epimorphismB_S → A_SwhereB_Shas Condition (P).

Proof

(1) ⇒ (2). We deal only with Condition (P), the proof for Condition (P_w) being similar. Now suppose that ψ : B_S → A_S is an S-poset epimorphism and suppose that a, a′ ∈ A, s, t ∈ S are such that as ≤ a′t in A_S. Since A_S satisfies Condition (P), there exist u, v ∈ S and a″ ∈ A such that a = a″u, a′ = a″v and us ≤ vt. Applying surjectivity of ψ, we obtain b ∈ B with ψ(b) = a″. Thus, we have a = a″u = ψ(bu), a′ = a″v = ψ(bv), and bus ≤ bvt in B_S, as required.

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

(3) ⇒ (1). Let A_S be a right S-poset. By (3) there exists a quasi-2-pure epimorphism ψ : B_S → A_S, where B_S satisfies Condition (P). Now suppose that as ≤ a′t in A_S for a, a′ ∈ A, s, t ∈ S. Then we see that bs ≤ b′t in B_S, ψ(b′) = a′ and ψ(b) = a for some b, b′ ∈ B. Also, because B_S satisfies Condition (P), from the inequality bs ≤ b′t, we obtain b″ ∈ B and u, v ∈ S with b = b″u, b′ = b″v and us ≤ vt. Consequently, a = ψ(b) = ψ(b″u) = ψ(b″)u and a′ = ψ(b′) = ψ(b″v) = ψ(b″)v in A_S, 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

LetSbe a pomonoid and letψ : B_S → A_Sbe anS-poset epimorphism in whichB_Shas Condition (P). Ifψis 2-pure thenA_Shas Condition (P).

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

Example 2.13

LetS = {1, x|x² = x} with the order ofSto be discrete. LetA = {a, b|a · 1 = a, a · x = b · x = b · 1 = bwith the ordera < b. Define anS-poset epimorphism f : S_S → A_Sbyf(1) = aandf(x) = b. Since a · 1 ≤ b · x, and we want to reach 1 · 1 ≤ x · x, but this is impossible. Hencefis not quasi-2-pure, and so by Proposition 2.11, A_Sfails to satisfy Condition (P). However, it is easy to see thatfis weakly quasi-2-pure, and soA_Ssatisfies Condition (P_w) 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

LetA_Sbe a rightS-poset. Then the following assertions are equivalent.

A_Sis strongly flat.
A_Shas Conditions (P) and (E).
Every surjectiveS-poset morphismB_S → A_Sis pure.
EveryS-poset morphismB_S → A_SwhereB_Sis finitely presented factors through a finitely generated, freeS-poset.
A_Sis isomorphic to a directed colimit of a family of finitely generated, freeS-posets.
A_Sis subpullback flat and subequalizer flat.
Every surjectiveS-poset morphismB_S → A_Sis 2-pure.
There exists a 2-pure epimorphismB_S → A_SwhereB_Sis strongly flat.
Every surjectiveS-poset morphismB_S → A_Sis quasi-2-pure and (quasi-)1-pure.
There exists a quasi-2-pure and (quasi-)1-pure epimorphismB_S → A_SwhereB_Sis 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

Letψ : B_S → A_Sbe a surjectiveS-poset morphism. We sayψis

(weakly) quasi-w-2-pure if, for all elementsa, a′ ∈ A, s, t ∈ S, allS-poset morphisms f : _S(Ss ∪ St) → _SS, andaf(s) ≤ a′f(t), there existb, b′ ∈ Band s′, t′ ∈ {s, t} such thatbf(s′) ≤ b′f(t′), a ⊗ s = (≤) ψ(b) ⊗ s′ andψ(b′) ⊗ t′ = (≤) a′ ⊗ tin A ⊗_S (Ss ∪ St).
(weakly) quasi-pw-2-pure if, for anya, a′ ∈ Aandas ≤ a′s, there existb, b′ ∈ Bsuch that bs ≤ b′s, a = (≤) ψ(b) andψ(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

LetA_Sbe a rightS-poset. Then the following assertions are equivalent.

A_Shas Condition (WP) (Condition (WP)_w).
Every surjectiveS-poset morphismB_S → A_Sis (weakly) quasi-w-2-pure.
There exists a (weakly) quasi-w-2-pure epimorphismB_S → A_SwhereB_Shas Condition (WP).

Proposition 2.17

LetA_Sbe a rightS-poset. Then the following assertions are equivalent.

A_Shas Condition (PWP) (Condition (PWP)_w).
Every surjectiveS-poset morphismB_S → A_Sis (weakly) quasi-pw-2-pure.
There exists a (weakly) quasi-pw-2-pure epimorphismB_S → A_SwhereB_Shas 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 ( Pw′) for S-posets.

Definition 2.18

LetA_Sbe a rightS-poset. We say that

A_Ssatisfies Condition (P′) if, for alla ∈ A, s, s′, z ∈ Swithas ≤ a′s′ andsz = s′z, there exista″ ∈ A, u, v ∈ Ssuch thata = a″u, a′ = a″vand us ≤ vs′.
A_Ssatisfies Condition ( Pw′) if, for alla ∈ A, s, s′, z ∈ Swithas ≤ a′s′ andsz = s′z, there exista″ ∈ A, u, v ∈ Ssuch that a ≤ a″u, a″v ≤ a′ and us ≤ vs′.

Now, we define two kinds of purity conditions of epimorphisms in order to consider Conditions (P′) and ( Pw′).

Definition 2.19

Letψ : B_S → A_Sbe a surjectiveS-poset morphism. We sayψ is

quasi-2′-pure if, for every a, a′ ∈ A and relations
as≤a′tandsz=tz,
there exist b, b′ ∈ Bsuch thata = ψ(b), ψ(b′) = a′, and bs ≤ bt.
weakly quasi-2′-pure if, for every a, a′ ∈ Aand relations
as≤a′tandsz=tz,
there exist b, b′ ∈ Bsuch that a ≤ ψ(b), ψ(b′) ≤ a′, and bs ≤ bt.

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

Proposition 2.20

LetA_Sbe a rightS-poset. Then the following assertions are equivalent.

A_Shas Condition (P′) (Condition (P′_w)).
Every surjectiveS-poset morphismB_S → A_Sis (weakly) quasi-2′-pure.
There exists a (weakly) quasi-2′-pure epimorphismB_S → A_SwhereB_Shas 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

⟺	Purity conditions of epimorphisms ψ : B_S → A_S
A_S is strongly flat	ψ is pure or 2-pure
A_S has Condition (E)	ψ is 1-pure or quasi-1-pure
A_S has Condition (E′)	ψ is 1′-pure or quasi-1′-pure
A_S has Condition (P)	ψ is quasi-2-pure
A_S has Condition (P′)	ψ is quasi-2′-pure
A_S has Condition (P_w)	ψ is weakly quasi-2-pure
A_S has Condition (P′_w)	ψ is weakly quasi-2′-pure
A_S has Condition (WP)	ψ is quasi-w-2-pure
A_S has Condition (WP)_w	ψ is weakly quasi-w-2-pure
A_S has Condition (PWP)	ψ is quasi-pw-2-pure
A_S has Condition (PWP)_w	ψ is weakly quasi-pw-2-pure

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 (A_i)_i∈I and a collection of right S-poset morphisms ϕ_i,j : A_i → A_j (i ≤ j) with the following properties:

ϕ_i,i = 1_{A_i} for all i ∈ I;
ϕ_j,k ∘ ϕ_i,j = ϕ_i,k, whenever i ≤ j ≤ k.

The colimit of the system (A_i, ϕ_i,j) is a right S-poset A_S together with right S-poset morphisms ϕ_i : A_i → A such that

ϕ_j ∘ ϕ_i,j = ϕ_i, whenever i ≤ j;
If B_S is a right S-poset and α_i : A_i → B are right S-poset morphisms such that α_j ∘ ϕ_i,j = α_i whenever i ≤ j, then there exists a unique S-poset morphism ψ : A → B such that the diagram
commutes for all i ∈ I.

Further, if the indexing set I satisfies the property that for all i, j ∈ I there exists k ∈ I such that k ≥ i, j, then we say that I is directed. In this case, we call the colimit (A, ϕ_i) 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 ((A_i)_i∈I, (ϕ_i,j)_i≤j) of rightS-posets exists, and may be represented as (A/θ, (ϕ_i)_i∈I), where

A=∐i∈IAi;
aθa′ (a ∈ A_i, a′ ∈ A_j) if and only ifϕ_i,k(a) = ϕ_j,k(a′) inA_kfor some k ≥ i, j;
[a]_θ ≤ [a′]_θ (a ∈ A_i, a′ ∈ A_j) if and only ifϕ_i,k(a) ≤ ϕ_j,k(a′) for some k ≥ i, j;
for eachi ∈ Ianda ∈ A_i, ϕ_i(a) = [a]_θ.

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

Lemma 3.2

([5, Proposition 2.6]). Let (A_i)_i∈I, (ϕ_i,j)_i≤jbe a direct system of right S-posets. Then the directed colimit (A, (ϕ_i)_i∈I) is characterized up to isomorphism by the conditions

eachϕ_i : A_i → A is an S-poset morphism;
ϕ_j ∘ ϕ_i,j = ϕ_i, wheneveri ≤ j;
A=⋃i∈Iϕi[Ai];
ϕi−1[R]⊆⋃j≥iϕi,j−1[Rj],whereR (resp. R_j) denotes the graph of the relation ≤ in theS-posetA (resp. A_j).

Actually, the property (4) of Lemma 3.2 states the order relation on A_S, that is, ϕ_i (a) ≤ ϕ_j(a′) if and only if ϕ_i,k(a) ≤ ϕ_j,k(a′) for some k ≥ i, j. Thereby, we obtain the fact that ϕ_i is an order-embedding if and only if ϕ_i,j is an order-embedding for all i ≤ j.

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

Lemma 3.3

Let (A_i, ϕ_i,j) be a direct system of rightS-posets with directed index setIand directed colimit (A, ϕ_i). For every familya₁, ⋯, a_m ∈ Aand relations

aαisi≤aβiti(i=1,⋯,n),

there exist somel ∈ Ianda1′,⋯,am′ ∈ A_lsuch thatϕ_l( aj′ ) = a_jfor all 1 ≤ j ≤ m, andaαi′si≤aβi′tifor all 1 ≤ i ≤ n.

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 (A_i, ϕ_i,j) and (B_i, φ_i,j) are direct systems of S-posets and S-poset morphisms. Suppose that for each i ∈ I there exists an S-poset morphism ψ_i: A_i → B_i and suppose that (A, ϕ_i) and (B, φ_i), the directed colimits of these systems, are such that the diagrams

commute for all i ≤ j ∈ I. Then we shall refer to ψ as the directed colimit of theψ_i.

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

Proposition 3.4

LetSbe a pomonoid. Directed colimits of (order-embeddings) epimorphisms ofS-posets are (order-embeddings) epimorphisms.

We now establish one of our main results.

Proposition 3.5

LetSbe a pomonoid. Directed colimits of pure epimorphisms of rightS-posets are pure.

Proof

Suppose that (A_i, ϕ_i,j) and (B_i, φ_i,j) are direct systems of right S-posets and S-poset morphisms. Suppose that for each i ∈ I there exists a pure epimorphism ψ_i: A_i → B_i and suppose that (A, ϕ_i) and (B, φ_i), the directed colimits of these systems, are such that the diagrams

commute for all i ≤ j ∈ I.

Suppose that there are b₁, ⋯, b_m ∈ B, s₁, ⋯, s_n, t₁, ⋯, t_n ∈ S and relations

bαisi≤bβiti(i=1,⋯,n).

In view of Lemma 3.3, there exist l ∈ I and b1′,⋯,bm′ ∈ B_l such that φ_l( bj′ ) = b_j for all 1 ≤ j ≤ m, and

bαi′si≤bβi′ti(i=1,⋯,n).

Since ψ_l is a pure epimorphism, there exist a₁, ⋯, a_m ∈ A_l with ψ_l(a_j) = bj′ for all 1 ≤ j ≤ m, and a_{α_i}s_i ≤ a_{β_i}t_i for all 1 ≤ i ≤ n. 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

LetSbe a pomonoid. Directed colimits of 1-pure (1′-pure) epimorphisms of rightS-posets are 1-pure (1′-pure).

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

Proposition 3.7

LetSbe a pomonoid. Directed colimits of quasi-2-pure epimorphisms of rightS-posets are quasi-2-pure.

Proof

Suppose that (A_i, ϕ_i,j) and (B_i, φ_i,j) are direct systems of right S-posets and S-poset morphisms. Suppose that for each i ∈ I there exists a quasi-2-pure epimorphism ψ_i: A_i → B_i and suppose that (A, ϕ_i) and (B, φ_i), the directed colimits of these systems, are such that the diagrams

commute for all i ≤ j ∈ I.

Suppose that b₁, b₂ ∈ B, s, t ∈ S are such that b₁s ≤ b₂t. By Lemma 3.2(3), there exist i, j ∈ I, b_i ∈ B_i and b_j ∈ B_j with b₁ = φ_i(b_i) and b₂ = φ_j(b_j). According to the order relation on B_S, we obtain k ≥ i, j with φ_i,k(b_i)s ≤ φ_j,k(b_j)t in B_k. Since ψ_k is quasi-2-pure, there exist a₁, a₂ ∈ A_k such that ψ_k(a₁) = φ_i,k(b_i), ψ_k(a₂) = φ_j,k(b_j) and a₁s ≤ a₂t. Now we can compute that ϕ_k(a₁)s ≤ ϕ_k(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(ϕ_k(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

LetSbe a pomonoid. Every directed colimit of a direct system of rightS-posets that are strongly flat (resp., have Conditions (E), (E′), (P), (P′), (P_w), (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

LetSbe a pomonoid. Every directed colimit of a direct system of projective rightS-posets is projective if and only ifSis 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 A_S is a strongly flat S-poset. It then follows from [5, Proposition 4.4] that A_S is isomorphic to a directed colimit of a family of finitely generated, free S-posets. So by assumption, A_S is projective since free S-posets are projective.

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

From [5, Proposition 4.6], it follows that an S-poset epimorphism φ: B_S → A_S is pure if and only if, for every finitely presented S-poset C_S and morphism η : C_S → A_S there exists a morphism μ : C_S → B_S 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 Θ_S and note that S_S → Θ_S 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

Letψ: A_S → B_Sbe a surjectiveS-poset morphism withB_Sis finitely presented. Thenψis pure if and only if it is split.

For future use, we record

Lemma 3.11

LetSbe a pomonoid, let

be a pullback diagram ofS-posets and suppose thatψis a pure epimorphism. Thenϕis 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

LetSbe a pomonoid and letψ: A_S → B_Sbe a surjectiveS-poset morphism. Thenψis pure if and only if it is a directed colimit of split epimorphisms.

Proof

Suppose that ψ: A_S → B_S is a pure epimorphism. From [5, Proposition 4.4] it follows that B_S is a directed colimit of finitely presented S-posets (B_i, ϕ_i,j) and so let ϕ_i: B_i → B be the canonical morphisms. For each B_i let

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

Ai={(bi,a)∈Bi×A| ϕi(bi)=ψ(a)},

ψ_i(b_i, a) = b_i and φ_i(b_i, a) = a, and that since ψ is onto then A_i ≠ ∅.

For i ≤ j define φ_i,j: A_i → A_j by φ_i,j(b_i, a) = (ϕ_i,j(b_i), a) and notice that φ_jφ_i,j = φ_i and ψ_jφ_i,j = ϕ_i,jψ_i. Next we show that (A, φ_i) is the directed colimit of (A_i, φ_i,j). So assume there exist an S-poset C_S and S-poset morphisms α_i: A_i → C with α_jφ_i,j = α_i for all i ≤ j. We now define α : A → C by α(a) = α_i(b_i, a) where i and b_i are chosen so that φ_i(b_i) = ψ(a). Then α is well-defined since if ψ(a) = ϕ_j(b_j) then there exists k ≥ i, j with ϕ_i,k(b_i) = ϕ_j,k(b_j) 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 αφ_i = α_i. Finally, if β: A → C is such that βφ_i = α_i for all i, then β(a) = βφ_i(b_i, a) = α_i(b_i, 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. f ≤ g for f, g: A → B if and only if f(a) ≤ g(a) for every a ∈ A). In such categories, a diagram

is called a subpullback diagram forαandβ if

αp₁ ≤ βp₂ and
for any diagram

with αp1′≤βp2′, there exists a unique morphism φ: P′ → P such that p_iφ = pi′ 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 A_S one gets the diagram

in Pos. Take

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

with p1′,p2′ being the restrictions of the projections. From the definition of subpullbacks it follows the existence of a unique monotonic mapping ϕ: A_S ⊗ _SP → P′ such that pi′ϕ = 1_A ⊗ p_ifori = 1, 2. This mapping is called theϕcorresponding to the subpullback diagramP(M, N, α, β, Q) for A_S. It can be checked that ϕ(a ⊗ (m, n)) = (a ⊗ m, a ⊗ n).

A subequalizer diagram for α and β

is defined similarly, where E = {m ∈ M |α(m) ≤ β(m)}. As we mentioned earlier, an S-poset A_S is called subpullback flat (subequalizer flat) if the functor A_S ⊗ − 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 A_S is an S-subposet of B_S and x, y are different elements not belonging to B_S, 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 a ∈ B_S ∖ A_S, s ∈ S and w ∈ {x, y}.

The order on B_S ∐^AB_S 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, b ∉ A_S. For w ∈ {x, y}, b ∉ A_S and a ∈ A,

(b,w)≤a⇔b<a and a≤(b,w)⇔a<b.

It is easily checked B_S ∐^AB_S is a right S-poset.

Lemma 4.1

Every regular monomorphismh : A_S → B_Sin Pos-Scan be consider as the subequalizer of the diagram

Proof

Let h : A_S → B_S be a regular monomorphism. Define α, β: B_S → B_S ∐^h(A)B_S as

α(b)=(b,x) ifb∉h(A)b otherwise

and

β(b)=(b,y) ifb∉h(A)b otherwise.

To show that (A_S, h) is a subequalizer diagram for α and β, clearly αh ≤ βh, and consider the diagram

where C_S and f are such that αf ≤ βf. By the construction of B_S∐^h(A)B_S we see that f(C_S) ⊆ h(A_S). Now, since h is a regular monomorphism, one can define a mapping f : C_S → A_S by f(c) = h⁻¹(f(c)) for c ∈ C_S. 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 (A_i, ϕ_i,j) be a direct system of S-psets and S-morphisms with directed index set I and directed colimit (A, ϕ_i). If every A_i 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 A_S we have the following diagram

and 1_A ⊗ l is a regular monomorphism. By the definition, the subequalizer of 1_A ⊗ α and 1_A ⊗ β 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 A ⊗ E ⊆ E′, next we want to show that E′ ⊆ A ⊗ E. Let a ∈ A, m ∈ M be such that a ⊗ m ∈ E′. So a ⊗ α(m) ≤ a ⊗ β(m) in A ⊗ N implies that the existence of u₁, v₁, …, u_n, v_n ∈ S, a₁, …, a_n ∈ A_S, y₂, …, y_n ∈ _SN, n ∈ ℕ such that

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

Denote a by a₀, then there exist a_{i_j} ∈ A_{i_j} such that a_j = ϕ_{i_j}(a_{i_j}), 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 l ≥ i₁, i₂, …, i_n 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 ϕ_i₀,l(a_i₀) ⊗ α(m) ≤ ϕ_i₀,l(a_i₀) ⊗ β(m) in A_l ⊗ N. Now, from the fact that A_l is subequalizer flat, and A_l ⊗ E is the subequalizer of 1_{A_l} ⊗ α and 1_{A_l} ⊗ β, it follows that ϕ_i₀,l(a_i₀) ⊗ m ∈ A_l ⊗ E. So m ∈ E, a ⊗ m ∈ A ⊗ E 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 (A_i, ϕ_i,j) be a direct system of S-posets with directed index set I and let (A, ϕ_i) be the directed colimit. Suppose that for each A_i 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 (x ⊗ m, y ⊗ n) belongs to the subpullback of P(A ⊗ M, A ⊗ N, 1_A ⊗ α, 1_A ⊗ β, A ⊗ Q), where x, y ∈ A, m ∈ M, n ∈ N. Then x ⊗ α(m) ≤ y ⊗ β(n) in A ⊗ Q. We should find a ∈ A, (m, n) ∈ P(m, N, α, β, Q) such that ϕ(A ⊗ (m, n)) = (x ⊗ m, y ⊗ n). Since x ⊗ α(m) ≤ y ⊗ β(n), there exist k ∈ ℕ and elements a₁, …, a_k ∈ A_S, q₂, …, q_k ∈ _SQ, u₁, v₁, …, u_k, v_k ∈ S such that

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

Denote x by a₀ and y by a_k+1, so there exist a_{i_j} ∈ A_{i_j} such that a_j = ϕ_{i_j} ( aij′ ), where i_j ∈ I 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 l ≥ i₀, i₁, …, i_k+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 ϕ_i₀,l( ai0′ ) ⊗ α(m) ≤ ϕ_{i_k+1,l}( aik+1′ ) ⊗ β(n) in A_l ⊗ Q. That is (ϕ_i₀,l (ai0′ ) ⊗ m, ϕ_{i_k+1,l}( aik+1′ ) belongs to the subpullback of P(A_l ⊗ M,A_l ⊗ N, 1_{A_l} ⊗ α, 1_{A_l} ⊗ β, A_l ⊗ Q). By the assumption, for A_l the mapping ϕ corresponding to the subpullback diagram P(M, N, α, β, Q) is surjective, so there exist a″ ∈ A_l, m′ ∈ M and n′ ∈ N such that ϕ(a″ ⊗ (m′, n′)) = (ϕ_i₀,l( ai0′ ) ⊗ m, ϕ_{i_k+1,l}( aik+1′ ) ⊗ n), where α(m′) ≤ β(n′). That is ϕ_i₀,l( ai0′ ) ⊗ m = a″ ⊗ m′ and ϕ_{i_k+1,l}( aik+1′ ) ⊗ n = a″ ⊗ n′. Since ϕ_i₀,l( ai0′ ) ⊗ m = a″ ⊗ m′, there exist elements c_i, cj′ ∈ A_l, m₂, …, m_p, m2′ , …, mp′′ ∈ _SM, s_i, t_i, sj′,tj′ ∈ S, 1 ≤ i ≤ p, 1 ≤ j ≤ p′ 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 ϕ_l on left column inequations and since x = ϕ_i₀ ( ai0′ ) = ϕ_lϕ_i₀,l( ai0′ ), 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 x ⊗ m = ϕ_l(a″) ⊗ m′ in A ⊗ M. Since ϕ_{i_k+1,l}( aik+1′ ) ⊗ n = a″ ⊗ n′, by a similar way we can prove y ⊗ n = ϕ_l(a″) ⊗ n′ in A ⊗ N. Therefore, ϕ(ϕ_l(a″) ⊗ (m′, n′)) = (ϕ_l(a″) ⊗ m′, ϕ_l(a″) ⊗ n′) = (x ⊗ m, y ⊗ n), as desired. □

Lemma 4.4

LetSbe a pomonoid, and (A_i, ϕ_i,j) be a direct system ofS-posets with directed index setIand let (A, ϕ_i) be the directed colimit. Suppose that for eachA_ithe mappingϕcorresponding to the subpullback diagramP(m, N, α, β, Q) is an order-embedding, then forAthe mappingϕcorresponding to the subpullback diagramP(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(A ⊗ M, A ⊗ N, 1_A ⊗ α, 1_A ⊗ β, A ⊗ Q), 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 A_S ⊗ _SP. Since a ⊗ m ≤ a′ ⊗ m′ in A_S ⊗ _SM and a ⊗ n ≤ a′ ⊗ n′ in A_S ⊗ _SN, there exist p, p′ ∈ ℕ and elements a_i, c_j ∈ A_S, m₂, …, m_p ∈ _SM, n₂, …, n_p′ ∈ _SN, u_i, v_i, uj′,vj′ ∈ S, 1 ≤ i ≤ p, 1 ≤ j ≤ p′ 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 l ∈ I and a0′,a1′,⋯,ap′,c1′,...,cp′′,c0′ ∈ A_l such that ϕl(a0′)=a,ϕl(ai′) = a, ϕl(cj′)=cj,ϕl(c0′)=a′ for all 1 ≤ i ≤ p, 1 ≤ j ≤ p′, 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 ϕ(a0′⊗(m,n))≤ϕ(c0′⊗(m′,n′)). Since for A_l the mapping ϕ corresponding to the subpullback diagram P(m, N, α, β, Q) is an order-embedding, we imply a0′⊗(m,n)≤c0′⊗(m′,n′) in A_l ⊗ _SP. So there exist d₁, …, d_r ∈ A_l, (m₂, n₂), …, (m_r,n_r) ∈ _SP, x₁, y₁, …, x_r, y_r ∈ S, r ∈ ℕ such that

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

Acting ϕ_l on left column inequations and since a=ϕl(a0′) anda′=ϕl(c0′), 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 A_S ⊗ _SP 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 ofS-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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Received: 2017-09-01

Accepted: 2018-04-11

Published Online: 2018-07-04

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Articles in the same Issue

https://doi.org/10.1515/math-2018-0061

Keywords for this article

S-poset; Flatness property; Pure epimorphism; Directed colimit

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