On a generalization of I-regularity

Husheng Qiao; Leting Feng

doi:10.1515/math-2025-0190

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On a generalization of I-regularity

Husheng Qiao and Leting Feng

Published/Copyright: September 5, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

Let S be a pomonoid. The projectivity and strong flatness of right S -posets have been central topics in the homological classification of pomonoids in recent decades. In 2005, Shi et al. introduced I -regular S -posets and proved that all its cyclic S -subposets are projective [Indecomposable, projective, and flat S-posets, Comm. Algebra 33 (2005), no. 1, 235–251]. Feng subsequently introduced CSF S -posets, where all its cyclic S -subposets are strongly flatness [CSF S -posets and C(P) S -posets, Master’s thesis, Northwest Normal University, Gansu, 2020]. In this article, we define a new class of strongly (SF)-cyclic S -posets, which lies strictly between the classes of I -regular S -posets and CSF S -posets. We then provide a homological classification of pomonoids based on this new property of their right (Rees factor) S -posets.

Keywords: pomonoids; S-posets; strongly (SF)-cyclic; homological classifications

MSC 2010: 06F05; 20M50

1 Definitions and preliminaries

Throughout this article, S will denote a pomonoid. We refer the reader to [1,2] for basic definitions and terminology relating to monoids and acts over monoids and to [3–5] for definitions and results on flatness of S -posets, which are used here.

A pomonoid is a monoid S together with a partial order relation, which is compatible with the binary operation. A right S-poset A S is a right S -act A equipped with a partial order ⩽ and, in addition, for all s , t ∈ S and a , b ∈ A , if s ⩽ t , then a s ⩽ a t , and if a ⩽ b , then a s ⩽ b s . Left S -posets S B are defined analogously, and Θ S = { θ } is the one-element right S -poset. A morphism of right S-posets is a monotone mapping f : A S → B S , which satisfies f ( a s ) = f ( a ) s for every a ∈ A and s ∈ S . Similarly, morphisms of left S -posets are defined. Morphisms of posets are just monotone mappings. In this way, the categories POS- S , S -POS, and POS of right S -posets, left S -posets, and posets naturally arise.

An order congruence on an S -poset is important in this article, and its essentials can be found in [3,6]. An equivalence θ on A S is called an order congruence if the quotient set A ⁄ θ is endowed with a natural order so that the canonical epimorphism is a morphism of right S -posets. For an act congruence θ on A S , we write a ⩽ θ a ′ if the so-called θ -chain

a ⩽ a 1 θ b 1 ⩽ a 2 θ b 2 … ⩽ a n θ b n ⩽ a ′

from a to a ′ exists in A S , where a i , b i ∈ A , 1 ⩽ i ⩽ n . It can be shown that an act congruence θ on A S is an order congruence if and only if a θ a ′ whenever a ⩽ θ a ′ ⩽ θ a . If φ : A S → B S is a morphism of right S -posets, ker → φ = { ( a , b ) ∈ A × A ∣ φ ( a ) ⩽ φ ( b ) } , then we obtain that ker → φ ∩ ker → φ − 1 = ker φ is an order congruence on A S (see [3], Proposition 2.3).

An S -poset A S is called cyclic if A = a S = { a s ∣ s ∈ S } for some a ∈ A S . An S -poset A S is cyclic if and only if it is isomorphic to the factor S -poset S ⁄ ρ of S S by an order congruence ρ = σ ∩ σ − 1 , σ = { ( s , t ) ∈ S × S ∣ a s ≤ a t } . A subpomonoid P of a pomonoid S is called convex, if P = [ P ] , where

[ P ] = { x ∈ S ∣ p ≤ x ≤ q for some p , q ∈ P } .

If K S is a convex right ideal of S , then there exists an S -poset congruence such that one of its classes is K and all the others are singletons. Moreover, the factor S -poset by this congruence is called the Rees factor S-poset of S by K, and denoted by S ⁄ K . For s ∈ S , the congruence class of s in S ⁄ K is denoted by [ s ] ρ K , or briefly [ s ] .

The investigation on flatness properties of S -posets was initiated by Fakhruddinin in 1980s [7,8]. Over the past several decades, this investigation has to do with various notions of flatness. In [3,9,10], definitions of Conditions ( P ) , ( E ) , strong flatness, and I -regularity in POS- S are formulated as follows:

A right S -poset A S satisfies Condition (P) if, for all a , a ′ ∈ A and s , s ′ ∈ S , a s ≤ a ′ s ′ implies a = a ′ ′ u , a ′ = a ′ ′ v for some a ′ ′ ∈ A , u , v ∈ S with u s ≤ v s ′ .

A right S -poset A S satisfies Condition (E) if, for all a ∈ A and s , s ′ ∈ S , a s ≤ a s ′ implies a = a ′ u for some a ′ ∈ A , u ∈ S with u s ≤ u s ′ .

A right S -poset A S is called strongly flat if it satisfies Conditions ( P ) and ( E ) .

A right S -poset A S satisfies Condition ( P w ) if, for all a 1 , a 2 ∈ A S and s , t ∈ S , a 1 s ⩽ a 2 t implies a 1 ⩽ a u , a v ⩽ a 2 for some a ∈ A , u , v ∈ S with u s ⩽ v t .

Since Tran 1985 introduced the concept of regularity of acts (extending the notion of von Neumann regularity in semigroups) in [11], numerous articles have explored how properties of regular acts characterize semigroups S – for example, [12]. In 2005, Shi et al. [9] defined a corresponding notion, called I -regularity, for S -posets. A right S -poset A is I-regular if for every a ∈ A , there exists a homomorphism f : a S → S such that a f ( a ) = a . A pomonoid S is termed a right PP pomonoid if all principal right ideals of pomonoid S are projective. As an application of I -regularity, the authors characterized right po-cancellative pomonoids and right P P pomonoids via the I -regularity and projectivity of S -posets. Moreover, [9, Propositions 3.2 and 4.2] establish that for an S -poset A S , the following are equivalent:

1. A S is I -regular;
2. All cyclic S -subposets of A S are projective;
3. For every a ∈ A , there exists an element z ∈ S such that ker → λ a = ker → λ z (where λ a : S → a S is the translation map) and the ideal z S is projective.

It is well known that strong flatness is an important generalization of projectivity. Thus, it is natural to consider some generalized concept of I -regularity around strong flatness. In 2020, Feng [13] introduced CSF S -posets, a broader framework encompassing I -regular S -posets. An S -poset A S is called CSF if all cyclic S -subposets of A S are strongly flat (extending the notion of CSF acts in [14]). However, unlike for I -regularity, Example 2.2 demonstrates that the CSF property of an S -poset A S is strictly weaker than the condition that for every a ∈ A , there exists an element z ∈ S such that ker → λ a = ker → λ z and z S is strongly flat. Motivated by this gap, we further investigate generalization of I -regularity. This article addresses this matter.

The outline of this article is as follows. In Section 2, we introduce a new generalization of I -regular S -posets, known as strongly (SF)-cyclic S -posets, and verify some basic properties. In Section 3, we characterize pomonoids over which all strongly (SF)-cyclic S -posets have other flatness properties, and vice versa. Finally, we investigate the homological classification of pomonoids by strongly (SF)-cyclic Rees factor S -posets in Section 4.

2 Basic properties of strongly (SF)-cyclic S -posets

As for acts, strong flatness for S -posets has been extensively studied with profound results, such as [3,10]. In this section, using strong flatness, we define a new class of S -posets called strongly (SF)-cyclic S -posets and take into account the correlation properties.

Definition 2.1

A right S -poset A is called strongly (SF)-cyclic if for every a ∈ A , there exists z ∈ S such that ker → λ a = ker → λ z and z S is strongly flat.

It is immediate from the definition and the preceding section obtaining that

I -regularity ⇒ strong (SF)-cyclicity ⇒ the CSF property .

The following examples show that these three implications are strict in general.

Example 2.2

(the CSF property ⇏ strong (SF)-cyclicity) Let S be a left zero pomonoid and Θ S an one element right S -poset. Then, Θ S satisfies Condition ( E ) . However, since for every z ∈ S , ker → λ z = Δ S ≠ S × S = ker → λ θ , Θ S is not strongly (SF)-cyclic.

Recalled that a pomonoid S is a right PSF pomonoid if all principal right ideals of pomonoid S are strongly flat.

Example 2.3

(strong (SF)-cyclicity ⇏ I -regularity) Let S = S 1 ∪ S 2 , where S 1 = { x , x 2 , x 3 , … } equipped with the order in which x < x 2 < x 3 < … , S 2 = { 1 , e 1 , e 2 , … } with e i ⋅ e j = e max { i , j } and 1 , e i are isolated. The multiplication in S is defined by s ⋅ x n = x n ⋅ s = x n for any s ∈ S 2 and n ∈ N . Thus, S is a right PSF pomonoid, but it is not a right P P pomonoid.

The following properties can be obtained directly by definitions.

Proposition 2.4

Let S be a pomonoid. Then, the following statements hold:

If { A i } i ∈ I is a family of S-posets, then ∐ i ∈ I A i is strongly (SF)-cyclic if and only if for every i ∈ I , A i is strongly (SF)-cyclic;
For any family { A i } i ∈ I , if for every i ∈ I , A i is strongly (SF)-cyclic, then ∏ i ∈ I A i is strongly (SF)-cyclic;
Every S-subposet of a strongly (SF)-cyclic right S-poset is strongly (SF)-cyclic.

Next, we note that strong (SF)-cyclicity and the CSF property coincide for the S -poset S S .

Proposition 2.5

Let S be a pomonoid. Then, the following statements are equivalent:

S S is a strongly (SF)-cyclic S-poset.
S S is a CSF S-poset.
S is a right PSF pomonoid.
Every projective S-poset is strongly (SF)-cyclic.

Proof

( 1 ) ⇔ ( 2 ) ⇔ ( 3 ) are routine matters.

( 1 ) ⇔ ( 4 ) . Let S be strongly (SF)-cyclic and A projective. Then, A ≅ ∐ i ∈ I e i S , where e i 2 = e i ∈ S , i ∈ I . But the S -subposets e i S , i ∈ I , of the strongly (SF)-cyclic S -poset S are strongly (SF)-cyclic. Therefore, A ≅ ∐ i ∈ I e i S is strongly (SF)-cyclic by Proposition 2.4. For the converse, the S -poset S is projective, and by assumption, S is strongly (SF)-cyclic.□

Example 2.6

If S is a left zero pomonoid, then S is a right PSF pomonoid. Since ( Θ S × S ) S ≅ S S and S S is a strongly (SF)-cyclic right S -poset, ( Θ S × S ) S is also strongly (SF)-cyclic, but Θ S is not strongly (SF)-cyclic.

Note that Example 2.6 shows that the converse of Proposition 2.4(2) is not necessarily true. Now, for a left collapsible pomonoid, we show strong (SF)-cyclicity can be transferred from products of acts to their components. The pomonoid S is called left collapsible, if for any s , t ∈ S , there exists u ∈ S such that u s = u t .

Proposition 2.7

Let S be a left collapsible pomonoid. If ∏ i ∈ I A i is strongly (SF)-cyclic, then A i is strongly (SF)-cyclic, for every i ∈ I .

Proof

Let ( s , t ) ∈ ker → λ a i for s , t ∈ S , a i ∈ A i . Since S is a left collapsible pomonoid, there exists u ∈ S such that u s = u t . Consider the fixed elements a j ∈ A j , z j ∈ S j for every i ≠ j , and take

c j = a j u , if j ≠ i , a i , if j = i , and x j = z j u , if j ≠ i z i , if j = i .

Then, ( c j ) I ∈ ∏ i ∈ I A i , ( x j ) I ∈ ∏ i ∈ I S i , and by the assumption, we have ( s , t ) ∈ ker → λ ( c j ) I = ker → λ ( x j ) I and ( x j ) I S is strongly flat. Thus, ( x j ) I s ⩽ ( x j ) I t , it follows that z i s ⩽ z i t and ( s , t ) ∈ ker → λ z i , and so ker → λ a i ⊆ ker → λ z i and z i S is strongly flat. The case of ker → λ z i ⊆ ker → λ a i is similar, and we obtain ker → λ a i = ker → λ z i . Therefore, A i is strongly (SF)-cyclic for every i ∈ I .□

Next to consider the relationship between retraction monomorphisms and strong (SF)-cyclicity, recall that a monomorphism f : A → B is a retraction if there exists an S -morphism g : B → A such that g f = 1 A .

Proposition 2.8

Let A, and B be two right S-posets. If a monomorphism f : A → B is a retraction and B is a strongly (SF)-cyclic S-poset, then A is a strongly (SF)-cyclic S-poset.

Proof

Suppose that ( x , y ) ∈ ker → λ a , a x ⩽ a y for any a ∈ A and x , y ∈ S . Then, f ( a ) x ⩽ f ( a ) y , where f ( a ) ∈ B , and so ( x , y ) ∈ ker → λ f ( a ) , which implies that ker → λ a ⊆ ker → λ f ( a ) . Since B is a strongly (SF)-cyclic S -poset, by the definition, there exists z ∈ S such that ker → λ f ( a ) = ker → λ z and z S is strongly flat. Since, f is a retraction, there exists an S -morphism g : B → A such that g f = 1 A . Suppose that ( x , y ) ∈ ker → λ f ( a ) , f ( a ) x ⩽ f ( a ) y . Then, we have g f ( a ) x ⩽ g f ( a ) y , so a x ⩽ a y implies ( x , y ) ∈ ker → λ a . Thus, ker → λ f ( a ) ⊆ ker → λ a and so ker → λ f ( a ) = ker → λ a . Hence, ker → λ a = ker → λ z , and this shows that A is a strongly (SF)-cyclic S -poset.□

3 Characterizations of pomonoids by strong (SF)-cyclicity of right S -posets

In this section, we characterize pomonoids over which all strongly (SF)-cyclic right S -posets have other properties, and vice versa. It is known that the amalgamated coproduct A ( I ) is an important tool to study the homological classification of pomonoids. Therefore, we first construct the amalgamated coproduct in the general sense.

Construction 3.1

Let S be a pomonoid, t S a strongly (SF)-cyclic right ideal of S , and I a right ideal of S such that I ⊂ t S , I ≠ t S . Assume x , y , and z are different elements not belonging to S , and let

M = { ( x , t s ) ∣ s ∈ S , t s ∉ I } ∪ { ( y , t s ) ∣ s ∈ S , t s ∉ I } ∪ { ( z , t s ) ∣ s ∈ S , t s ∈ I } .

Define a right multiplication on M as follows:

( z , t s ) r = ( z , t s r ) , for all r ∈ S .

( x , t s ) r = ( x , t s r ) , if t s r ∉ I , ( z , t s r ) , if t s r ∈ I ,

( y , t s ) r = ( y , t s r ) , if t s r ∉ I , ( z , t s r ) , if t s r ∈ I .

The order on M is defined as

( w 1 , s ) ⩽ ( w 2 , t ) ⇔ ( w 1 = w 2 , s ⩽ t ) or ( w 1 ≠ w 2 , s ⩽ i ⩽ t for some i ∈ I ) ,

where { w 1 , w 2 } = { x , y } .

It is easy to check that M is a right S -poset with two generating elements ( x , t ) and ( y , t ) . Let

M 1 = { ( x , t s ) ∣ s ∈ S , t s ∉ I } ∪ { ( z , t s ) ∣ s ∈ S , t s ∈ I }

and

M 2 = { ( y , t s ) ∣ s ∈ S , t s ∉ I } ∪ { ( z , t s ) ∣ s ∈ S , t s ∈ I } .

Then, M 1 ≅ t S ≅ M 2 and M 1 , M 2 are S -subposets of M . Since t S is strongly (SF)-cyclic, hence by (2) of Proposition 2.4, M is also strongly (SF)-cyclic as required.

In Construction 3.1, when t = 1 , we obtain the concept of the amalgamated coproduct A ( I ) [15,16].

The following results describe the relationships between strongly (SF)-cyclicity and other flatnesses. We begin with the weakest flatness property weak torsion freeness. To do this, we need the following preparations.

Recall that a right S -poset is called simple if it has no proper S -subposet. A right S -poset is called completely reducible if it is a coproduct of simple S -posets. A right S -poset A S is weakly torsion free if, for any a , b ∈ A and any right po-cancellable element c ∈ S , a c = b c implies a = b . In the following theorems, we suppose that there exists at least a strongly (SF)-cyclic right S -poset and U is the greatest strongly (SF)-cyclic right ideal of S .

Lemma 3.2

Let S be a pomonoid. If there exists a strongly (SF)-cyclic right S-poset, then there exists the greatest strongly (SF)-cyclic right ideal U of S.

Proof

By assumption, there exists a strongly (SF)-cyclic right S -poset A . Thus, for every a ∈ A , there exists z ∈ S such that ker → λ a = ker → λ z , and so a S ≅ z S . Since a S is a S -subposet of A , it is strongly (SF)-cyclic, z S is also strongly (SF)-cyclic, and so we have at least one strongly (SF)-cyclic right ideal of S . Now, the union of all strongly (SF)-cyclic right ideals of S is the greatest strongly (SF)-cyclic right ideal of S , and it is denoted by U , which by (2) of Proposition 2.4 is strongly (SF)-cyclic.□

Theorem 3.3

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic S-posets are weakly torsion free;
For every u ∈ U and every right po-cancellable element c of S, there exists an element m ∈ S such that u = u c m .

Proof

( 1 ) ⇒ ( 2 ) Suppose that all strongly (SF)-cyclic S -posets are weakly torsion free, and let u ∈ U , c ∈ S , where c is right po-cancellable. We prove that u S = u c S . If u c S ⊂ u S and so by Construction 3.1, since u S ⊆ U and U is strongly (SF)-cyclic, M is strongly (SF)-cyclic. Thus, by assumption, M is weakly torsion free. Since u c = ( u , x ) c = ( u , y ) c , we have ( u , x ) = ( u , y ) , which is a contradiction. Thus, u S = u c S , and so there exists m ∈ S such that z = z c m .

( 2 ) ⇒ ( 1 ) Suppose that A S is a strongly (SF)-cyclic S -poset. Let a c = b c for a , b ∈ A S and c is a right po-cancellable element of S . By assumption, there exist z 1 , z 2 ∈ S such that ker → λ a = ker → λ z 1 , ker → λ b = ker → λ z 2 , and so a S ≅ z 1 S and b S ≅ z 2 S . Since A S is a strongly (SF)-cyclic S -poset, then by (3) of Proposition 2.4, a S and b S are strongly (SF)-cyclic and so z 1 S and z 2 S are also strongly (SF)-cyclic. Since z 1 S ∪ z 2 S ⊆ U , by assumption, there exists m ∈ S such that z 1 = z 1 c m . Thus, if z 1 ⩽ z 1 c m , we have ( 1 , c m ) ∈ ker → λ z 1 = ker → λ a , i.e., a ⩽ a c m . For the case z 1 c m ⩽ z 1 , similarly, one can obtain a c m ⩽ a , and so a = a c m implies a c = a c m c .

From a c = a c m c , it follows that b c = b c m c . If b c ⩽ b c m c , it implies ( c , c m c ) ∈ ker → λ b = ker → λ z 2 ; hence, z 2 c ⩽ z 2 c m c , z 2 ⩽ z 2 c m , so ( 1 , c m ) ∈ ker → λ z 2 = ker → λ b implies b ⩽ b c m = a c m = a . If b c m c ⩽ b c , then we obtain ( c m c , c ) ∈ ker → λ b = ker → λ z 2 , so z 2 c m c ⩽ z 2 c , z 2 c m ⩽ z 2 . Thus, ( c m , 1 ) ∈ ker → λ z 2 = ker → λ b , it follows a = a c m = b c m ⩽ b . Therefore, a = b , and this shows that A S is weakly torsion free.□

Recall that a right S -poset A is (po-)torsion free if, for any a , b ∈ A and a right (po-)cancellable element c ∈ S , a c = b c ( a c ⩽ b c ) implies a = b ( a ⩽ b ) .

Theorem 3.4 gives the characterization of a pomonoid when strongly (SF)-cyclic right S -posets are po-torsion free.

Theorem 3.4

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right S-posets are po-torsion free;
All finitely generated strongly (SF)-cyclic right S-posets are po-torsion free;
All cyclic right strongly (SF)-cyclic right S-posets are po-torsion free;
For every u ∈ U , u S is a po-torsion free right ideal of S.

Proof

( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) ⇒ ( 4 ) are obvious.

( 4 ) ⇒ ( 1 ) Suppose that A is a strongly (SF)-cyclic right S -poset. Then, by definition, for any a ∈ A , there exists u ∈ S such that ker → λ a = ker → λ u and so a S ≅ u S . Since a S is strongly (SF)-cyclic, which is an S -subposet of A , u S is strongly (SF)-cyclic and so u ∈ U . Then, by assumption, u S is po-torsion free and so a S is po-torsion free; hence, by [10, Proposition 3.19], A = ∐ a ∈ A a S is po-torsion free.□

From Example 3.5, we note that strongly (SF)-cyclicity and projectivity are independent of each other.

Example 3.5

Let S = { 0, 1 , x } with x 2 = 0 , where 0 < 1 and x cannot be compared with others. Then, S S as a right S -poset is free, so it is projective, but S S is not strongly (SF)-cyclic. In fact, x S = { 0 , x } as a cyclic S -subposet of S S which is not strongly flat; otherwise, x ⋅ x ⩽ x ⋅ 0 would imply that there exist u ∈ S such that x = x u and u ⋅ x ⩽ u ⋅ 0 . It is a contradiction.

Therefore, we naturally consider when strongly (SF)-cyclicity implies projectivity. To do this, we need the following lemma.

Lemma 3.6

Let S be a pomonoid and A S a right S-poset. If all cyclic S -subposets of A S are simple, then for any a , a ′ ∈ A S , either a S ∩ a ′ S = ∅ or a S = a ′ S .

Proof

A similar argument as in [17] for acts can be used.□

In [18], we say that Condition ( M R ) means that S satisfies the descending chain condition on principal right ideals, which is required in the next theorem.

Theorem 3.7

Let S be a pomonoid satisfying Condition ( M R ) . Then, the following statements are equivalence:

All strongly (SF)-cyclic right S-posets satisfy Condition ( P ) ;
All strongly (SF)-cyclic S-posets are projective.

Proof

( 1 ) ⇒ ( 2 ) Let u ∈ U . Then, u S is strongly (SF)-cyclic. Suppose I is a right ideal of S such that I ⊆ u S and I ≠ u S . Let M be the strongly (SF)-cyclic S -poset constructed in Construction 3.1. By assumption, M satisfies Condition ( P ) . By [13, Proposition 2.11], M must be a coproduct of cyclic S -subposets, which is impossible because ( u , x ) S ∩ ( u , y ) S = { ( u s , z ) } . Hence, u S must be a minimal right ideal.

Suppose that A is a strongly (SF)-cyclic right S -poset. Then, by definition, for every a ∈ A , there exists u ∈ S such that a S ≅ u S and u S is strongly flat. So a S is strongly flat. Since u S is a minimal right ideal of S for u ∈ U , u S is simple and so for every a ∈ A , a S is simple. Thus, by Lemma 3.6 for any a , a ′ ∈ A , either a S ∩ a ′ S = ∅ or a S = a ′ S . Hence, there exists A ′ ⊆ A such that A = ∪ a ∈ A ′ ˙ a S . By assumption, S satisfies ( M R ) , and by [18, Theorem 3.10], every strongly flat cyclic S -poset is projective. So by [9, Proposition 3.1], a S is projective and A is also projective.

( 2 ) ⇒ ( 1 ) It is obvious.□

Lemma 3.8

Let S be a pomonoid, uS a strongly (SF)-cyclic right ideal of S, and I S a right ideal of S such that I S ⊂ u S . Then, A S = u S ∐ I S u S is strongly (SF)-cyclic.

Proof

A S = ( u , x ) S ∪ ˙ I S ∪ ˙ ( u , y ) S , where B S = ( u , x ) S ∪ ˙ I S ≅ z S ≅ ( z , y ) S ∪ ˙ I S = C S . By assumption, z S is strongly (SF)-cyclic and A S = B S ∪ C S . Hence, A S is strongly (SF)-cyclic by (3) of Proposition 2.4.□

Recall that a right S -poset A S satisfies Condition ( P w ) if, for a 1 , a 2 ∈ A S and s , t ∈ S with a 1 s ⩽ a 2 t , there exist a ∈ A and u , v ∈ S such that u s ⩽ v t , a 1 ⩽ a u , a v ⩽ a 2 .

Furthermore, we obtain the conditions that a pomonoid satisfies when strongly (SF)-cyclic S -posets satisfy Condition ( P w ) .

Theorem 3.9

For any pomonoid S, if all strongly (SF)-cyclic right S-posets satisfy Condition ( P w ) , then for every u ∈ U , u S is a minimal right ideal of S.

Proof

Let u ∈ U . We say that u S is a minimal right ideal of S , otherwise, there exists a right ideal I of S such that I ⊂ u S . Then, by Lemma 3.8, A S = u S ∐ I S u S is strongly (SF)-cyclic and so A S satisfies Condition ( P w ) . Now let u c ∈ I . By the definition of A S , u c = ( u , x ) c = ( u , y ) c and so ( u , x ) c ⩽ ( u , y ) c . Then, there exist a ∈ A S , w 1 , w 2 ∈ S such that ( u , x ) ⩽ a w 1 , a w 2 ⩽ ( u , y ) and w 1 u ⩽ w 2 u . Now, ( u , x ) ⩽ a w 1 implies that a = ( t , x ) for some t ∈ u S \ I S ; similarly a = ( t ′ , y ) for some t ′ ∈ u S \ I S . So, we obtain a contradiction.□

Recall that a right S -poset A is called (regular) divisible if A = A c for any left (po-)cancellable element c ∈ S .

Theorem 3.10

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right S-posets are regular divisible;
All finitely generated strongly (SF)-cyclic right S-posets are regular divisible;
All cyclic strongly (SF)-cyclic right S-posets are regular divisible;
For every u ∈ U , u S is regular divisible.

Proof

( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) It is clear.

( 3 ) ⇒ ( 4 ) For any u ∈ U . Then, u S ⊆ U and so u S is strongly (SF)-cyclic which is an S -subposet of U . By assumption, u S is regular divisible.

( 4 ) ⇒ ( 1 ) Suppose that A is a strongly (SF)-cyclic right S -poset. By definition, for any a ∈ A , there exists u ∈ S such that ker → λ a = ker → λ u and so a S ≅ u S . Since a S is an S -subposet of A , it is strongly (SF)-cyclic. So u S is also strongly (SF)-cyclic and u ∈ U . By assumption, u S is regular divisible and so a S is regular divisible; hence, for any left (po-)cancellable element c ∈ S , a S c = a S , we have A c = ( ∪ a ∈ A a S ) c = ∪ a ∈ A a S c = ∪ a ∈ A a S = A . Thus, A is regular divisible.□

As we have seen in the previous section, I -regularity can imply strong (SF)-cyclicity, but not the converse. So it is natural to ask for pomonoids over which all strongly (SF)-cyclic S -posets are I -regular.

Theorem 3.11

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right S-posets are I-regular;
All finitely generated strongly (SF)-cyclic right S-posets are I-regular;
All cyclic strongly (SF)-cyclic right S-posets are I-regular;
For every u ∈ U , u S is I-regular.

Proof

( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) It is clear.

( 3 ) ⇒ ( 4 ) It is similar to the previous one.

( 4 ) ⇒ ( 1 ) Suppose that A is a strongly (SF)-cyclic right S -poset. By definition, for any a ∈ A there exists u ∈ S such that ker → λ a = ker → λ u and so a S ≅ u S . Since a S is an S -subposet of A , it is strongly (SF)-cyclic, so u S is also strongly (SF)-cyclic and so u ∈ U . Thus, by assumption, u S is I -regular and so a S is I -regular; hence, by [9, Lemma 4.5], A is I -regular.□

We already know that every regular pomonoid is right PSF; now let us use strong (SF)-cyclicity to fill this gap.

Theorem 3.12

Let S be a pomonoid. Then, the following conditions are equivalent:

S is a right PSF pomonoid and all strongly (SF)-cyclic right S-posets are principally weakly po-flat;
S is a right PSF pomonoid and all strongly (SF)-cyclic right S-posets are principally weakly flat;
S is a regular pomonoid.

Proof

( 1 ) ⇒ ( 2 ) It is clear.

( 2 ) ⇒ ( 3 ) Take any s ∈ S . If s S = S , then s is a regular element. If I = s S ≠ S , then s S is a proper right ideal of S ; we consider A ( s S ) . Since A ( s S ) = ( x , 1 ) S ∪ ( y , 1 ) S and ( x , 1 ) S ≅ S ≅ ( y , 1 ) S , it implies that A ( s S ) is a strongly (SF)-cyclic S -poset. Hence, by assumption, A ( s S ) is principally weakly flat, so by ([16], Lemma 2.2), for any i ∈ I , there exists j ∈ I such that i = i j i . Especially, let i = s , there exists t ∈ S such that s = s t s . Thus, S is a regular pomonoid.

( 3 ) ⇒ ( 1 ) It follows from [16, Theorem 2.3].□

Recall that a right S -poset A S is called faithful (strongly faithful) [9] if from a s ⩽ a t , s , t ∈ S , for all (some) a ∈ A , it follows that s ⩽ t .

Theorem 3.13

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right S-posets are strongly faithful;
S is left po-cancellative.

Proof

( 1 ) ⇒ ( 2 ) Suppose that A is a strongly (SF)-cyclic right S -poset and that for any s , t , z ∈ S , z s ⩽ z t . Let a ∈ A . Then, ( a z ) s ⩽ ( a z ) t . Since A is strongly faithful, s ⩽ t and so S is left po-cancellative.

( 2 ) ⇒ ( 1 ) Suppose that A is a strongly (SF)-cyclic right S -poset and that for a ∈ A , s , t ∈ S , a s ⩽ a t . By definition, there exists z ∈ S such that ker → λ a = ker → λ z . Then, a s ⩽ a t implies that ( s , t ) ∈ ker → λ a = ker → λ z and so z s ⩽ z t . Since S is left po-cancellative, hence s ⩽ t and so A is strongly faithful as required.□

Finally, we mainly discuss the pomonoids by strong (SF)-cyclicity of other right S -posets.

Theorem 3.14

Let S be a pomonoid. Then, the following conditions are equivalent:

All projective right S-posets are strongly (SF)-cyclic;
All free right S-posets are strongly (SF)-cyclic;
S is a right PSF pomonoid.

Proof

( 1 ) ⇒ ( 2 ) Obviously.

( 2 ) ⇒ ( 3 ) S S is a free S -poset, and so by the assumption, it is strongly (SF)-cyclic; thus, by ( 1 ) of Proposition 2.4, S is a right PSF pomonoid.

( 3 ) ⇒ ( 1 ) It follows from Propositions 2.4 and 2.5.□

Lemma 3.15

[4] Let S be any pomonoid. Then, the following statements hold:

Θ S is free if and only if ∣ S ∣ = 1 .
Θ S is projective if and only if S has a left zero element.
Θ S satisfies condition (E) if and only if S is left collapsible.

Theorem 3.16

Θ S is strongly (SF)-cyclic if and only if S contains a left zero element.

Proof

Suppose Θ S = { θ } is strongly (SF)-cyclic. Then, by definition, there exists z ∈ S such that ker → λ θ = ker → λ z . Since ker → λ θ = S × S . We take ( s , 1 ) ∈ S × S , then z s = z , so z is a left zero element.

For the converse, suppose that S contains a left zero element z . Then, ker → λ θ = ker → λ z = S × S . Also, S is left collapsible; hence, by Lemma 3.15, Θ S satisfies Condition ( E ) , and so z S = { z } is strongly flat.□

Theorem 3.17

All completely reducible right S-posets are strongly (SF)-cyclic if and only if S contains a left zero element.

Proof

The one-element right S -poset Θ S = { θ } is obviously completely reducible. Hence, Θ S = { θ } is strongly (SF)-cyclic. By Theorem 3.16, S contains a left zero element.

For the converse, suppose that S contains a left zero element. Then, the only simple right S -posets are one-element. By assumption, one-element right S -poset are strongly flat, then they are strongly (SF)-cyclic. Thus, every completely reducible right S -poset is strongly (SF)-cyclic by ( 2 ) of Proposition 2.4.□

4 Classifications of pomonoids by strong (SF)-cyclicity of right Rees factor S -posets

In this section, we mainly discuss the homological classification problems of pomonoids by strong (SF)-cyclicity of their right Rees factor S -posets.

We start with a characterization of Rees factor S -posets having strong (SF)-cyclicity. In what follows, we need the following result.

Lemma 4.1

[19] Let K be a convex, proper right ideal of the pomonoid S. Then, the following assertions are equivalent:

S ⁄ K is free;
S ⁄ K is projective;
S ⁄ K is strongly flat;
S ⁄ K satisfies condition (P);
∣ K ∣ = 1 .

Theorem 4.2

Let S be a pomonoid and K S a convex right ideal of the pomonoid S. Then, S ⁄ K S is a strongly (SF)-cyclic S-poset if and only if K S = S and S contains a left zero element or ∣ K S ∣ = 1 and S is a right PSF pomonoid.

Proof

( ⇒ ) Let K S be a convex right ideal of the pomonoid S , and suppose that S ⁄ K S is a strongly (SF)-cyclic S -poset. Then, there are two cases as follows:

Case (i) If K S = S , then, S ⁄ K S ≅ Θ S is strongly (SF)-cyclic, and so by Theorem 3.16, S contains a left zero element;

Case (ii) If K S is a proper right ideal of S , by assumption, S ⁄ K S is strongly (SF)-cyclic, then S ⁄ K S is strongly flat and by Lemma 4.1, ∣ K S ∣ = 1 and so S ⁄ K S ≅ S . Thus, S is a strongly (SF)-cyclic S -poset if and only if S is a right PSF pomonoid.

( ⇐ ) Suppose that K S is a convex right ideal of the pomonoid S . If K S = S and S contains a left zero element, then S ⁄ K S ≅ Θ S . By Theorem 3.16, S ⁄ K S is strongly (SF)-cyclic, if ∣ K S ∣ = 1 and S is a right PSF pomonoid. Since S ⁄ K S ≅ S , it is a strongly (SF)-cyclic S -poset.□

For the convenience of the following description, we refer to A S as having property (M) if its property is weaker than freeness.

Theorem 4.3

Let S be a pomonoid. Then, the following conditions are equivalent:

All right Rees factor S-posets having property (M) are strongly (SF)-cyclic S-posets;
All right Rees factor S-posets having property (M) are strongly flat, and if S contains a left zero, then S is a right PSF pomonoid, and if Θ S satisfies property (M), then S contains a left zero.

Proof

( 1 ) ⇒ ( 2 ) Since all right Rees factor S -posets having property ( M ) are strongly (SF)-cyclic, then all right Rees factor S -posets having property ( M ) are strongly flat;

Suppose that S contains a left zero element z . K S = z S = { z } , then S ⁄ K S ≅ S S . Since S S is free, so S ⁄ K S is free. Thus, S ⁄ K S satisfies property ( M ) , and by assumption, S ⁄ K S is strongly (SF)-cyclic. Thus, S S is strongly (SF)-cyclic and so by (1) of Proposition 2.4, S is a right PSF pomonoid;

If Θ S ≅ S ⁄ S S satisfies property ( M ) , then by assumption Θ S is strongly (SF)-cyclic, and by Theorem 3.16 S is collapsible and contains a left zero element.

( 2 ) ⇒ ( 1 ) For any convex ideal K S of S , let S ⁄ K S satisfies property ( M ) . Now there are two cases as follows:

Case (i) If K S = S , then, S ⁄ K S ≅ Θ S , and by assumption, S contains a left zero. Thus, by Theorem 3.16, S ⁄ K S is a strongly (SF)-cyclic S -poset;

Case (ii) If K S is a proper right ideal of S , since by assumption S ⁄ K S are strongly flat, by Lemma 4.1, ∣ K S ∣ = 1 and so K S = z S = { z } for some z ∈ S . Thus, z is a left zero, and by assumption, S is a right PSF pomonoid. Hence, by (1) of Proposition 2.4, S ⁄ K S ≅ S is a strongly (SF)-cyclic S -poset.□

Corollary 4.4

Let S be a pomonoid. Then, the following conditions are equivalent:

All free right Rees factor S-posets are strongly (SF)-cyclic S-posets;
All Rees factor projective generators are strongly (SF)-cyclic S-posets;
All projective right Rees factor S-posets are strongly (SF)-cyclic S-posets;
S contains no left zero or S is a right PSF pomonoid.

Proof

( 3 ) ⇒ ( 2 ) ⇒ ( 1 ) are obvious.

( 1 ) ⇒ ( 4 ) Suppose that S contains a left zero element. Then, by Theorem 4.3, S is a PSF pomonoid.

( 4 ) ⇒ ( 3 ) By Theorem 4.3, if Θ S is projective, then S has a left zero element and it follows from Lemma 3.15.□

Corollary 4.5

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly flat right Rees factor S-posets are strongly (SF)-cyclic S-posets;
S is not left collapsible or S contains a left zero and S is a right PSF pomonoid.

Proof

( 1 ) ⇒ ( 2 ) If S is left collapsible, then by Lemma 3.15, Θ S satisfies Condition ( E ) and so it is strongly flat. Thus, by Theorem 4.3, S contains a left zero and also S is a right PSF pomonoid.

( 2 ) ⇒ ( 1 ) If Θ S is strongly flat, then Θ S satisfies Condition ( E ) , by Lemma 3.15, S is left collapsible. The result follows from assumption and Theorem 4.3.□

By [20], let K S be a right ideal of the pomonoid S . K is called left stabilizing if any k ∈ K S , there exists l ∈ K S such that k ∈ [ l k ] , and let K S be a convex, proper right ideal of the pomonoid S . K S is called strongly left stabilizing if for every k ∈ K S and s ∈ S , such that k ⩽ l implies there exists k ′ ∈ K S and k ′ l ⩽ l or l ⩽ k implies there exists k ′ ∈ K S and l ⩽ k ′ l .

And by [4], a pomonoid S is called right weakly reversible, if for any p , q ∈ S , there exist u , v ∈ S such that u p ⩽ v q .

Proposition 4.6

Let S be a pomonoid. Then, the following conditions are equivalent:

All right Rees factor S-posets satisfying condition ( P ) are strongly (SF)-cyclic S-posets;
Every weakly right reversible pomonoid S is left collapsible and if S contains a left zero, then S is a right PSF pomonoid.

Proof

It follows from Theorem 4.3 and [20, Theorem 3.15].□

Proposition 4.7

Let S be a pomonoid. Then, the following conditions are equivalent:

All weakly flat right Rees factor S-posets are strongly (SF)-cyclic S-posets;
All flat right Rees factor S-posets are strongly (SF)-cyclic S-posets;
Every weakly right reversible pomonoid S is left collapsible and has no proper, left stabilizing convex right ideal K S such that ∣ K S ∣ > 1 . And if S contains a left zero, then S is a right PSF pomonoid.

Proof

It follows from Theorem 4.3 and [20, Theorem 3.16].□

Proposition 4.8

Let S be a pomonoid. Then, the following conditions are equivalent:

All principally weakly flat right Rees factor S-posets are strongly (SF)-cyclic S-posets;
S is left collapsible and has no proper, left stabilizing convex right ideal K S such that ∣ K S ∣ > 1 . And if S contains a left zero, then S is a right PSF pomonoid.

Proof

It follows from 4.3 and [20, Theorem 3.18].□

Proposition 4.9

Let S be a pomonoid. Then, the following conditions are equivalent:

All weakly po-flat right Rees factor S-posets are strongly (SF)-cyclic S-posets;
Every weakly right reversible pomonoid S is left collapsible and has no proper, strongly left stabilizing convex right ideal K S such that ∣ K S ∣ > 1 . And if S contains a left zero, then S is a right PSF pomonoid.

Proof

It follows from Theorem 4.3 and [20, Theorem 3.17].□

Proposition 4.10

Let S be a pomonoid. Then, the following conditions are equivalent:

All principally weakly po-flat right Rees factor S-posets are strongly (SF)-cyclic S-posets;
S is left collapsible and has no proper, strongly left stabilizing convex right ideal K S such that ∣ K S ∣ > 1 . And if S contains a left zero, then S is a right PSF pomonoid.

Proof

It follows from Theorem 4.3 and [20, Theorem 3.19].□

Theorem 4.11

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right Rees factor S-posets have property (M);
If S contains a left zero element, then Θ S has property ( M ) .

Proof

( 1 ) ⇒ ( 2 ) Suppose that S contains a left zero element. By Theorem 3.16, Θ S is a strongly (SF)-cyclic S -poset. Since Θ S ≅ S ⁄ S S , S ⁄ S S is strongly (SF)-cyclic S -poset. By assumption, S ⁄ S S has property ( M ) , so implies Θ S has property ( M ) ;

( 2 ) ⇒ ( 1 ) Suppose that S ⁄ K S is a strongly (SF)-cyclic S -poset for K S is a convex ideal of S . Then, there are two cases as follows:

Case (i) If K S = S , then S ⁄ K S ≅ Θ S is strongly (SF)-cyclic S -poset, and by Theorem 3.16, S is left collapsible and contains a left zero element. Thus, by assumption S ⁄ K S ≅ Θ S has property ( M ) ;

Case (ii) If K S is a proper ideal of S , by assumption, S ⁄ K S is strongly flat, by Theorem 4.2, ∣ K S ∣ = 1 , so S ⁄ K S ≅ S S . Thus, S ⁄ K S is free and so it has property ( M ) .□

Corollary 4.12

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right Rees factor S-posets are free;
If S is left collapsible, then ∣ S ∣ = 1 .

Proof

It follows from Theorem 4.11 and Lemma 3.15 (1).□

Corollary 4.13

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right Rees factor S-posets are projective;
If S is left collapsible, then S contains a left zero.

Proof

It follows from Theorem 4.11 and Lemma 3.15 (2).□

Corollary 4.14

Let S be a pomonoid. Then, the following conditions are equivalent:

All strongly (SF)-cyclic right Rees factor S-posets are strongly flat;
S is left collapsible.

Proof

It follows from Theorem 4.11 and Lemma 3.15(3).□

Let ρ be a right-order congruence on pomonoid S and s ∈ S . By s ρ , we denoted the right-order congruence on pomonoid S defined by x ( s ρ ) y ⇔ s x ρ s y , and the order relation on the quotient set is defined by [ x ] s ρ ⩽ [ y ] s ρ ⇔ [ s x ] ρ ⩽ [ s y ] ρ for x , y ∈ S . Now, we give conditions under which a cyclic act strongly (SF)-cyclic.

Theorem 4.15

Let ρ be a right-order congruence on pomonoid S. Then, S ⁄ ρ is strongly (SF)-cyclic if and only if for any s ∈ S , there exists z ∈ S such that z is strongly flat and s ρ = k e r λ z .

Proof

Suppose that S ⁄ ρ is a strongly (SF)-cyclic S -poset. By definition, there exists z ∈ S such that ker → λ [ s ] ρ = ker → λ z for any [ s ] ρ ∈ S ⁄ ρ and z S is strongly flat, and so the map f : [ s ] ρ S ⟶ z S , f ( [ s ] ρ t ) = z t is an isomorphism. Thus, it follows that x ( s ρ ) y , which implies ( s x ) ρ ( s y ) . Then, we have z x = z y , and so x ( k e r λ z ) y . Thus, s ρ ⊆ k e r λ z . We can obtain k e r λ z ⊆ s ρ at the same time. Therefore, s ρ = k e r λ z .

For the converse, let s ∈ S , by assumption, there exists z ∈ S such that z S is strongly flat and s ρ = k e r λ z . By [21, Corollary 3.7], z S ≅ S ⁄ k e r λ z and by [21, Lemma 3.34], S ⁄ ( s ρ ) ≅ [ s ] ρ S . Thus, z S ≅ [ s ] ρ S , which follows that ker → λ [ s ] ρ = ker → λ z . Hence, S ⁄ ρ is strongly (SF)-cyclic.□

It is also known that pomonoid S is a right PSF pomonoid if and only if for each s ∈ S s S is strongly flat. Furthermore, if we denote by Con A S the set of all congruence on the S -poset A S , then clearly (Con A S , ∩ ) is a pomonoid with the identity congruence A S . It can be routinely verified that for ( a , b ) ∈ A S × B S , ker → λ ( a , b ) = ker → λ a ∩ ker → λ b .

The next theorem gives a characterization of pomonoids over which D ( S ) is strongly (SF)-cyclic.

Theorem 4.16

Let S be a pomonoid. Then, the following conditions are equivalent:

The diagonal S-poset D ( S ) is strongly (SF)-cyclic;
S is a right PSF pomonoid and the set R = { ker → λ z ∣ z ∈ S } ∪ ( S S × S S ) is a subpomonoid of T = ( C o n S S , ∩ ) .

Proof

( 1 ) ⇒ ( 2 ) Take s ∈ S . Since D ( S ) is strongly (SF)-cyclic, ker → λ s = ker → λ ( s , s ) = ker → λ z and z S is strongly flat for some z ∈ S . Thus, S is a right PSF pomonoid. On the other hand, by assumption for each pair of elements z 1 , z 2 ∈ S , there exists z ′ ∈ S such that ker → λ z 1 ∩ ker → λ z 2 = ker → λ ( z 1 , z 2 ) = ker → λ z ′ and z ′ S is strongly flat.

( 2 ) ⇒ ( 1 ) Suppose that ( s , t ) ∈ D ( S ) for any s , t ∈ S . So there exist z 1 , z 2 ∈ S such that ker → λ s = ker → λ z 1 , ker → λ t = ker → λ z 2 and z 1 S , z 2 S are strongly flat. Since R is a subpomonoid of T , there exists z ∈ S such that ker → λ z 1 ∩ ker → λ z 2 = ker → λ z , z S is strongly flat since S is a right PSF pomonoid. Now, we obtain ker → λ ( s , t ) = ker → λ s ∩ ker → λ t = ker → λ z , and hence, D ( S ) is strongly (SF)-cyclic.□

The following theorem asserts equivalent conditions for every finite product of strongly (SF)-cyclic S -posets to be strongly (SF)-cyclic for a right PSF pomonoid.

Theorem 4.17

Let S be a pomonoid. Then, the following conditions are equivalent:

S n is strongly (SF)-cyclic for every n ∈ N ;
The diagonal S-poset D(S) is strongly (SF)-cyclic.

Proof

( 1 ) ⇒ ( 2 ) Take n = 1, 2 respectively.

( 2 ) ⇒ ( 1 ) Suppose that A and B are two strongly (SF)-cyclic S -posets. Take ( a , b ) ∈ A × B . Let ker → λ a = ker → λ z 1 , ker → λ b = ker → λ z 2 , and z 1 S , z 2 S are strongly flat for some z 1 , z 2 ∈ S . By Theorem 4.16, there exists z ∈ S , ker → λ ( a , b ) = ker → λ a ∩ ker → λ b = ker → λ z 1 ∩ ker → λ z 2 = ker → λ z and z S is strongly flat since S is a right PSF pomonoid. Now, by induction, we obtain the desired result.□

To conclude this article, we introduce a new flatness called strong (SF)-cyclicity and show that the class of S -posets having this property lies strictly between the classes of I -regular S -posets and CSF S -posets. Moreover, we characterize some important pomonoids by using this property, such as regular pomonoids (see Theorem 3.12), PSF pomonoids (see Theorem 3.14), and so on. In particular, if the order of pomonoids S and S -posets are discrete, then we can obtain the definition of strongly (SF)-cyclic acts and the corresponding results for right acts. Meanwhile, according to [14] and [11], strong (SF)-cyclicity actually lies exactly between regularity and the CSF property for acts. These concepts and conclusions further enrich the theory of S -acts and S -posets.

Acknowledgments

The authors would like to give many thanks to the referee for his/her invaluable comments and suggestions.

Funding information: This study was supported by Innovative Fundamental Research Group Project of Gansu Province, Grant no. 23JRRA684 and The National Natural Science Foundation of China, Grant no. 12461003.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflicts of interest.

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Received: 2024-11-17

Revised: 2025-06-18

Accepted: 2025-07-25

Published Online: 2025-09-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0190

Keywords for this article

pomonoids; S-posets; strongly (SF)-cyclic; homological classifications

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