Open Mathematics Research Article

Xingliang Liang*, Xinyang Feng, and Yanfeng Luo

On homological classification of pomonoids by GP-po-flatness of S-posets

DOI 10.1515/math-2016-0070

Received November 13, 2015; accepted May 10, 2016.

Abstract: In this paper, we introduce GP-po-flatness property of S-posets over a pomonoid S, which lies strictly between principal weak po-flatness and po-torsion freeness. Furthermore, we investigate the homological classification problems of pomonoids by using this new property. Finally, we consider direct products of GP-po-flat S-posets. As an application, characterizations of pomonoids over which direct products of nonempty families of principally weakly po-flat S-posets are principally weakly po-flat are obtained, and some results of Khosravi, R. in a certain extent are generalized.

Keywords: S-poset, Principal weak po-flatness, GP-po-flatness, Direct product MSC: 06F05, 20M50

Open Access

1 Introduction

Let S be a monoid. It is well-known that flatness properties of S-acts play an important role in studying the homological classification problems of monoids. Different so-called flatness properties (freeness, projectivity, strong flatness, Conditions (P), (WP), (P WP), flatness, weak flatness, principal weak flatness, torsion freeness) of S-acts have been widely used in the homological classification of monoids. A recent and complete treatment of these flatness properties of S-acts appears in the monograph [1].

The study of flatness properties of partially ordered acts over a pomonoid S, or S-posets, was initiated by Fakhruddin, S. M. in the 1980s, see [2, 3]. During recent years, the ordered versions of various flatness properties of acts are defined (in a natural way) and studied [4-7], and also some new properties such as Conditions (Pw), (WP)w and (PWP) w are discovered in the studying process, see [4]. More particularly, some classes of pomonoids, such as (po-)cancellable, left PP, left PSF, (order) regular, regularly almost regular and poperfect pomonoids etc., are characterized by using flatness properties of S-posets.

In [8], Qiao and Wei introduced GP-flatness of acts and showed that the class of acts having this property lies strictly between the classes of principally weakly flat acts and torsion free acts. Moreover, using GP-flatness, some important monoids are generalized, such as regular monoids, left almost regular monoids and so on, and also a new class of monoids, called generally regular monoids, are characterized. Our aim in this paper is to carry over some of these results to the setting of S-posets over a pomonoid S. Firstly, in Section 2, we define GP-po-flat S-posets, and describe GP-po-flatness by certain subpullback diagrams. We then give an equivalent condition under which the amalgamated coproduct A(I) of two copies of S over a proper ideal I is GP-po-flat in Section 3. In Section 4, we characterize pomonoids S over which all (cyclic, Rees factor) S -posets are GP-po-flat, and pomonoids S over

Corresponding Author: Xingliang Liang: Department of Mathematics, Shaanxi University of Science and Technology, Shaanxi 710021, China and School of Mathematics and Statistics, Lanzhou University, Gansu 730000, China, E-mail: lxl_119@126.com Xinyang Feng: School of Mathematics and Statistics, Lanzhou University, Gansu 730000, China, E-mail: fxy1012@126.com Yanfeng Luo: School of Mathematics and Statistics, Lanzhou University, Gansu 730000, China, E-mail: luoyf@lzu.edu.cn

© 2016 Liang etal, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Unauthenticated

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which all po-torsion free S -posets are GP-po-flat. Moreover, we present examples which distinguish between GP-po-flatness and principal weak po-flatness (respectively, po-torsion freeness).

Flatness properties of product acts over a monoid have been extensively studied in recent decades, see [9-11]. However, the research on flatness properties of product S-posets over a pomonoid S is so far less advanced. To our knowledge, the work on this aspect first appeared in [12]. In that paper, the author gave conditions on a pomonoid S under which the S-poset S1 is principally weakly po-flat for each nonempty set I. Moreover, the author proved that direct products of S-posets satisfying Condition (P) (Conditions (E) and (Pw)) again satisfy that condition, if and only if the S-poset S1 is so for each nonempty set I. However, the situation for GP-po-flatness and principal weak po-flatness is markedly different. Thereby, in Section 5, we determine a condition under which principal weak po-flat and GP-po-flat S-posets are preserved under direct products, and extend some results from [12].

2 Definitions and general properties

Throughout this paper, S always stands for a pomonoid and N for the set of natural numbers. A nonempty poset (A, <) is called a right S-poset, usually denoted As, if there exists a mapping A x S ! A, (a,s) ! as, which satisfies the conditions: (1) the action is monotonic in each variable, (2) a(ss0) = (as)s0 and al = a for all a e A and s,s0 e S. Left S-posets sB are defined analogously, and by ©s = } we denote the one-element right S-poset. A nonempty subset I of S is called a left ideal of S if I satisfies SI c I, whereas an ordered left ideal I of S is a left ideal I of S for which a < b e I implies a e I for all a, b e S. Similarly, (ordered) right ideals of S are defined.

Various flatness properties are defined in terms of tensor products. To define the tensor product A (s B of a right S-poset As and a left S-poset s B [7], we first equip the Cartesian product A x B with component-wise order. Let A (s B = (A x B)/p, where p is the order-congruence on the right S-poset A x B (on which S acts trivially) generated by the relation

H = {((as, b), (a,sb)) | a e A, b e B,s e S}.

The equivalence class of (a, b) in A (s B is denoted by a ( b. The order relation on A (s B will be described in Lemma 2.3. In this way, a functor As ( — from the category of left S-posets into the category of posets is obtained. It is easily established, as for S-acts, that A (s S can be equipped with a natural right S-action, and A (s S = As for all S-posets As.

In S-acts, principal weak flatness and GP-flatness are formulated as follows.

• An S-act As is called principally weakly flat if the functor As (— (from the category of left S-acts to the category of sets) preserves all embeddings of principal left ideals of a monoid S into S. In the language of elements this means that, for any s e S and a, a0 e A, a ( s = a0 ( s in A (s S implies a ( s = a0 ( s in A (s Ss (see [1, III, Lemma 10.1]).

• An S-act As is called GP-flat [8] if for any a, a0 e A and s e S, a ( s = a0 ( s in A (s S implies that there exists n e N such that a ( sn = a0 ( sn in A (s Ssn.

In [6], Shi introduced an ordered version of principal weak flatness as follows.

• An S-poset As is called principally weakly po-flat if the functor As ( — preserves order embeddings of principal left ideals I of a monoid S into S. This means, for any s e S and a, a0 e A, a ( s < a0 ( s in A (s S implies a ( s < a0 ( s in A (s Ss.

Inspired by the work of [6] and generalizing [8], we define here GP-po-flatness property in S-posets.

Definition 2.1. A right S -poset As is called GP-po-flat if for any a, a0 e A and s e S, a ( s < a0 ( s in A (s S implies that there exists n e N such that a ( sn < a0 ( sn in A (s Ssn.

Indeed, the example from [7] shows that GP-po-flat S-posets do exist. For any pomonoid S, let A = {a, a0} be a two-elements chain with a < a0 and as = a, a0s = a0 for every s e S. Then A is a right S-poset. We can verify that A is GP-po-flat by Definition 2.1.

Remark 2.2. In Definition 2.1, if n = 1, then every GP-po-flat S -poset is in fact principally weakly po-flat. So principal weak po-flatness implies GP-po-flatness, but in Section 4 we will show that this implication is strict.

Similar to principal weak flatness of S-posets, GP-flatness for S-posets can be defined by replacing "<" by "=" in Definition 2.1. It is obvious that every GP-po-flat S-poset is GP-flat, but the converse is not true by [13, Example 8].

In what follows, we will provide some basic properties about GP-po-flat S-posets. We start with a description of GP-po-flatness, and the following lemma is needed.

Lemma 2.3 ([7]). Let As be a right S-poset, and s B a left S-poset. Then a ® b < a' ® b' in A ®s B for a, a' 2 A, 2 B if and only if there exist a i, a2, • • • , a« € A, b2, • • • , bn 2 B and s \, t \, • • • , sn, tn 2 S such that

a < aisi ai ti < a2s2 si b < ti b2

a«tn < a' s«b« < t«b'.

Applying Definition 2.1 and Lemma 2.3, the following result holds.

Lemma 2.4. A right S -poset As is GP-po-flat if and only if for any a, a' 2 A and s 2 S, as < a's in As implies that there exist m, n 2 N, ai,a2, ••• , am 2 A and s i, ti, • • • , sm, tm 2 S such that

a < aisi

ai ti < a2s2 si sn < ti sn

a m tm < a 0 sm sn < tm sn .

In the above lemma, the natural numbers m and n are called the length and degree of the scheme connecting (a, sn) to (a', sn), respectively. In particular, the minimum length and degree of the existing schemes will be denoted by (a, a') and (a, a'), respectively. Recall that an element c of a pomonoid S is called rightpo-cancellable if, for any s, t 2 S, sc < tc implies s < t .A right S-poset As is called po-torsion free if, for any a, b 2 A and any right po-cancellable element c of S, ac < bc implies a < b.

The following result, which counterpart is true for S-acts, establishes a connection between GP-po-flatness and po-torsion freeness for S-posets.

Proposition 2.5. For any pomonoid S, every GP-po-flat S -poset is po-torsion free.

Proof. Using Lemma 2.4, the proof is routine. □

Note that Example 4.18 below illustrates in particular that the necessary condition in the above proposition is not sufficient. But, for a right po-cancellable pomonoid, GP-po-flatness coincides with po-torsion freeness.

Corollary 2.6. Let S be a right po-cancellable pomonoid and As a right S-poset. Then the following statements are equivalent.

(1) As satisfies Condition (PWP)w.

(2) As is principally weakly po-flat.

(3) AS is GP-po-flat.

(4) As is po-torsion free.

Proof. This follows from Proposition 2.5 and [14, Corollary 2.3]. □

At the end of this section, we give a characterization of GP-po-flatness by using subpullback diagrams. For information on subpullback diagrams in the category of S-posets, we refer the reader to [4, 15].

Proposition 2.7. Let As be a right S -poset. The following statements are equivalent.

(1) AS is GP-po-flat.

(2) Every subpullback diagram P(Ss, Ss, t, t, S), where s e S and t : s (Ss) !s S is an order-embedding of left S -posets, satisfies the following condition:

(8a,a0 e A)(Vm, u,s e S)

[a (t(us) < a0 ( t(us)] =)[(3n e N)(9a00 e A)(9u0, u0 e S)(t(u0sn) < t(u0sn)

A a ( usn < a00 ( u0sn A a00 ( u0sn < a0 ( usn].

Proof. (1) ) (2). Let a ( t(us) < a0 ( t(us) in A (s S, for any a, a0 e A, u, u,s e S, and an order-embedding t : s(Ss) !sS. Denoting t(s) = t we have a ( ut < a0 ( ut in A (s S, and so aui < a0ut in As. Since As is GP-po-flat, by Lemma 2.4, there exists a scheme

au < aisi aiti < a2s2 sitn < t1tn

amtm < a u smt < tmt ,

where m,n e N, a, e A, s,, t, e S, i = 1, ••• ,m. Since t is an order-embedding, from t(s) = t, we can see that the above scheme implies au ( sn < a0u ( sn in A (s (Ssn). Then we have a ( usn < a ( usn and a ( usn = au ( sn < a0u ( sn = a0 ( usn in A (s (Ssn), exactly as needed.

(2) ) (1). Assume a, a0 e A and s e S are such that a ( s < a0 ( s in A (s S. Now consider the order-embedding t : s(Ss) ! sS. Then we have a ( t(s) < a0 ( t(s) in A (s S. By (2), there exist n e N, a00 e A and u, u e S such that a ( sn < a00 ( usn and a00 ( usn < a0 ( sn in A (s (Ssn), and t(usn) < t(usn). Since t is an order-embedding, the last inequality implies usn < usn. Thus we may compute that a ( sn < a00 ( usn < a00 ( usn < a0 ( sn in A (S (Ssn). This means that AS is GP-po-flat. □

3 GP-po-flatness of the S-poset A(I)

Let I be a proper right ideal of a pomonoid S. As it is known, the amalgamated coproduct A(7) of two copies of S over I is an important tool to study the homological classification of pomonoids. In this section, we will investigate GP-po-flatness of the S -poset A(I).

Suppose that I is a proper right ideal of a pomonoid S. For any x, y, z 2 S, let A(I) = ({x, y g x (S — I)) U ({z} x I). Define a right S-action on A(I) by

if us 21, if us e 1,

if us 21, if us e 1,

The order on A(1) is defined by

(wi,s) < (w2,i) ^^ (wi = W2,s < i) or (wi ^ W2, s < i < i for some i e 1).

In [16] it is proved that A(1) is a right S -poset.

We now present an equivalent condition under which A(1) is GP-po-flat. This condition will be useful to characterize pomonoids over which all S -posets are GP-po-flat

(x, u)s = (y, u)s =

(x, us), (z, us),

(y, us), (z, us), (z, u)s = (z, us).

Proposition 3.1. Let I be a proper right ideal of a pomonoid S. Then the right S -poset A(I) is GP-po-flat if and only if, for every w, v, s e S and i e I,

us < i < vs =) (3n e N)(3 j e I)(wsn < jsn < vsn).

Proof. Necessity. Suppose that the right S-poset A(I) is GP-po-flat. If ws < i < vs for w, v, s e S and i e I, then there are two cases to be considered:

Case 1. w e I or v e I. Then we have ws e I or vs e I, and so it suffices to take j = w or j = v. Case 2. w e I and v e I. Then we have two possibilities.

Subcase 1. ws e I or vs e I .If ws e I, then (x, w)s < (y, w)s in A(I). Since A(I) is GP-po-flat, by Lemma 2.4, there exist m, n e N, and (wi, w i), • • • , (wm, wm) e A(I), si, ii, • • • , sm, tm e S such that

(x,w) < (wi,wi)si (w 1 , w 1 )t 1 < (W2,M2)s2 sisn < iisn

(W2, M2)i2 < (W3, M3)s3 s2sn < i2sn

(wm,wm)tm < (y,w) sms < ¿ms .

Denote x by wo and y by wn+i, then there exists k e {0,1, • • • , mg such that w^ ^ w^+i, and so, according to the order relation on A(I), there exists j e I such that wkik < j < wk+isk+i. Thus we can compute that

ws < wisis <•••< Mkiks < js < wk+isk+is <•••< wmims < ws .

But the order is antisymmetric, we have wsn = jsn, the result follows. If vs e I, a similar argument can be used.

Subcase 2. ws e I and vs e I .By definition of the order on A(I), we have (x, w)s < (y, v)s. The remainder of the proof is similar to the Subcase 1.

From what has been discussed above, we obtain the desired conclusion.

Sufficiency. Assume (wi, w), (w2, v) e A(I) where wi, w2 e {x, y, zg, and w, v, s e S are such that (wi, w) ® s < (w2, v) ® s in A(I) (g>s S. Then we have four cases as follows:

Case 1. (wi,w), (w2, v) e (x, 1)S. Since (x, 1)S ^ S is free, it follows that (x, 1)S is GP-po-flat, and so (wi, w) ( sn < (w2, v) ( sn holds in (x, 1)S (s Ssn for some n e N, and hence also in A(I) (s Ssn, exactly as needed.

Case 2. (wi, w), (w2, v) e (y, 1)S. This case is analogous to the previous one.

Case 3. wi = x and w2 = y. In this case, it necessarily implies w, v e S — I. Then we have (x, w)s < (y, v)s in A(I), and so there exists i e I such that ws < i < vs. By the assumed condition, there exist n e N and j e I such that wsn < jsn < vsn, then we can calculate that, in A(I) (s Ssn,

(x, w) ( sn = (x, 1) ( wsn < (x, 1) ( jsn = (z, j) ( sn

= (y, 1)j ( sn = (y, 1) ( jsn < (y, 1) ( vsn = (y, v) ( sn.

Case 4. wi = y and w2 = x. This is similar to the Case 3.

In conclusion, A(I) is GP-po-flat, and the proof is complete. □

4 Homological classification of pomonoids

Based on the preparation of the previous section, in this section, we are going to consider the homological classification of pomonoids by GP-po-flatness of (cyclic, Rees factor) S-posets.

Recall that a pomonoid S is called regular, if for every s e S, there exists x e S such that s = sxs.

Definition 4.1. A pomonoid S is called generally regular, if for every s e S, there exist n e N and x e S such that

nn sn = sxs'

It is obvious that every regular pomonoid is generally regular. But, [8, Example 3.3] shows that the converse is not true in general.

Using the amalgamated coproduct A(I), we first give a characterization of pomonoids S over which all S-posets are GP-po-flat. Its corresponding result for S-acts is true (see [8, Theorem 3.4]).

Theorem 4.2. For any pomonoid S, the following statements are equivalent.

(1) All right S-posets are GP-po-flat.

(2) All right S-posets satisfying Condition (E) are GP-po-flat.

(3) S is a generally regular pomonoid.

Proof. The implication (1) ) (2) is clear.

(2) ) (3). For any s e S, if sS = S, then there exists x e S such that s = sxs. Otherwise, I = sS is a proper right ideal of S, by [16, Lemma 2.4], A(I) satisfies Condition (E), and so A(I) is GP-po-flat. In view of Proposition 3.1, from the inequalities 1 • s < s < 1 • s we obtain n e N and j e I with sn < jsn < sn. This means that there exists x e S such that j = sx, and so sn = sxsn, as required.

(3) ) (1). It is straightforward to verify. □

From Theorem 4.2 we can deduce the following.

Corollary 4.3. For a commutative pomonoid S, the following statements are equivalent.

(1) All right S-posets are GP-po-flat.

(2) For every s e S, there exist n e N and x e S such that sn = snxsn.

We stated that in view of Proposition 2.5, GP-po-flatness implies po-torsion freeness, but the converse is not true. So we naturally consider the question of when all po-torsion free S -posets are GP-po-flat.

The following definition is a generalization of the duality for regularly right almost regular pomonoids which is introduced by Zhang and Laan in [17].

Definition 4.4. An element s of S is called generally regularly left almost regular, if there exist natural numbers m,n e N, elements r, ri, ••• , rm, si,s2, ••• ,sm, s0,s2, ••• ,s^ e S, and right po-cancellable elements c i, C2, • • • , cm e S such that

sici < sri < s0ci s2C2 < sir2 < s0 r2 < s2C2

smcm < sm—irm < sm_irm < smcm sn s sn s0 sn:

s = sm s = sm s :

In particular, when n = 1, we say the element s of S is regularly left almost regular.

A pomonoid S is (generally) regularly left almost regular (denoted by (G)RLAR for short) if all its elements are (generally) regularly left almost regular.

It is easy to see that every (generally) regular pomonoid is (G)RLAR, and every RLAR pomonoid is GRLAR. But Example 4.5 below and [8, Example 3.3] illustrate that these two implications are both strict, respectively.

Example4.5. ((G)RLAR ) (generally) regular). Let S = (e,s,c | e2 = e,es = se = ec = ce = s,sc = cs = s2}. Equip S with the order induced by the relations e < s and s < s2, thereby, obtaining a commutative pomonoid S. Actually, S = {1,e,sk ,ck (k e N)g, and the elements of the form ck (k e N) and 1 are the only po-cancellable elements. It is not difficult to see that e and 1 are the only regular elements. But since eck = sk and sk = esk, the elements of the form sk (k e N) are also RLAR, although they are not generally regular. This shows that there exists a pomonoid, which is not (generally) regular, is (G)RLAR.

Proposition 4.6. If S is a GRLAR pomonoid, then all po-torsion free right S -posets are GP-po-flat.

of As implies as2 < a's2. In this way we finally arrive at asm < a's^, and so we can now compute that a ® sn = a ® smsn = asm ® s < a'si ® s = a' ® si,sn = a' ® sn

Proof. Let S be a GRLAR pomonoid. Assume As is apo-torsion free S-poset. Let as < a's for a, a' e A ands e S. Since s is GRLAR, there exist m,n e N, r, ri, • • • , rm, si , s2, • • • , sm, s' ,s2, • • • , s^ e S, and right po-cancellable elements c i, c2, • • • , cm e S such that

sici < sri < s'ci s2C2 < sir2 < s'' r2 < s2C2

smcm < sm—irm < sm—irm < smcm s — — *

Using the first inequality we get asici < asri < a'sri < a's'ci. Since As is po-torsion free, we see that asi < a's'. Further, for the second inequality, we have as2c2 < asir2 < a's'r2 < a's2c2. So po-torsion freeness 2. In this way we finally arrive at asm < a's^,,

msn — asm ® s < a'sm ® s — a ^ s

in A (g>s Ssn. This means that As is GP-po-flat. □

In particular, when n — 1 in the proof of the above proposition, we can deduce

Corollary 4.7. If S is a RLAR pomonoid, then all po-torsion free right S -posets are principally weakly flat.

In addition, from [4] we remark that Condition implies GP-po-flatness, but [4, Example 6.3] shows

that this implication is strict. So it is natural to ask for pomonoids over which GP-po-flatness of S-posets implies Condition (PWP)

w. To reach the target, we need some more preliminary material. Recall that a pomonoid S is called left PSF if all principal left ideal of S is strongly flat (as a left S-poset). It is shown in [6] that a pomonoid S is left PSF if and only if for s, t, u e S, su < ím implies that there exists r e S such that ru — u and sr < tr.

Lemma 4.8. The following statements on a pomonoid S are equivalent.

(1) For every proper right ideal I of S there exists j e I — Ij.

(2) For every infinite sequence (x0,xi,x2, •••) with x¿ — x¿ + ix¿, x¿ e S, i — 0,1, •••, there exists a positive integer n such that xn — xn+1 — • • • — 1.

Proof. A similar argument as [18, Proposition 2.1] can be used. □

The following proposition is the ordered analogue of [10, Proposition 2.5]. The technique for the proof is taken from the unordered case.

Lemma 4.9. Let S be a left PSF pomonoid. Then the following statements are equivalent.

(1) AS is GP-po-flat.

(2) For any a, a' e A, s e S, as < a's implies that there exist n e N and u e S such that usn — sn and au < a'u. Now we can address the above matter.

Theorem 4.10. Let S be a left PSF pomonoid and 1 the identity of S, in which 1 is incomparable with every other element of S. If for every proper right ideal I of S there exists i e I — li, then all GP -po-flat right S -posets satisfy Condition (PWP)w.

Proof. Suppose that As is a GP-po-flat right S-poset. Let as < a's for a, a' e A and s e S. Then by Lemma 4.9, there exist n e N, u e S such that au < a'u and usn — sn. Since S is left PSF, from usn < sn we get xi e S with xisn — sn and uxi < xi. Further, from the inequality uxi < xi we obtain x2 e S with x2xi — xi and ux2 < x2. By continuing this process, letting xo — sn we can find an infinite sequence (xo, xi, • • •), such that

x¿ + ix¿ — x¿, ux¿ < x¿, i — 0,1, • • • .

By Lemma 4.8, there exists a positive integer m such that xm = xm+i = • • • = 1. Thus, we get u < 1. But 1 is isolated, we obtain u = 1 and so a < a0. This shows that As satisfies Condition (PWP)w. □

Notice that the proof of the above theorem also allows us to deduce the following.

Theorem 4.11. Let S be a left PSF pomonoid and 1 the identity of S, in which 1 is either the minimal or the maximal element of S. If for every proper right ideal I of S there exists i e I — li, then all GP -po-flat right S-posets satisfy Condition (PWP)w.

Next, we turn our attention to GP-po-flatness of cyclic S-posets. We need some more preliminary material.

Recall that a relation a on an S-poset As is called a pseudo-order on As if it is transitive, compatible with the S-action, and contains the relation < on As . For information pertaining to pseudo-orders on S-posets, we refer the reader to [19], and for further information about order congruence on S-posets to [13, 20]. Suppose p is a right order congruence on a pomonoid S. Define a relation p by

s pi ^^ [s]p < [i]p in S/p.

It is clear that p is a pseudo-order on Ss .

The following lemma is useful in dealing with GP-po-flat cyclic S-posets.

Lemma 4.12. Let p be a right order congruence on S ands e S.Then [u]p ( sn < [v]p ( sn in S/p (s Ssn for u, v e S and n e N, if and only if (u, v) e p u kerpsn.

Proof. It is similar to that of [20, Lemma 3.18]. □

Proposition 4.13. Let p be a right order congruence on S. Then S/p is GP-po-flat if and only if for u, v,s e S, [us]p < [vs]p implies (u, v) e p u kerpsn for some n e N.

Proof. Necessity. Let [us]p < [vs]p in S/p for u, v,s e S .Then we have [u]ps < [v]ps,andso [u]p ( s < [v] p ( s in S/p (s S. Since S/p is GP-po-flat, we have [u]p (sn < [v]p (sn in S/p (s Ssn for some n e N. This implies that (u, v) e p u kerpsn by Lemma 4.12.

Sufficiency. If [u]ps < [v]ps in S/p, then [us]p < [vs]p, and so by assumption, (u, v) e p u kerpsn for some n e N. Lemma 4.12 implies that [u]p ( sn < [v]p ( sn in S/p (S Ssn. Therefore, S/p is GP-po-flat. □

Proposition 4.13 immediately implies the following fact about one-element S-posets.

Corollary 4.14. For any pomonoid S, the one-element S -poset ©s is GP-po-flat.

Recall that a subpomonoid P of a pomonoid S is called convex, if P = [P] where

[P] = {x e S | e P, p < x < qg. For Rees factor S-posets, we have the following description of GP-po-flatness.

Proposition 4.15. Let K be a convex, proper right ideal of a pomonoid S. Then the right Rees factor S -poset S/K is GP-po-flat if and only if, for every k e K and u, s e S,

k < us =) (9n e N)(9k0 e K)(k0sn < usn) and us < k =) (9n e N)(9k00 e K)(usn < k00sn).

Proof. Necessity. Assume first that the right Rees factor S-poset S/K is GP-po-flat. Let k e K and u,s e S with k < us. Then we see [ks]PK < [us]PK in S/K and so, GP-po-flatness of S/K implies that [1]PK ( ksn < [1]PK ( usn in S/K (s Ssn for some n e N. In view of [13, Lemma 4], if ksn < usn, there is nothing to prove. Otherwise, there exists an array

ksn < kisn

k'sn < k2sn

for some k,, k0 e K, i = 1, • • • , m. The last line of this array gives what we want. In case us < k, a similar argument can be used.

Sufficiency. Suppose that u, v,s e S are such that [m]pk ( s < ( s in S/K S. Then we see

[us]pK < [vs]^K in S/K. In light of [13, Lemma 3.1], if us < vs, then immediately [1]^K ( us < [1]^K ( vs, the result follows. Otherwise, us < k and 1 < vs for some k, 1 e K .By the assumed condition, there exist n i, n2 e N and k0, 10 e K such that usni < k0sni and 10sn2 < vsn2. Set n = max{ni,n2}. Then usn < k0sn and 10sn < vsn, and so we may now compute that

[uW ( sn = [1]^^ ( usn < [1]^^ ( k0sn = [k0]^^ ( sn = [1 0]^k ( sn < [1] ( 10sn = [v]^^ ( sn

in S/K (s Ssn, and the proof is complete. □

In [8], Qiao and Wei proved that generally regular monoids are precisely the monoids over which all Rees factor S-acts are GP-flat. We shall prove an analogue of this result for S-posets.

Theorem 4.16. For any pomonoid S, the following statements are equivalent.

(1) All Rees factor right S -posets are GP-po-flat.

(2) For every s e S, there exist n e N, s0, s00 e S such that ss0sn < sn < ss00sn.

Proof. (1) ) (2). For every s e S, [sS] is a convex right ideal of S. If [sS] = S, then there exist w, v e S such that sw < 1 < sv. Postmultiplying by sn for any n e N we obtain swsn < sn < svsn, exactly as needed. If [sS] ^ S then [sS] is a convex, proper right ideal of S. Obviously, s e [sS] and s = 1 • s, from Proposition 4.15 we obtain n e N and k,k0 e [sS] with ksn < sn andsn < k0sn. Also, since k,k0 e [sS], there exist s0,s00,si,s2 e S such that

ss0 < k < ssi and ss2 < k0 < ss00:

Thus we may compute that

ss0sn < ksn < sn < k0sn < ss00sn:

(2) ) (1). Let K be a convex right ideal of S. If K = S, then by Corollary 4.14, S/K ^ ©S is GP-po-flat. If K is proper, we will use Proposition 4.15 to check that S/K is GP-po-flat. So for every k e K and u, s e S, for s by (2) there exist w, v e S such that swsn < sn < svsn. If k < us, then we get (kw)sn < uswsn < usn. If us < k, then we have usn < usvsn < (kv)sn. Setting k0 = kw or k00 = kv, the desired result is obtained. □

As we saw in Section 2, principally weakly po-flat ) GP-po-flat ) po-torsion free. Now our crucial thing is to verify the distinctness of these properties.

Example 4.17. (GP-po-flat) p. w. po-flat) Let S = K U {I g with

I / 0 m n

k = o o t l\ 0 0 0

I @100

e N ; , I = I 010

The order on S is defined by

fa b c\ S a' b' c0 \

I 0 de I < I 0 d ' e' I ^^ a < a',è < è',c < c',d < d ',e < e' and f < /'.

\00fj \0 0 f'J

Then K is a convex, proper right ideal of the pomonoid S. By Proposition 4.15, we can verify that the Rees factor

/ 0 10 \

S-poset S/K is GP-po-flat. On the other hand, note that k = ( 0 0 1 ) € K, but there is no k' e K such that k < k'k. From [13, Proposition 10] it follows that S/K is not principally weakly po-flat.

Example 4.18 ([13, Example 7]). (po-t. free ) GP-po-flat) Let S denote an infinite monogenic monoid {1,s,s2, •••}, equipped with the order in which

2 3 4 s2 < s3 < s4 < • • •

and s and 1 are isolated. Let K = {s2,s3,s4, • • • }. Then by [13, Example 7], S/K is po-torsion free. However, because s2 < s • s, there cannot exist n e N and k e K such that k • sn < s • sn, and by Proposition 4.15, S/K is not GP-po-flat.

5 GP-po-flatness of product S-posets

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 As = U ; e/ A;, where A,, i e I, is a strongly convex S -subposet of As . Then As is GP-po-flat if and only if every A, is GP-po-flat, i e 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 ) is a directed system of GP-po-flat right S -posets over a directed index set I with directed colimit (A, a, ). Let as < a's in As , for a, a' e A and s e S .In view of [15, Proposition 2.6 (3)], there exist i, j e I, a, e A,, ay e Ay such that a = a, (a, ) and a' = ay (ay/.Since I is directed, from [15, Proposition 2.6 (4)] we obtain k > i, j with (a, )s < k(ay )s in Ak. Further, since Ak is GP-po-flat, Lemma 2.4 implies that there exist m, n e N, a i , a2, ••• , am e Ak and s i, 11, • • • , sm, tm e S such that

0;,k(a,) < aisi

aiii < a2s2 sisn < iisn

am'm < 0j,k(aj) sms < tms .

Acting each inequality in the left hand column of the above scheme by ak, we can establishes that

ak(&,k(a, )) g sn < ak(0y,k(ay )) g sn in A gs Ssn. Therefore, we can deduce that

a (g) sn = a, (a, ) g sn = ak(a, ) g sn < ak0yjk(ay ) g sn = ay (ay ) g sn = a' g sn in A gS Ssn. This shows that AS is GP-po-flat. □

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

Lemma 5.3. Let {A, }, 2/ be a family of right S -posets and s B be a left S -poset. If (a, ) g b < (a, ) g b' in (HI;'e/ A/) (S B for any (a, ), (a') e ]"[,2/ A, and b,b' e B, then a, g b < a' g b' in A, gs B for each i e I.

Proposition 5.4. For any family {A, },<=/ of right S -posets, if\\ A, is GP-po-flat, then A, is GP-po-flat for every i e I.

Proof. Let aj s < ajs for s e S and a j, aj e A j, j e I. For each i ^ j in I, choose b, e A,. Then we define

c, = ( 1 if i * j' and c0 = ! ■ if i * j aj, if i = j, ' a., if i = j.

This implies that (c, )s < (c0)s in f"[,e/ A,, so by assumption, (c,) ® sn < (c0) ® sn in (n,e/ A,) ®s Ssn for some n e N. The result now follows by Lemma 5.3. □

Corollary 5.5. For any family {A/ 2/ of right S-posets, if Y\¿e/ A, is principally weakly po-flat, then A, is principally weakly po-flat for every i e 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.

(1) HIn=l A/ is GP-po-flat.

(2) For any s e S and a, ,a0 e A,, 1 < i < n, if (ai, ••• ,an)s < (a0 , ••• , an )s in f"[ ¿ = i A,, then there exist m e N and u e S such that usm = sm and (ai, • • • , an)u < (ai, • • • , a'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.

(1) HIn=i A, is principally weakly po-flat.

(2) For any s e S and a,, a0 e A,, 1 < i < n, if (ai, • • • , an)s < (a0 , • • • , an)s in f"[n=i A,, then there exists u e S such that us = s and (ai, • • • , an)u < (ai, • • • , an)u.

It is shown in [10] that, for a left PSF monoid, ]"[n=i A, is GP-flat if and only if A, 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 \\n=i A, is GP-po-flat if and only if A, 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 Sn is principally weakly po-flat for each n e N.

Corollary 5.9. Let S be a left PSF pomonoid. Then \\n=i A, is principally weakly po-flat if and only if A, 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 as = n¿el A, for a family {A, }, 2/ of right S-posets. If the S-poset S1 is GP-po-flat and (u, )s < (v )s for u,, v ,s e S, then for each i e I and a, e A,, (a, u,) ® s < (a, v) ® 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.

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

(2) (a) S1 is GP-po-flat for each nonempty set I, and

(b) for each s 2 S, there exist m, n 2 N such that for every GP-po-flat right S -poset As , if as < a's for any a, a0 2 A then (a, a') < m and (a, a') < n.

Proof. (1) ) (2). Part (a) is obvious. Now we prove part (b) by contradiction. Assume there is s 2 S such that, (i) for each i 2 N there exists a GP-po-flat S-posets (A, )s such that a, s < b,s and (a,, b,) > i for some a,, b, 2 A,, or (ii) for each j 2 N there exists a GP-po-flat S-posets (Ay)s such that ay s < bys and (ay, by) > j for some ay, by 2 Ay. If case (i) holds, then by (1), 1 i A,- is GP-po-flat. Thereby, (a,-)s < (b, )s in 1 i A, implies the existence of a scheme of length m (and degree k) in ]"[ 11 A, x Ssk. That is, for each i 2 N, a scheme of length m connecting (a,, sk) to (b,, sk) in A, x Ssk, which contradicts (am+i, bm+i) > m C 1. Case (ii) can be disposed of similarly.

(2) ) (1). Suppose that {A, g, e/ is a family of GP-po-flat right S-posets and As = ]"[,e/ A/. Let (a, )s < (a' )s in As .Then a, s < a's for each i 2 I, and by (b), there exist m,n 2 N such that (a, ,sn) and (a' ,sn) are connected by a scheme of length m and degree n in A, x Ssn for every i 2 I. This implies that for every i 2 I, there exists a scheme of the form

a; < ai,si, aizii,' < a2,s2z si,sn < ii,sn

amZ ^mZ < a/ smZs < ^mZs ,

where ai,, ••• ,am, 2 A,, and si,, ••• ,sm, ,ii,, ••• ,im, 2 S. From the right-hand of the above schemes, we get syi)sn < (iyi

(sy, )sn < (rji)sn in S1 for all 1 < j < m. Further, in light of Lemma 5.10, we see that for every 1 < j < m,

(ay,sy,) ® snk < (ay,iy,) ® snk

in A ®S Ssnk for some k 2 N. Therefore, we may compute that

(a,) ® snk < (ai,ii,) ® snk < (ai,ii,) ® snk < (a2,i2,) ® snk < (a2,i2,) ® snk < ••• < (amiim,') ® snk < (am,-im/) ® snk < (a') ® snk

in A ® s Ssnk, and the proof is complete. □

Observing the proof of Theorem 5.11, when |I | < 1, 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 2 N, Sn is GP-po-flat right S-poset if and only if Y\1=i A, is GP-po-flat where A,, 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 S1 is GP-po-flat for each nonempty set I.

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

(1) S1 is GP-po-flat for each nonempty set I.

(2) For any s 2 S, there exist m, n 2 N and (si, ii), • • • , (sm, im) 2 D(S) such that

(a) s, sn < i, sn for all 1 < i < m, and

(b) if us < vs for some u, v 2 S, then there exist u i, • • • , um 2 S such that

u < uisi

Miii < M2S2

Mm— \tm—1 < Mmsm <

Proof. (1) ) (2). Let L = {(u,u) e D(S)|us < us}, and index L by L = {(m/ , u/)|i e I}. Then we see (m/ )s < (u/)s in S1. Since S1 is GP-po-flat, from Lemma 2.4 we obtain m,n e N, (mi/), • • • , (um/) e S1 and si,ii, • • • , sm, im e S such that

(m/) < (mi/)si (uii)ii < (M2/)s2 sisn < iisn

(um/)tm < (u/) sms < ^ms •

Hence we see that we have reached the desired conclusion.

(2) ) (1). Let I ^ 0, and let (m,), (u) e S1 be such that (m, ) ( s < (u) ( s in S1 (S S. Then we see (m/)s < (u/)s in S1. By (2), there exist m,n e N and (s i, 11), ••• ,(sm,im) e D(S) such that sy sn < iy sn for all 1 < j < m, and there exist m i/, • • • , um/ for all i e I such that

m/ < ui/si Mi/ii < M2/s2

Mm/im < u/ •

Then we can compute

(m/) ( sn < (Mi/)si ( sn = (Mi/) ( sisn < (Mi/) ( iisn

= (Mi/)ii ( sn < (M2/ )s2 ( sn < ••• < (Mm/)im ( sn < (u/) ( sn

in S1 (s Ssn, and this shows that S1 is GP-po-flat, as required. □

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

Lemma 5.14. A right S -poset As is principally weakly po-flat if and only if for any a, a0 e A and s e S, as < a0 s in As implies that there exist m e N, ai, a2, • • • , am e A and si, ii, • • • , sm, im e S such that

a < aisi

ai ii < a2s2 si s < ii s amim < a sms < ims-

In the above lemma, we define (a, a0) to be the minimum length of the existing schemes connecting (a,s) to

(a0, 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.

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

(2) (a) S1 is principally weakly po-flat for each nonempty set I, and

(b) for each s e S, there exists n e N such that for every principally weakly po-flat right S-poset As, if as < a0s for any a, a0 e A then (a, a0) < 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 (Pw). 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 = (xi ,X2, •••|xn+ixn = Xn+i = XnXn+i), 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 xk (k 2 N). Then we could directly apply Examples 2.12 from [11] and obtain that, for every principally weakly po-flat S -poset As , if as < a's for a,a' 2 A and s 2 S then there exists an S-tossing of length 1 connecting the pairs (a,s) and (a',s) in A x 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 S2 is GP-po-flat. Then, according the above statement, for (1, xi)x2 = (xi, xi)x2 in S2, there exists an S-tossing of length 1 and degree 1 connecting the pairs ((1, xi), x2) and ((xi, xi), x2) in S2 x Sx2, but this is impossible. Thus S2 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 2 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 2 S there exists an idempotent e 2 S such that es = s and us < vs implies ue < ve for u, v 2 S. Further, using Proposition 3.3 and Corollary 3.7 of[20],itis straightforward to prove that for every x 2 S, Sx is projective if and only if [1]ker,ox contains a right zero, where kerpx = {(u, v) 2 S x S | ux = vxg.

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.

(1) S is a left PP pomonoid.

(2) S is a left PSF pomonoid and S1 is principally weakly po-flat for any nonempty set I.

(3) S is a left PSF pomonoid and S1 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 S1 is principally weakly po-flat, assume that (s, )s < (i, )s in S1. Since S is left PP, there exists e2 = e 2 S such that es = s and s, e < i, e for each i 2 I. This means that (s, )e < (i, )e in S1, and the desired result is readily obtained.

(2) ) (3). It is clear.

(3) ) (1). Suppose s 2 S for a left PSF pomonoid S. Based on the above discussion, it is enough to find a right zero in [1]ker^s. Assume that [1]ker^s is represented by an index set I as [1]ker^s = {u, |i 2 I g. Then (u, )s = s in S1. Since S1 is principally weakly po-flat, by Lemma 4.9, we obtain u, w 2 S with us = s, u, u < u, and ws = s, w < u, w for any i 2 I. Further, since S is commutative, we can compute that for each i 2 I,

u, uw < uw < uu, w = u, uw,

that is, u, uw = uw. Therefore, uw 2 [1]ker^s is a right zero. □

From Proposition 5.8, we remark that, for any left PSF pomonoid S, Sn is GP-po-flat for each n 2 N. However, the example below shows that the converse is not true in general. Indeed, let S denote the monoid {0, x, 1 g in which x2 = 0. The order of S is discrete. We can verify that S2 is GP-po-flat. On the other hand, note that 0 • x = x • x, there are no elements r 2 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 Sn 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 e S, implies the existence of u', v' e S such that mm' < 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.

(1) S is a left PSF pomonoid.

(2) S is a left P(P) pomonoid and Sn is principally weakly po-flatfor each n e N.

(3) S is a left P(P) pomonoid and Sn is GP-po-flatfor each n e 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 e S. Then we see (1, u)s < (1, v)s in S2 and by Lemma 2.4, there exists a scheme realizing the inequality (1, u) ( s < (1, v) ( s in S2 ( S Ss of the form

(1,u) < (xi,yi)si (xi,yi)ii < (x2,y2)s2 sisn < tisn (X2, y2>2 < (X3, y3>3 s2sn < i2sn

(xm>ym)'m < (1,v) sms < ims

of length m, where m, « e N, xi, • • • , xm, yi, • • • , ym, si, • • • ,sm, 11, • • • , tm e 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

(1,u) < (xi,yi)si (xi,yi)ii < (1, v) sisn < iisn,

thereby from the left hand of the above scheme, we get 1 < xisi and xiii < 1. But 1 is isolated and we obtain xi = si = ti = 1, and then usi < yisi < vsi and sis = s, as desired.

Assume m > 1. The inequalities (xi,yi)ti < (x2,y2)s2 and (x2,y2)i2 < (x3,y3)s3 yield xiii < x2s2, yiti < y2s2, x2i2 < x3s3 and y2i2 < y3s3. Since S is a left P(P) pomonoid, from the inequality s2sn < i2sn we obtain ui, vi e S with s2ui < i2vi and uisn = visn = sn. Then we see xiiiui < x2s2ui < x2i2vi < x3s3vi and yiiiui < y2s2ui < y2i2vi < y3s3vi. This shows the following is a scheme of length m — 1 realizing the inequality (1, u) ( s < (1, v) ( s in S2 (s Ss:

(1,u) < (xi,yi)si (xi,yi)iiMi < (x3,y3)s3vi sisn < (iiui)sn (x3,y3)i3 < (x4,y4)s4 (s3vi)sn < i3sn

(xm,ym)im < (1,v) sms < ims •

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 S1 is principally weakly po-flat (hence S1 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 Sn 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 Sn.

As the concluding result, we have

Proposition 5.20. For a right po-cancellative pomonoid S and any family {A, }, e/ of right S -posets, the following statements are equivalent.

(1) HI/e/ A/ is principally weakly po-flat.

(2) n/e/ A, is GP-po-flat.

(3) f[¿e/ A, ispo-torsion free.

(4) S is a po-group.

Proof. (1) ) (2) ) (3) and (4) ) (1) are obvious.

(3) ) (4). It is true by Theorem 4.2. □

Acknowledgement: The authors would like to give many thanks to the anonymous referee for their invaluable comments and suggestions. This research was partially supported by Natural Science Foundation of Shaanxi University of Science and Technology (Grant No. 2016BJ-26) and NSFC (Grant No. 11371177).

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