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 for* *α* *and* *β* if

*α* *p*_{1} ≤ *βp*_{2} and

for any diagram

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

$$\begin{array}{}{\displaystyle P=\{(m,n)\in M\times N|\text{\hspace{0.17em}}\alpha (m)\le \beta (n)\}.}\end{array}$$

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

$$\begin{array}{}{\displaystyle {P}^{\prime}=\{(a\otimes m,a\otimes n)\in ({A}_{S}\otimes {}_{S}M)\otimes ({A}_{S}\otimes {}_{S}N)|\text{\hspace{0.17em}}a\otimes \alpha (m)\le a\otimes \beta (n)\}}\end{array}$$

with
$\begin{array}{}{\displaystyle {p}_{1}^{\prime},{p}_{2}^{\prime}}\end{array}$
being the restrictions of the projections. From the definition of subpullbacks it follows the existence of a unique monotonic mapping *ϕ*: *A*_{S} ⊗ _{S}P → *P*′ such that
$\begin{array}{}{p}_{i}^{\prime}\varphi \end{array}$
= 1_{A} ⊗ *p*_{i} *for* *i* = 1, 2. This mapping is called *the* *ϕ* *corresponding to the subpullback diagram* *P*(*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

$$\begin{array}{}{\displaystyle {B}_{S}{\coprod}^{A}{B}_{S}=\{(a,x)|\text{\hspace{0.17em}}a\notin {A}_{S}\}\dot{\cup}{A}_{S}\dot{\cup}\{(a,y)|\text{\hspace{0.17em}}a\notin {A}_{S}\}.}\end{array}$$

Define an *S*-action

$$\begin{array}{}{\displaystyle (a,w)s=\left\{\begin{array}{}(as,w)& \text{\hspace{0.17em}if}\phantom{\rule{1em}{0ex}}as\notin {A}_{S}\\ as& \text{\hspace{0.17em}otherwise,}\end{array}\right.}\end{array}$$

for every *a* ∈ *B* _{S} ∖ *A*_{S}, *s* ∈ *S* and *w* ∈ {*x*, *y*}.

The order on *B*_{S} ∐^{A} *B*_{S} is given by

$$\begin{array}{}{\displaystyle (a,{w}_{1})\le (b,{w}_{2})\iff ({w}_{1}={w}_{2},\text{\hspace{0.17em}}a\le b)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}({w}_{1}\ne {w}_{2},\text{\hspace{0.17em}}a\le {a}^{\prime}\le b\text{\hspace{0.17em}}\text{for some}\text{\hspace{0.17em}}{a}^{\prime}\in {A}_{S}),}\end{array}$$

where {*w*_{1}, *w*_{2}} = {*x*, *y*} and *a*, *b* ∉ *A*_{S}. For *w* ∈ {*x*, *y*}, *b* ∉ *A*_{S} and *a* ∈ *A*,

$$\begin{array}{}{\displaystyle (b,w)\le a\iff b<a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\le (b,w)\iff a<b.}\end{array}$$

It is easily checked *B*_{S} ∐^{A} *B*_{S} is a right *S*-poset.

#### Lemma 4.1

*Every regular monomorphism* *h* : *A*_{S} → *B*_{S} *in Pos*-*S* *can 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

$$\begin{array}{}{\displaystyle \alpha (b)=\left\{\begin{array}{}(b,x)& \text{\hspace{0.17em}if}\phantom{\rule{1em}{0ex}}b\notin h(A)\\ b& \text{\hspace{0.17em}otherwise}\end{array}\right.}\end{array}$$

and

$$\begin{array}{}{\displaystyle \beta (b)=\left\{\begin{array}{}(b,y)& \text{\hspace{0.17em}if}\phantom{\rule{1em}{0ex}}b\notin h(A)\\ b& \text{\hspace{0.17em}otherwise.}\end{array}\right.}\end{array}$$

To show that (*A*_{S}, *h*) is a subequalizer diagram for *α* and *β*, clearly *αh* ≤ *β* *h*, 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

$$\begin{array}{}{\displaystyle {E}^{\prime}=\{a\otimes m\in A\otimes M|\text{\hspace{0.17em}}({1}_{A}\otimes \alpha )(a\otimes m)\le ({1}_{A}\otimes \beta )(a\otimes m),a\in A,m\in M\}.}\end{array}$$

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 *u*_{1}, *v*_{1}, …, *u*_{n}, *v*_{n} ∈ *S*, *a*_{1}, …, *a*_{n} ∈ *A*_{S}, *y*_{2}, …, *y*_{n} ∈ _{S} *N*, *n* ∈ ℕ such that

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a\le {a}_{1}{u}_{1}}\\ {a}_{1}{v}_{1}\le {a}_{2}{u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\alpha (m)\le {v}_{1}{y}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\vdots \\ {a}_{n}{v}_{n}\le a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{u}_{n}{y}_{n}\le {v}_{n}\beta (m).\end{array}$$

Denote *a* by *a*_{0}, then there exist *a*_{ij} ∈ *A*_{ij} such that *a*_{j} = *ϕ*_{ij}(*a*_{ij}), *j* = 0, 1, .., *n*. Then we get

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{\varphi}_{{i}_{0}}({a}_{{i}_{0}})\le {\varphi}_{{i}_{1}}({a}_{{i}_{1}}){u}_{1}}\\ {\varphi}_{{i}_{1}}({a}_{{i}_{1}}){v}_{1}\le {\varphi}_{{i}_{2}}({a}_{{i}_{2}}){u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\alpha (m)\le {v}_{1}{y}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \\ {\varphi}_{{i}_{n}}({a}_{{i}_{n}}){v}_{n}\le {\varphi}_{{i}_{0}}({a}_{{i}_{0}})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}{u}_{n}{y}_{n}\le {v}_{n}\beta (m).\end{array}$$

Since *I* is directed, there exists *l* ≥ *i*_{1}, *i*_{2}, …, *i*_{n} such that

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{\varphi}_{{i}_{0},l}({a}_{{i}_{0}})\le {\varphi}_{{i}_{1},l}({a}_{{i}_{1}}){u}_{1}}\\ \phantom{\rule{thinmathspace}{0ex}}{\varphi}_{{i}_{1},l}({a}_{{i}_{1}}){v}_{1}\le {\varphi}_{{i}_{2},l}({a}_{{i}_{2}}){u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\alpha (m)\le {v}_{1}{y}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \\ {\varphi}_{{i}_{n},l}({a}_{{i}_{n}}){v}_{n}\le {\varphi}_{{i}_{0},l}({a}_{{i}_{0}})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{u}_{n}{y}_{n}\le {v}_{n}\beta (m).\end{array}$$

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_{Al} ⊗ *α* and 1_{Al} ⊗ *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 *a*_{1}, …, *a*_{k} ∈ *A*_{S}, *q*_{2}, …, *q*_{k} ∈ _{S}Q, *u*_{1}, *v*_{1}, …, *u*_{k}, *v*_{k} ∈ *S* such that

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x\le {a}_{1}{u}_{1}}\\ {a}_{1}{v}_{1}\le {a}_{2}{u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\alpha (m)\le {v}_{1}{q}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\vdots \\ {a}_{k}{v}_{k}\le y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{u}_{k}{q}_{k}\le {v}_{k}\beta (n).\end{array}$$

Denote *x* by *a*_{0} and *y* by *a*_{k+1}, so there exist *a*_{ij} ∈ *A*_{ij} such that *a*_{j} = *ϕ*_{ij} (
$\begin{array}{}{\displaystyle {a}_{{i}_{j}}^{\prime}}\end{array}$
), where *i*_{j} ∈ *I* and *j* = 0, 1, …, *k*, *k* + 1. Hence we have

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{\varphi}_{{i}_{0}}({a}_{{i}_{0}}^{\prime})\le {\varphi}_{{i}_{1}}({a}_{{i}_{1}}^{\prime}){u}_{1}}\\ {\varphi}_{{i}_{1}}({a}_{{i}_{1}}^{\prime}){v}_{1}\le {\varphi}_{{i}_{2}}({a}_{{i}_{2}}^{\prime}){u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\alpha (m)\le {v}_{1}{q}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \\ {\varphi}_{{i}_{k}}({a}_{{i}_{k}}^{\prime}){v}_{k}\le {\varphi}_{{i}_{k+1}}({a}_{{i}_{k+1}}^{\prime})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}{u}_{k}{q}_{k}\le {v}_{k}\beta (n).\end{array}$$

Since *I* is directed, there exists *l* ≥ *i*_{0}, *i*_{1}, …, *i*_{k+1} such that

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{\varphi}_{{i}_{0}}({a}_{{i}_{0}}^{\prime})\le {\varphi}_{{i}_{1}}({a}_{{i}_{1}}^{\prime}){u}_{1}}\\ {\varphi}_{{i}_{1}}({a}_{{i}_{1}}^{\prime}){v}_{1}\le {\varphi}_{{i}_{2}}({a}_{{i}_{2}}^{\prime}){u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\alpha (m)\le {v}_{1}{q}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \\ {\varphi}_{{i}_{k}}({a}_{{i}_{k}}^{\prime}){v}_{k}\le {\varphi}_{{i}_{k+1}}({a}_{{i}_{k+1}}^{\prime})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}{u}_{k}{q}_{k}\le {v}_{k}\beta (n).\end{array}$$

Then *ϕ*_{i0,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
) ⊗ *α*(*m*) ≤ *ϕ*_{ik+1,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{k+1}}^{\prime}}\end{array}$
) ⊗ *β*(*n*) in *A*_{l} ⊗ *Q*. That is (*ϕ*_{i0,l}
($\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
) ⊗ *m*, *ϕ*_{ik+1,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{k+1}}^{\prime}}\end{array}$
) belongs to the subpullback of *P*(*A*_{l} ⊗ *M*,*A*_{l} ⊗ *N*, 1_{Al} ⊗ *α*, 1_{Al} ⊗ *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}(
$\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
) ⊗ *m*, *ϕ*_{ik+1,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{k+1}}^{\prime}}\end{array}$
) ⊗ *n*), where *α*(*m*′) ≤ *β*(*n*′). That is *ϕ*_{i0,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
) ⊗ *m* = *a*″ ⊗ *m*′ and
*ϕ*_{ik+1,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{k+1}}^{\prime}}\end{array}$
) ⊗ *n* = *a*″ ⊗ *n*′. Since *ϕ*_{i0,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
) ⊗ *m* = *a*″ ⊗ *m*′, there exist elements *c*_{i},
$\begin{array}{}{\displaystyle {c}_{j}^{\prime}}\end{array}$
∈ *A*_{l}, *m*_{2}, …, *m*_{p},
$\begin{array}{}{\displaystyle {m}_{2}^{\prime}}\end{array}$
, …,
$\begin{array}{}{\displaystyle {m}_{{p}^{\prime}}^{\prime}}\end{array}$
∈ _{S}M, *s*_{i}, *t*_{i},
$\begin{array}{}{\displaystyle {s}_{j}^{\prime},{t}_{j}^{\prime}}\end{array}$
∈ *S*, 1 ≤ *i* ≤ *p*, 1 ≤ *j* ≤ *p*′ for *p*, *p*′ ∈ ℕ such that

$$\begin{array}{}{\displaystyle {\varphi}_{{i}_{0},l}({a}_{{i}_{0}}^{\prime})\le {c}_{1}{s}_{1}}\end{array}$$

$$\begin{array}{}{\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1}{t}_{1}\le {c}_{2}{s}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{s}_{1}m\le {t}_{1}{m}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{p}{t}_{p}\le {a}^{\u2033}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{t}_{p}{m}_{p}\le {t}_{p}{m}^{\prime},\\ \text{\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{a}^{\u2033}\le {c}_{1}^{\prime}{s}_{1}^{\prime}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1}^{\prime}{t}_{1}^{\prime}\le {c}_{2}^{\prime}{s}_{2}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{s}_{1}^{\prime}{m}^{\prime}\le {t}_{1}^{\prime}{m}_{2}^{\prime}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \\ {c}_{{p}^{\prime}}^{\prime}{t}_{{p}^{\prime}}^{\prime}\le {\varphi}_{{i}_{0},l}({a}_{{i}_{0}}^{\prime})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}_{{p}^{\prime}}^{\prime}{m}_{{p}^{\prime}}^{\prime}\le {t}_{{p}^{\prime}}^{\prime}m.\end{array}$$

Acting *ϕ*_{l} on left column inequations and since *x* = *ϕ*_{i0} (
$\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
) = *ϕ*_{l}ϕ_{i0,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{0}}^{\prime}}\end{array}$
), we get

$$\begin{array}{}{\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}x\le {\varphi}_{l}({c}_{1}){s}_{1}}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{l}({c}_{1}){t}_{1}\le {\varphi}_{l}({c}_{2}){s}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{1}m\le {t}_{1}{m}_{2}\\ \text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\vdots \\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{l}({c}_{p}){t}_{p}\le {\varphi}_{l}({a}^{\u2033})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}{t}_{p}{m}_{p}\le {t}_{p}{m}^{\prime},\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{l}({a}^{\u2033})\le {\varphi}_{l}({c}_{1}^{\prime}){s}_{1}^{\prime}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{l}({c}_{1}^{\prime}){t}_{1}^{\prime}\le {\varphi}_{l}({c}_{2}^{\prime}){s}_{2}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{1}^{\prime}{m}^{\prime}\le {t}_{1}^{\prime}{m}_{2}^{\prime}\\ \text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\vdots \\ {\varphi}_{l}({c}_{{p}^{\prime}}^{\prime}){t}_{{p}^{\prime}}^{\prime}\le x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{t}_{{p}^{\prime}}^{\prime}{m}_{{p}^{\prime}}^{\prime}\le {t}_{{p}^{\prime}}^{\prime}m.\end{array}$$

Thus *x* ⊗ *m* = *ϕ*_{l}(*a*″) ⊗ *m*′ in *A* ⊗ *M*. Since *ϕ*_{ik+1,l}(
$\begin{array}{}{\displaystyle {a}_{{i}_{k+1}}^{\prime}}\end{array}$
) ⊗ *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

*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 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*(*A* ⊗ *M*, *A* ⊗ *N*, 1_{A} ⊗ *α*, 1_{A} ⊗ *β*, *A* ⊗ *Q*), where *a*, *a*′ ∈ *A* and (*m*, *n*), (*m*′, *n*′) ∈ _{S}P. So we have *α*(*m*) ≤ *β*(*n*), *α*(*m*′) ≤ *β*(*n*′), and

$$\begin{array}{}{\displaystyle a\otimes m\le {a}^{\prime}\otimes {m}^{\prime}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{A}_{S}\otimes {}_{S}M,\text{\hspace{0.17em}}a\otimes n\le {a}^{\prime}\otimes {n}^{\prime}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{A}_{S}\otimes {}_{S}N.}\end{array}$$

We will show that *a* ⊗ (*m*, *n*) ≤ *a*′ ⊗ (*m*′, *n*′) in *A*_{S} ⊗ _{S}P. Since *a* ⊗ *m* ≤ *a*′ ⊗ *m*′ in *A*_{S} ⊗ _{S} *M* and *a* ⊗ *n* ≤ *a*′ ⊗ *n*′ in *A*_{S} ⊗ _{S}N, there exist *p*, *p*′ ∈ ℕ and elements *a*_{i}, *c*_{j} ∈ *A*_{S}, *m*_{2}, …, *m*_{p} ∈ _{S}*M*, *n*_{2}, …, *n*_{p′} ∈ _{S}N, *u*_{i}, *v*_{i},
$\begin{array}{}{\displaystyle {u}_{j}^{\prime},{v}_{j}^{\prime}}\end{array}$
∈ *S*, 1 ≤ *i* ≤ *p*, 1 ≤ *j* ≤ *p*′ such that

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a\le {a}_{1}{u}_{1}}\\ {a}_{1}{v}_{1}\le {a}_{2}{u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}m\le {v}_{1}{m}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \\ \phantom{\rule{thinmathspace}{0ex}}{a}_{p}{v}_{p}\le {a}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}{u}_{p}{m}_{p}\le {v}_{m}{m}^{\prime},\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a\le {c}_{1}{u}_{1}^{\prime}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1}{v}_{1}^{\prime}\le {c}_{2}{u}_{2}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{u}_{1}^{\prime}n\le {v}_{1}^{\prime}{n}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\vdots \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \\ {c}_{{p}^{\prime}}{v}_{{p}^{\prime}}^{\prime}\le {a}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{u}_{{p}^{\prime}}^{\prime}{n}_{{p}^{\prime}}\le {v}_{{p}^{\prime}}^{\prime}{n}^{\prime}.\end{array}$$

By Lemma 3.3, there exist *l* ∈ *I* and
$\begin{array}{}{\displaystyle {a}_{0}^{\prime},{a}_{1}^{\prime},\cdots ,{a}_{p}^{\prime},{c}_{1}^{\prime},...,{c}_{{p}^{\prime}}^{\prime},{c}_{0}^{\prime}}\end{array}$
∈ *A*_{l} such that
$\begin{array}{}{\displaystyle {\varphi}_{l}({a}_{0}^{\prime})=a,{\varphi}_{l}({a}_{i}^{\prime})}\end{array}$
= *a*,
$\begin{array}{}{\displaystyle {\varphi}_{l}({c}_{j}^{\prime})={c}_{j},{\varphi}_{l}({c}_{0}^{\prime})={a}^{\prime}}\end{array}$
for all 1 ≤ *i* ≤ *p*, 1 ≤ *j* ≤ *p*′, and

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{a}_{0}^{\prime}\le {a}_{1}^{\prime}{u}_{1}}\\ {a}_{1}^{\prime}{v}_{1}\le {a}_{2}^{\prime}{u}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}{u}_{1}m\le {v}_{1}{m}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \\ {a}_{p}{v}_{p}\le {c}_{0}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}{u}_{p}{m}_{p}\le {v}_{m}{m}^{\prime},\\ \phantom{\rule{1em}{0ex}}{a}_{0}^{\prime}\le {c}_{1}^{\prime}{u}_{1}^{\prime}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1}^{\prime}{v}_{1}^{\prime}\le {c}_{2}^{\prime}{u}_{2}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{u}_{1}^{\prime}n\le {v}_{1}^{\prime}{n}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \\ \phantom{\rule{thinmathspace}{0ex}}{c}_{{p}^{\prime}}^{\prime}{v}_{{p}^{\prime}}^{\prime}\le {c}_{0}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}{u}_{{p}^{\prime}}^{\prime}{n}_{{p}^{\prime}}\le {v}_{{p}^{\prime}}^{\prime}{n}^{\prime}.\end{array}$$

This means that

$$\begin{array}{}{\displaystyle {a}_{0}^{\prime}\otimes m\le {c}_{0}^{\prime}\otimes {m}^{\prime}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{A}_{l}\otimes {}_{S}M,\text{\hspace{0.17em}}\alpha (m)\le \beta (n),}\\ {a}_{0}^{\prime}\otimes n\le {c}_{0}^{\prime}\otimes {n}^{\prime}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{A}_{l}\otimes {}_{S}N,\text{\hspace{0.17em}}\alpha ({m}^{\prime})\le \beta ({n}^{\prime}).\end{array}$$

Hence
$\begin{array}{}{\displaystyle \varphi ({a}_{0}^{\prime}\otimes (m,n))\le \varphi ({c}_{0}^{\prime}\otimes ({m}^{\prime},{n}^{\prime})).}\end{array}$
Since for *A*_{l} the mapping *ϕ* corresponding to the subpullback diagram *P*(*m*, *N*, *α*, *β*, *Q*) is an order-embedding, we imply
$\begin{array}{}{\displaystyle {a}_{0}^{\prime}\otimes (m,n)\le {c}_{0}^{\prime}\otimes ({m}^{\prime},{n}^{\prime})}\end{array}$
in *A*_{l} ⊗ _{S} *P*. So there exist *d*_{1}, …, *d*_{r} ∈ *A*_{l}, (*m*_{2}, *n*_{2}), …, (*m*_{r},*n*_{r}) ∈ _{S}P, *x*_{1}, *y*_{1}, …, *x*_{r}, *y*_{r} ∈ *S*, *r* ∈ ℕ such that

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{a}_{0}^{\prime}\le {d}_{1}{x}_{1}}\\ {d}_{1}{y}_{1}\le {d}_{2}{x}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}(m,n)\le {y}_{1}({m}_{2},{n}_{2})\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\vdots \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \\ \phantom{\rule{thinmathspace}{0ex}}{d}_{r}{y}_{r}\le {c}_{0}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{r}({m}_{r},{n}_{r})\le {y}_{r}({m}^{\prime},{n}^{\prime}).\end{array}$$

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

$$\begin{array}{}{\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}a\le {\varphi}_{l}({d}_{1}){x}_{1}}\\ {\varphi}_{l}({d}_{1}){y}_{1}\le {\varphi}_{l}({d}_{2}){x}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}(m,n)\le {y}_{1}({m}_{2},{n}_{2})\\ \text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vdots \\ {\varphi}_{l}({d}_{r}){y}_{r}\le {a}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phantom{\rule{2em}{0ex}}{x}_{r}({m}_{r},{n}_{r})\le {y}_{r}({m}^{\prime},{n}^{\prime}).\end{array}$$

Therefore, *a* ⊗ (*m*, *n*) ≤ *a*′ ⊗ (*m*′, *n*′) in *A*_{S} ⊗ _{S}P 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*.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.