In this section, we first show that GP-po-flatness is preserved under coproducts and directed colimits, respectively. Furthermore, we mainly consider the question of when GP-po-flat transfers from *S*-posets to their products. As an application, we also consider the same question to principally weakly po-flat, and extend some results from [12].

The following two propositions show that GP-po-flatness is closed under coproducts and directed colimits, respectively. For more information about coproducts and directed colimits in the category of *S*-posets, the reader is referred to [15, 21].

**Proposition 5.1:** *Let* ${A}_{S}={\coprod}_{i\in I}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{A}_{j}$, *where A*_{j}, i ∈ I, is a strongly convex S-subposet of A_{S}. Then A_{S} is GP-po-flat if and only if every A_{j} is GP-po-flat, i ∈ I.

**Proof:** *It is a direct consequence of the definition.*

*
***Proposition 5.2:** *Every directed colimit of a directed system of GP-po-flat right S-posets, is GP-po-flat*.

**Proof:** *Suppose that (**A*_{j}, *ϕ*_{i, j}) is a directed system of GP-po-flat right *S*-posets over a directed index set *I* with directed colimit (*A, α*_{i}). Let *as* ≤ *a′s* in *A*_{S}, for *a, a′* ∈ *A* and *s* ∈ *S*. In view of [15, Proposition 2.6 (3)], there exist *i, j* ∈ *I, a*_{i} ∈ *A*_{j}, *a*_{j} ∈ *A*_{j} such that *a* = *α*_{i}(*a*_{i}) and *a′*= *α*_{j}(*a*_{j}). Since *I* is directed, from [15, Proposition 2.6 (4)] we obtain *k* ≤ *i, j* with *ϕ*_{i, k}(*a*_{j})*s* ≤ *ϕ*_{j, k}(*a*_{j})*s* in *A*_{k}. Further, since *A*_{k} is GP-po-flat, Lemma 2.4 implies that there exist *m, n* ∈ ℕ, *a*_{1}, *a*_{2}, ⋯, *a*_{m} ∈ *A*_{k} and *s*_{1}, *f*_{1}, ⋯, *s*_{m}, *t*_{m} ∈ *S* such that
$$\begin{array}{cc}{\varphi}_{i,k}({a}_{i})\le {a}_{1}{s}_{1}& \\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{a}_{1}{t}_{1}\le {a}_{{2}^{S}2}& {s}_{1}{s}^{n}\le {t}_{1}{s}^{n}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\cdots & \cdots \\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{a}_{m}{t}_{m}\le {\varphi}_{j,k}({a}_{j})& \phantom{\rule{thickmathspace}{0ex}}{s}_{m}{s}^{n}\le {t}_{m}{s}^{n}\end{array}$$
Acting each inequality in the left hand column of the above scheme by *α*_{k}, we can establishes that
$${\alpha}_{k}({\varphi}_{i,k}({a}_{i}))\otimes {s}^{n}\underset{\xaf}{<}{\alpha}_{k}({\varphi}_{j,k}({a}_{j}))\otimes {s}^{n}$$
in *A*⊗_{s}*Ss*^{n}. Therefore, we can deduce that
$$a\otimes {s}^{n}={\alpha}_{i}({a}_{i})\otimes {s}^{n}={\alpha}_{k}{\varphi}_{i,k}({a}_{i})\otimes {s}^{n}\le {\alpha}_{k}{\varphi}_{j,k}\text{(}{a}_{j})\otimes {s}^{n}={\alpha}_{j}({a}_{j})\otimes {s}^{n}={a}^{\prime}\otimes {s}^{n}$$
in *A*⊗_{s}*Ss*^{n}. This shows that *A*_{S} is GP-po-flat.

*
*The following will be used frequently in this section, and its proof is straightforward.

**Lemma 5.3:** *Let* {*A*_{i}}_{i ∈ I} *be a family of right S-posets and* _{S}*B* *be a left S-poset. If* $({a}_{i})\otimes b\le ({{a}^{\prime}}_{i})\otimes {b}^{\prime}$ *in* $(\prod _{i\in I}{A}_{j}){\otimes}_{S}B$ *for any* $({a}_{i}),({{a}^{\prime}}_{i})\in {\prod}_{i\in I}{A}_{i}$ *and* *b, b*' ∈ *B, then* ${a}_{i}\otimes b\le {{a}^{\prime}}_{i}\otimes {b}^{\prime}$ *in A*_{i} ⊗_{S}*B for each i* ∈ *I*.

**Proposition 5.4:** *For any family* {*A*_{j}}_{i ∈ I} *of right S-posets, if* ${\prod}_{i\in I}{A}_{i}$ *is GP-po-flat, then* *A*_{i} *is GP-po-flat for every i* ∈ *I*.

**Proof:** *Let ${a}_{j}s\le {{a}^{\prime}}_{j}s$ for **s* ∈ *S* and ${a}_{j},{{a}^{\prime}}_{j}\in {A}_{j},j\in I$. For each *i* ≠ *j* in *I*, choose *b*_{j} ∈ *A*_{j}. Then we define
$${c}_{i}=\{\begin{array}{l}{b}_{i},\text{if\hspace{0.17em}}i\ne j,\\ {a}_{j},\text{if\hspace{0.17em}}i\ne j,\end{array}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}}{{c}^{\prime}}_{i}=\{\begin{array}{l}{b}_{i},\text{if\hspace{0.17em}}i\ne j,\\ {{a}^{\prime}}_{j},\text{if\hspace{0.17em}}i\ne j.\end{array}$$
This implies that $({c}_{i})s\le ({{c}^{\prime}}_{i})s$ in ${\prod}_{i\in I}{A}_{j}$, so by assumption, $({c}_{i})\otimes {s}^{n}\le ({{c}^{\prime}}_{i})\otimes {s}^{n}$ *in* $({\prod}_{i\in I}){\otimes}_{S}S{s}^{n}$ for some *n* ∈ ℕ. The result now follows by Lemma 5.3.

*
***Corollary 5.5:** *For any family* {*A*_{j}}_{i} ∈ *I*} *of right S-posets, if* ${\prod}_{i\in I}{A}_{j}$ *is principally weakly po-flat, then* *A*_{j} *is principally weakly po-flat for every i* ∈ *I*.

**Proof:** *Apply Proposition 5.4 for **n* = 1.

*
*Observing Proposition 5.4 (Corollary 5.5), we remark that pomonoids *S* need no condition for transferring GP-po-flatness (principal weak po-flatness) from products to their components. However, [10, Example 2.9] shows that direct products do not necessarily preserve these two properties.

Bearing in mind the above, a question naturally arises: when is GP-po-flatness of *S*-posets preserved under direct products? We first consider the case of finite direct products for this question.

The following is an easy consequence of Lemma 4.9.

**Corollary 5.6:** *For any left ***PSF** pomonoid, the following statements are equivalent.

*${\prod}_{i=1}^{n}{A}_{i}$ **is GP-po-flat*.

*For any s* ∈ *S and* ${a}_{i},{{a}^{\prime}}_{i}\in {A}_{j},1\le i\le n$ *if* $({a}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{a}_{n})s\le ({{a}^{\prime}}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{{a}^{\prime}}_{n})s$ *in* $\prod}_{i=1}^{n}{A}_{i$, *then there exist* *m* ∈ ℕ *and* *u* ∈ *S such that* *us*^{m} = *s*^{m} *and* $({a}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{a}_{n})u\le ({{a}^{\prime}}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{{a}^{\prime}}_{n})u$.

*
*It follows the same outline as the corresponding result of [10].

Applying [6, Theorem 3.13], the following is an evident result for principal weak po-flatness.

**Corollary 5.7:** *For any left PSF pomonoid, the following statements are equivalent*.

*$\prod}_{i=1}^{n}{A}_{i$ **is principally weakly po-flat*.

*For any s* ∈ *S and* ${a}_{j},{{a}^{\prime}}_{i}\in {A}_{j},1\le i\le n$, *if* $({a}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{a}_{n})s\le ({{a}^{\prime}}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{{a}^{\prime}}_{n})s$ *in* $\prod}_{i=1}^{n}{A}_{i$, *then there exists u* ∈ *S such that us* = *s and* $({a}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{a}_{n})u\le ({{a}^{\prime}}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{{a}^{\prime}}_{n})u$.

*
*It is shown in [10] that, for a left PSF monoid, $\prod}_{i=1}^{n}{A}_{i$ is GP-flat if and only if *A*_{j} is GP-flat, 1 ≤ *i* ≤ *n*. For *S*-posets, the corresponding statement is also valid.

**Proposition 5.8:** *Let S be a left PSF pomonoid. Then* $\prod}_{i=1}^{n}{A}_{i$ *is GP-po-flat if and only if* *A*_{j} *is GP-po-flat*, 1 ≤ *i* ≤ *n*.

**Proof:** *Applying Proposition 5.4, and a similar argument as for acts it can easily be proved.*

*
*Specifically, the following corollary generalizes a result of [12], which says that for any left *PSF* pomonoid *S* the *S*-poset *S*^{n} is principally weakly po-flat for each *n* ∈ ℕ.

**Corollary 5.9:** *Let S be a left PSF pomonoid. Then* $\prod}_{i=1}^{n}{A}_{i$ *is principally weakly po-flat if and only if* *A*_{j} *is principally weakly po-flat for every* 1 ≤ *i* ≤ *n*.

**Proof:** *This follows from Corollary 5.5 and [6, Theorem 3.13].*

*
*Our next task is to discuss the case of infinite products for the question mentioned above. The inspiration for some of the following results comes from [11].

By virtue of Lemma 2.4, the following is now immediate.

**Lemma 5.10:** *Let* ${A}_{S}={\displaystyle {\prod}_{i=1}^{n}{A}_{i}}$ *for a family* {*A*_{i}}_{i ∈ I} *of right S-posets. If the S-poset* *S*^{I} *is GP-po-flat and* (*u*_{i})*s* ≤ (*v*_{i}})*s* for *u*_{i}, *v*_{i}, *s* ∈ *S, then for each* *i* ∈ *I* *and* *a*_{i} ∈ *A*_{i}, (*a*_{i}*u*_{i}) ⊗ *s* ∈ (*a*_{i} *v*_{i}) ⊗ *s* in *A* ⊗_{S} *Ss*.

Now we intend to present the main results of this paper.

**Theorem 5.11:** *The following statements are equivalent for a pomonoid S*.

*The direct product of every nonempty family of GP-po-flat right S-posets is GP-po-flat*.

*
*

*S*^{I} *is GP-po-flat for each nonempty set I, and*

*for each s* ∈ *S, there exist m, n* ∈ ℕ *such that for every GP-po-flat right S-poset* *A*_{S}, if as ≤ a's for any a, a′ ∈ *A then l*_{s}(*a, a′*) ≤ *m* and *d*_{s} (*a, a′*) ≤ *n*.

Observing the proof of Theorem 5.11, when |*I*|<∞, we can readily obtain the condition (b) of the part (2) in Theorem 5.11. Thereby, we have the following.

**Corollary 5.12:** *For n* ∈ ℕ, *S*^{n} *is GP-po-flat right* *S*-*it poset if and only if* ${\prod}_{i=1}^{n}{A}_{i}$ *is GP-po-flat wherews A*_{i}, 1 ≤ *i* ≤ *n, are GP-po-flat right S-posets*.

In order to make Theorem 5.11 more specific, we give a description of pomonoids *S* over which *S*^{I} is GP-po-flat for each nonempty set *I*.

**Proposition 5.13:** *The following statements are equivalent for a pomonoid S*.

*S*^{I} *is GP-po-flat for each nonempty set I*.

*For any s* ∈ *S, there exist m, n* ∈ ℕ and (*s*_{1}, *t*_{1}), ⋯, (*s*_{m}, *t*_{m}) ∈ *D*(*S*) *such that*

*s*_{i}*s*^{n} ≤ *t*_{i}*s*^{n} *for all* 1 ≤ *i* ≤ *m, and*

*if us* ≤ *vs for some* *u, v* ∈ *S, then there exist* *u*_{1}, ⋯, *u*_{m} ∈ *S* *such that*
$$\begin{array}{l}u\le {u}_{1}{s}_{1}\\ {u}_{1}{t}_{1}\le {u}_{2}{s}_{2}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\cdots \\ {u}_{m-1}{t}_{m-1}\le {u}_{m}{s}_{m}\\ {u}_{m}{t}_{m}\le \upsilon \end{array}$$

*
*

*
*In Lemma 2.4, particularly when *n* = 1, we have the following result.

**Lemma 5.14:** *A right S-poset A*_{S} is principally weakly po-flat if and only iffor any a, a′ ∈ *A and s* ∈ *S, as* ≤ *a′s* *in* *A*_{S} *implies that there exist* *m* ∈ ℕ, *a*_{1}, *a*_{2}, ⋯, *a*_{m} ∈ *A* *and* *s*_{1}, *t*_{1}, ⋯, *s*_{m}, *t*_{m} ∈ *S* *such that*
$$\begin{array}{cc}a\le {a}_{1}{s}_{1}& \\ {a}_{1}{t}_{1}\le {a}_{2}{s}_{2}& {s}_{1}s\le {t}_{1}s\\ \cdots & \cdots \\ {a}_{m}{t}_{m}\le {a}^{\prime}& {s}_{m}s\le {t}_{m}s\end{array}$$

In the above lemma, we define ${{d}^{\prime}}_{s}(a,{a}^{\prime})$ to be the minimum length of the existing schemes connecting (*a, s*) to (*a′, s*).

Now by a similar argument as in the proof of Theorem 5.11, for principal weak po-flatness we have the following result, which extends Proposition 2.4 of [12].

**Corollary 5.15:** *The following statements are equivalent for a pomonoid S*.

*The direct product of every nonempty family of principally weakly po-flat right **S*-posets is principally weakly po-flat.

*
*

*S*^{I} *is principally weakly po-flat for each nonempty set I, and*

*for each s* ∈ *S, there exists n* ∈ ℕ *such that for every principally weakly po-flat right S-poset A*_{S}, *if as ≤ a′s for any* *a, a′* ∈ *A then* ${{d}^{\prime}}_{s}(a,\text{\hspace{0.17em}}{a}^{\prime})\le n$.

We pointed that in Section 1, for GP-po-flatness and principal weak po-flatness, the direct product case is different from that of Conditions (*P*), (*E*) and (*P*_{w}). In other words, we need to identify that the two conditions in Theorem 5.11 (2) or Corollary 5.15 (2) are independent. Indeed, from [11, Example 2.13] we can see that the condition (a) in Theorem 5.11 (2) (resp., Corollary 5.15(2)) does not imply the condition (b). Also, the following example shows that the converse is not true.

**Example 5.16 ([11, Examples 2.12]):** *Let S* = 〈 *x*_{1}, *x*_{2}, ⋯s |*x*_{n+1} *x*_{n} = *x*_{n+1}= *x*_{n} *x*_{n + 1}〉, *where the order of S is discrete. It is not hard to see, that S is a commutative pomonoid consisting of the elements of the form 1 and* ${x}_{i}^{k}$ (*k* ∈ ℕ). *Then we could directly apply Examples 2.12 from [11] and obtain that, for every principally weakly po-flat S-poset A*_{S}, if as ≤ a′s for a, a′ ∈ *A and s ∈ S then there exists an S-tossing of length 1 connecting the pairs (a, s) and (a′, s) in A × Ss. This shows that S satisfies the condition (b) in Corollary 5.15, and so it also satisfies the condition (b) of Theorem 5.11 (it suffices in Theorem 5.11 to take m = 1.) On the other hand, we assume S*^{2} is GP-po-flat. Then, according the above statement, for (1, x_{1})x_{2} = (x_{1}, x_{1})x_{2} in S^{2}, there exists an S-tossing of length 1 and degree 1 connecting the pairs ((1, x_{1}), x_{2}) and ((x_{1}, x_{1}), x_{2}) in S^{2} × Sx_{2}, but this is impossible. Thus S^{2} is not GP-po-flat, and so it is not principally weakly flat.

Recall that a pomonoid *S* is called *left PP* if the *S*-subposet *Sx* is projective for all *x* ∈ *S*. (Note, however, that *Sx* may not be an ideal of *S* in the ordered sense.) According to [7, Proposition 4.8], a pomonoid *S* is left *PP* if and only if for every *s* ∈ *S* there exists an idempotent *e* ∈ *S* such that *es* = *s* and *us* ≤ *vs* implies *ue* ≤ *ve* for *u, v* ≤ *S*. Further, using Proposition 3.3 and Corollary 3.7 of [20], it is straightforward to prove that for every *x* ∈ *S, Sx* is projective if and only if [1]_{kerρX} contains a right zero, where *ker*ρ_{X} = {(*u, v*) ∈ *S* × *S*|*ux* = *vx*}.

Note that, every left *PP* pomonoid is left *PSF*, and but the converse is not true. The next proposition shows that, for a commutative pomonoid this intervening gap between these two classes of pomonoids can be filled by GP-po-flatness.

**Proposition 5.17:** *The following are equivalent for a commutative pomonoid S*.

*S is a left PP pomonoid*.

*S is a left PSF pomonoid and S*^{I} is principally weakly po-flat for any nonempty set I.

*S is a left PSF pomonoid and S*^{I} is GP-po-flat for any nonempty set I.

*
***Proof:** *(1) ⇒ (2). Let **S* be a left *PP* pomonoid. Then *S* is left *PSF*. To show that *S*^{I} is principally weakly po-flat, assume that (*s*_{i})*s* ≤ (*t*_{i})*s* in *S*^{I}. Since *S* is left *PP*, there exists *e*^{2} = *e* ∈ *S* such that *es* = *s* and *s*_{i}*e*≤*t*_{i}*e* for each *i* ∈ *I*. This means that (*s*_{i})*e* ≤ (*t*_{i})*e* in *S*^{I}, and the desired result is readily obtained.(2) ⇒ (3). It is clear.(3) ⇒ (1). Suppose *s* ∈ *S* for a left *PSF* pomonoid *S*. Based on the above discussion, it is enough to find a right zero in [1]_{kerρs}. Assume that [1]_{kerρs} is represented by an index set *I* as [1]_{kerρs} = {*u*_{i}|*i* ∈ *I*}. Then (*u*_{i})*s* = *s* in *S*^{I}. Since *S*^{I} is principally weakly po-flat, by Lemma 4.9, we obtain *u, w* ∈ *S* with *us* = *s, u*_{i}*u* ≤ *u*, and *ws* = *s, w* ≤ *u*_{i}*w* for any *i* ∈ *I*. Further, since *S* is commutative, we can compute that for each *i* ∈ *I*,
$${u}_{i}uw\le uw\le u{u}_{j}w={u}_{i}uw,$$
that is, *u*_{i}*uw* = *uw*. Therefore, *uw* ∈ [1]ker*ρ*_{S} is a right zero.

From Proposition 5.8, we remark that, for any left *PSF* pomonoid *S, S*^{n} is GP-po-flat for each *n* ∈ ℕ. However, the example below shows that the converse is not true in general. Indeed, let *S* denote the monoid {0, *x*, 1} in which *x*^{2} = 0. The order of *S* is discrete. We can verify that *S*^{2} is GP-po-flat. On the other hand, note that 0 · *x* = *x* · *x*, there are no elements *r* ∈ *S* such that *r* · *x* = *x* and 0 · *r* = *x* · *r*. Hence *S* is not a left *PSF* pomonoid.

It is natural to ask when GP-po-flatness of *S*^{n} implies that *S* is a left *PSF* pomonoid. To reach this target, we need to introduce a corresponding notion, known as left *P*(*P*) monoids, for *S*-posets.

**Definition 5.18:** *We call a pomonoid S left P(P) if every principal left ideal of S satisfies Condition (P)*.

It can be readily checked that a pomonoid *S* is left *P*(*P*) if and only if *us* ≤ *vs* for *u, v, s* ∈ *S*, implies the existence of *u', v'* ∈ *S* such that *uu'* ≤ *vv'* and *u's* = *v's* = *s*. Clearly, every left *PSF* pomonoid is left *P*(*P*). But, from [22, Example 2.4] we see that the converse is not true in general.

Now we can establish one of our main results.

**Theorem 5.19:** *Let S be a pomonoid and* 1 *the identity of S, in which* 1 *is isolated. Then the following conditions on pomonoids are equivalent*.

*S is a left PSF pomonoid*.

*S is a left P(P) pomonoid and S*^{n} is principally weakly po-flat for each n ∈ ℕ.

*S is a left P(P) pomonoid and S*^{n} is GP-po-flat for each n ∈ ℕ.

*
***Proof:** *(1) ⇒ (2). It follows directly from [12, Proposition 2.3].(2) ⇒ (3). It is obvious.(3) ⇒ (1). Let **us* ≤ *vs* for *u, v, s* ∈ *S*. Then we see (1, *u*)*s* ≤ (1, *v*)*s* in *S*^{2} and by Lemma 2.4, there exists a scheme realizing the inequality (1, *u*) ⊗ *s* ≤ (1, *v*) ⊗ *s* in *S*^{2} ⊗_{S} *Ss* of the form
$$\begin{array}{cc}(1,u)\le ({x}_{1},{y}_{1}){s}_{1}& \\ ({x}_{1},{y}_{1}){t}_{1}\le ({x}_{2},{y}_{2}){s}_{2}& {s}_{1}{s}^{n}\le {t}_{1}{s}^{n}\\ ({x}_{2},{y}_{2}){t}_{2}\le ({x}_{3},{y}_{3}){s}_{3}& {s}_{2}{s}^{n}\le {t}_{2}{s}^{n}\\ \cdots & \cdots \\ ({x}_{m},{y}_{m}){t}_{m}\le (1,v)& {s}_{m}{s}^{n}\le {t}_{m}{s}^{n}\end{array}$$
of length *m*, where *m, n* ∈ ℕ, *x*_{1}, ⋯, *x*_{m}, *y*_{1}, ⋯, *y*_{m}, *s*_{1}, ⋯, *s*_{m}, *t*_{1}, ⋯, *t*_{m} ∈ *S*. Without loss of generality, suppose that the length *m* of this scheme is minimal. We claim that *m* = 1 and hence our scheme would be of the form
$$\begin{array}{cc}(1,u)\le ({x}_{1},{y}_{1})& \\ ({x}_{1},{y}_{1}){t}_{1}\le (1,v)& {s}_{1}{s}^{n}\le {t}_{1}{s}^{n},\end{array}$$
thereby from the left hand of the above scheme, we get 1 ≤ *x*_{1}*s*_{1} and *x*_{1}*t*_{1} ≤ 1. But 1 is isolated and we obtain *x*_{1} = *s*_{1} = *t*_{1} = 1, and then *us*_{1} ≤ *y*_{1} *s*_{1} ≤ *vs*_{1} and *s*_{1} *s* = *s*, as desired.Assume *m* > 1. The inequalities (*x*_{1}, *y*_{1}) *t*_{1} ≤ (*x*_{2}, *y*_{2})*s*_{2} and (*x*_{2}, *y*_{2})*t*_{2} ≤ (*x*_{3}, *y*_{3})*s*_{3} yield *x*_{1}*t*_{1}≤ *x*_{2}*s*_{2}, *y*_{1}*t*_{1} ≤ *y*_{2}*s*_{2}, *x*_{2}*t*_{2} ≤ *x*_{3}*s*_{3} and *y*_{2}*t*_{2} ≤ *y*_{3}*s*_{3}. Since *S* is a left *P*(*P*) pomonoid, from the inequality *s*_{2}*s*^{n} ≤ *t*_{2}*s*^{n} we obtain *u*_{1}, *v*_{1} ∈ *S* with *s*_{2}*u*_{1} ≤ *t*_{2}*v*_{1} and *u*_{1}*s*^{n}=*v*_{1}*s*^{n} = *s*^{n}. Then we see *x*_{1}*t*_{1}*u*_{1} ≤ *x*_{2}*s*_{2}*u*_{1} ≤ *x*_{2}*t*_{2}*v*_{1} ≤ *x*_{3}*s*_{3}*v*_{1} and *y*_{1}*t*_{1}*u*_{1} ≤ *y*_{2}*s*_{2}*u*_{1} ≤ *y*_{2}*t*_{2}*v*_{1} ≤ *y*_{3}*s*_{3}*v*_{1}. This shows the following is a scheme of length *m* — 1 realizing the inequality (1, *u*) ⊗ *s* ≤ (1, *v*) ⊗ *s* in *S*^{2} ≤_{S} *Ss*:
$$\begin{array}{cc}(1,\text{\hspace{0.17em}}u)\le ({x}_{1},\text{\hspace{0.17em}}{y}_{1}){s}_{1}& \\ ({x}_{1},\text{\hspace{0.17em}}{y}_{1}){t}_{1}{u}_{1}\le ({x}_{3},\text{\hspace{0.17em}}{y}_{3}){s}_{3}{v}_{1}& {s}_{1}{s}^{n}\le ({t}_{1}{u}_{1}){s}^{n}\\ ({x}_{3},\text{\hspace{0.17em}}{y}_{3}){t}_{3}\le ({x}_{4},\text{\hspace{0.17em}}{y}_{4}){s}_{4}& ({s}_{3}{v}_{1}){s}^{n}\le {t}_{3}{s}^{n}\\ \cdots & \cdots \\ ({x}_{m},\text{\hspace{0.17em}}{y}_{m}){t}_{m}\le (1,\text{\hspace{0.17em}}v)& {s}_{m}{s}^{n}\le {t}_{m}{s}^{n}\end{array}$$
This contradicts the minimality of *m*.

Here we prove that the two conditions in the second part and in the third part of Theorem 5.19 are independent from each other. On the one hand, note from [11, Example 2.13] that there is a pomonoid *S* (the order of *S* is discrete) over which *S*^{I} is principally weakly po-flat (hence *S*^{I} is GP-po-flat) for each nonempty set *I*, but *S* is not a left *PSF* pomonoid. Thus in view of Theorem 5.19, principal weak po-flatness (GP-po-flatness) of *S*^{n} does not imply that *S* is a left *P*(*P*) pomonoid. On the other hand, from [22, Example 2.4] there is a left *P*(*P*) pomonoid which is not a left *PSF* pomonoid. This shows that being a left *P*(*P*) pomonoid does not imply GP-po-flatness (principal weak po-flatness) of *S*^{n}.

As the concluding result, we have

**Proposition 5.20:** *For a right po-cancellative pomonoid S and any family* {*A*_{i}}_{i} ∈ *I* *of right S-posets, the following statements are equivalent*.

*${\prod}_{i=1}^{n}{A}_{i}$ **is principally weakly po-flat*.

*${\prod}_{i=1}^{n}{A}_{i}$ **is GP-po-flat*.

*${\prod}_{i=1}^{n}{A}_{i}$ **is po-torsion free*.

*S is a po-group*.

*
***Proof:** *(1) ⇒ (2) ⇒ (3) and (4) ⇒ (1) are obvious.(3) ⇒ (4). It is true by Theorem 4.2.*

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