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Kyungpook Mathematical Journal 2022; 62(4): 657-671

Published online December 31, 2022 https://doi.org/10.5666/KMJ.2022.62.4.657

Copyright © Kyungpook Mathematical Journal.

On Generators in the Category of Actions of Pomonoids on Posets and its Slices

Farideh Farsad and Ali Madanshekaf∗

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P. O. Box 35131-19111, Semnan, Iran
e-mail : faridehfarsad@yahoo.com and amadanshekaf@semnan.ac.ir

Received: November 29, 2021; Revised: May 22, 2022; Accepted: October 10, 2022

Where S is a pomonoid, let Pos-S be the category of S-posets and S-poset maps. We start off by characterizing the pomonoids S for which all projectives in this category are either generators or free. We then study the notions of regular injectivity and weakly regularly d-injectivity in this category. This leads to homological classification results for pomonoids. Among other things, we get find relationships between regular injectivity in the slice category Pos-S/BS, for any S-poset BS, and generators and cyclic projectives in Pos-S.

Keywords: S-poset, generator, projective, slice category, regular injective.

General ordered algebraic structures play a key role in a wide range of areas, including analysis, logic, theoretical computer science, and physics [2]. One of these structures, which is of interest to mathematicians, is the category of representations of a pomonoid by order preserving maps of partially ordered sets (see for example [3, 4, 5, 6, 7, 8, 9, 14, 16, 18,19]). Although there exist many papers which investigate various properties of generator acts over a fixed monoid (see [10, 11, 12, 17] for example), among them there seems to be very little known on generator S-posets, where S is a pomonoid. In [14], V. Laan investigated some properties of generator S-posets. Furthermore, in [9] some homoligical characterizations of pomonoids by properties of generators were presented. Continuing this study, in this paper, after some introductory results in Section 1, we attempt in Section 2 to collect new results on generators in Pos-S to apply to the question of the homological classification of pomonoids.

M-injective objects in the slice category C/B, for any B in C, form the right part of a weak factorization system that has morphisms of M as the left part (see [1]). Here, we consider the same case in the slice category Pos-S/BS of right S-poset maps over BS, where BS is an arbitrary S-poset. In Section 3, we first find conditions for when all generators are regular d-injective or weakly regularly injective. Then, we prove that every M-injective object in Pos-S/BS is a split epimorphism, where M=Emb is the class of all order-embeddings of S-posets. Also, we investigate the relationship between regular injectivity in Pos-S and Pos-S/BS and generators and cyclic projectives which becomes evident when passing to acts over their endomorphism monoids.

For the rest of this section, we give some preliminaries about S-acts, S-posets and slice category which we will need in the sequel. The reader is referred to [13] and [1], respectively, for information on general properties of S-acts and S-posets that are not fully explained here.

Let S be a monoid with identity 1. Recall that a (right) S-act is a set A equipped with a map μ:A×SA called its action, such that, denoting μ(a,s) by as, we have a1 = a and a(st)=(as)t, for all a ∈ A, and s, t ∈ S. The category of all S-acts, with action-preserving (S-act) maps (f:AB with f(as) = f(a)s, for sS,aA), is denoted by Act-S. For instance, take any monoid S and a non-empty set A. Then A becomes a right S-act by defining as = a for all a ∈ A, s ∈ S; we call that A an S-act with trivial action. Clearly S itself is an S-act with its operation as the action.

On a monoid S we define the following relations: for every s, t ∈ S

  • 1. sRt iff sS=tS.

  • 2. sJt   iff SsS=StS.

  • 3. sDt iff there exists u∈ S with sS=uS and St=Su.

These relations are called Green's relations on S (see [13]). Here, we consider these notions for a pomonoid S and supply some suitable results. A monoid S is said to be a partially ordered monoid (briefly a pomonoid) if it is also a poset whose partial order ≤ is compatible with the binary operation, i.e., st,stimplysstt (see [2]). In this paper S denotes a pomonoid with an arbitrary order, unless otherwise stated.

Let S be a pomonoid and A be a poset. Then A×S becomes a poset with componentwise order. A poset A is said to be a (right) S-poset} over a pomonoid S if it is an S-act and the action is monotone ((a1,s1)(a2,s2) implies, a1s1a2s2, where a1,a2A and s1,s2S). We denote it by AS. The category of all S-posets with action preserving monotone maps is denoted by Pos-S. Clearly S itself is an S-poset with its operation as the action. A left S-poset A can be defined analogously (see [3]) and denoted by SA. Also, we denote the category of all left S-posets with action preserving monotone maps by S-Pos. As in the unordered case, the coproduct in Pos-S is simply the disjoint union, with S-action and order given componentwise, and as usual the coproduct of a family {AiiI} will be denoted by iIAi. Let T and S be pomonoids. Then a poset A is called a T-S-biposet if it is a left T-poset and a right S-poset and (ta)s=t(sa) for every sS,tT and a∈ A. We denote it by TAS.

We recall the following results from [14]:

(i) For every AS in Pos-S, consider the set End(AS)=PosS(A,A) as a pomonoid with respect to composition and pointwise order. We define the left End(AS)-action on A by fa=f(a), for every f ∈ End(AS), a ∈ A. Note that this action is monotone because if f,gEnd(AS) and a , b ∈ A are such that f ≤ g and a ≤ b then we have fa=f(a)f(b)g(b)=gb. Thus one has End(AS)AS.

(ii) The following two mappings are pomonoid homomorphisms:

ρ:SEnd(AS);sρs,
λ:TEnd(TA);tλt.

Here, ρs:ASAS,aas and λt: TA TA,ata are morphisms in Pos-S and T-Pos, respectively.

(iii) For every T-S-biposet T AS recall that if B ∈,Pos-S then the set PosS(B , A) of all S-poset maps from BS to AS is an object in T-Pos with the action defined by tf=λtf for every tT,fPosS(B,A). Consequently, we have a functor

PosS(,A):PosSTPos

by taking

PosS(,A)(B)=PosS(B,A)

for every B∈ Pos-S.

An S-poset GS is a generator in the category Pos-S if for any distinct S-poset maps α,β:XSYS there exists an S-poset map f:GSXS such that αfβf.

For any category C and an object B of C, there is a slice category (also called comma category) C/B. The objects of C/B are morphisms of C with codomain B, and morphisms in C/B from one such object f:FB to another g:EB are commutative triangles in C:

i.e, gh=f. We write h:fg.The composition in C/B is defined from the composition in C, in the obvious way- the triangles are pasted together (for more details see [15]).

A poset is said to be complete if each of its subsets has an infimum and a supremum, in particular, a complete poset is bounded, that is, it has a least (bottom) element and a greatest (top) element .

In this section, we discuss the properties of generators and projective generators in Pos-S. Recall that a projective S-poset AS which is also a generator is called a projective generator S-poset. A cyclic S-poset is an S-poset A for which there exists an element a ∈ A such that A = aS. By a cyclic projective S-poset we mean a cyclic S-poset which is also projective.

As we mentioned in the introduction, generators for the category Pos-S were characterized in [14] with the following two propositions.

Proposition 2.1. Cyclic projectives in Pos-S are precisely retracts of SS.

Proposition 2.2. An S-poset AS is a cyclic projective generator in Pos-S if and only if ASeSS for an idempotent e∈ S with eJ1.

The following is immediate from Proposition 2.2:

Proposition 2.3. Let S be a commutative pomonoid. Then all cyclic projective generators in Pos-S are isomorphic to SS.

We will also need the following characterization of cyclic projective S-posets from [19, Proposition 4.2].

Proposition 2.4. Let AS be an S-poset and a ∈ A. Then the following statements are equivalent:

(i) aSS is projective.

(ii) aSS ≅ eSS for some idempotent e ∈ S.

We state the following two facts about projectives and generators from [19] and [14] respectively. They will be used throughout the paper.

Theorem 2.5. An S-poset PS is projective if and only if PS iIeiS where ei2=eiS,iI.

Theorem 2.6. The following assertions are equivalent for a right S-poset AS.

  • 1. For all XS,YS Pos-S and f,gPosS(X,Y),fg whenever fkgk for all kPosS(A;X).

  • 2. AS is a generator.

  • 3. For every XS Pos-S there exists a set I and an epimorphism h:IAX in Pos-S.

  • 4. There exists an epimorphism π:AS in Pos-S.

  • 5. SS is a retract of AS.

Now we can prove the following result.

Theorem 2.7. Every S-poset PS is projective generator if and only if PS= iIPi where PieiS for every i ∈ I, and at least one Pj, j ∈ I is a generator with ejJ1.

Proof. On the one hand, let the S-poset PS be a projective generator. By Theorem 2.5 we have PS iIeiS where ei2=eiS,iI. And by Theorem 2.6 there exists a surjective S-poset epimorphism π:PSSS, so 1=π(a)for some aejS,jI. Now π|ejS:ejS SS is also an epimorphism in Pos-S, because for any s ∈ S we have s=1s=π(a)s=π(as) and asejS. Hence, ejS is a generator and by Proposition 2.2, ejJ1.

On the other hand, assume that PS has the factorisation in the statement of the theorem. By Theorem 2.5, PS is projective. That Pj is generator, implies that there exists an S-poset epimorphism πj:Pj SS. Now, for the following diagram

take qj=πj and qi the composite S-poset map PieiSS for every iI, i=j. By the property of the coproduct S-poset PS= iIPi, corresponding to the S-poset epimorphisms {qiiI}, there exists a unique S-poset map π:PSS such that π|Pi=qi for all iI. In particular, π|Pj=πj and πj is an S-poset epimorphism, so π is also an S-poset epimorphism. Hence, PS is generator.

Notice that for every pomonoid S and idempotent e∈ S, the sub S-poset eSS of SS is projective according to Proposition 2.4, but it is not a generator because eJ1 does not necessarily hold. For example, if we take a periodic monoid S endowed it with discrete order then we have a pomonoid. Now if we take an idempotent 1eS, then eJ1 does not hold (see [13, Proposition I.3.26 on page 32] for more details).

Next, we have the following result.

Theorem 2.8. For any pomonoid S the following statements are equivalent:

(i) All projective right S-posets are generators in Pos-S.

(ii) All cyclic projective right S-posets are generators in Pos-S.

(iii) eJ1 for every idempotent e ∈ S.

Proof. That (i) implies (ii) is clear. To see that (ii) implies (iii) observe that for any idempotent e∈ S, the right S-poset eSS is cyclic, hence it is a genreator by assumption. The result thus follows by Proposition 2.2. .

For the implication (iii) (i), let PS be an S-poset. By Theorem 2.5 we have PS iIeiS where ei2=eiS,iI. By the assumption we have eiJ1 for every i ∈ I and so PS is a generator by Theorem 2.7.

Recall [4] that a right poideal of a pomonoid S is a (possibly empty) subset I of S if it is both a monoid right ideal (ISI) and a down set (ab,bI imply that aI). It is principal if it is generated (as a monoid right ideal of S) by a single element. For example

rS={tS : sS, trs}

is a principal poideal of S, for every rS.

In the following we shall characterize pomonoids for which all principal right poideals are generators.

Proposition 2.9. Let S be a pomonoid and e∈ S satisfy e2=e. If the cyclic projective sub S-poset eSS of SS is a generator in Pos-S, then eS is also a generator.

Proof. By assumption there exists an S-poset epimorphism f:eSSSS. Define the mapping g: SSS by g(x):=f(ex) for every xeS. It is easy to see that g is an S-poset map. Also, for every s ∈ S there exists u ∈ S such that f(eu)=s. Then we have

g(eu)=f(eeu)=f(eu)=s.

This means that g is an epimorphism. By Theorem 2.6 we conclude that eS is a generator, as required.

Lemma 2.10. Let S be a pomonoid and z∈ S. If the principal right poideal zS is a generator in Pos-S, then there exist x,yS such that 1yx, and zazb, a, b∈ S implies yayb.

Proof. Since zS is a generator in Pos-S, by Theorem 2.6, there exists an epimorphism g: zSSS. Hence, there are elements u zS and t ∈ S such that uzt and g(u)=1. Let y=g(z) and x=t. Then yx=g(z)x=g(zx). Since uzx, the monotonicity of g implies that g(u)g(zx). Consequently, 1=g(u)g(zx)=yx. Now, suppose za≤ zb, a, b∈ S. Then ya=g(z)a=g(za)g(zb)=g(z)b=yb.

Next we answer the question about the conditions under which the assumptions of Proposition 2.9 are satisfied.

Proposition 2.11. Let S be a pomonoid in which the identity element is the top element. If all poideals of S are generators in Pos-S, then the sub S-poset eSS of SS is a generator in Pos-S, for every idempotent e ∈ S.

Proof. Assume that all poideals of S are generators in Pos-S. Then for every idempotent e∈ S, eS is a generator in Pos-S. By Lemma 2.10, there exist x, y∈ S such that 1yx, and eaeb, a, b ∈ S, always implies yayb. In particular, since e1ee we have y ≤ ye, so 1yxyex. As we have yex1 by the hypothesis, we get yex=1, which means that eJ1. So eSS is a projective generator by Proposition 2.2, as needed.

Theorem 2.12. Let S be a pomonoid in which the identity element is the top element. The following statements are equivalent:

(i) All projective right S-posets are generators in Pos-S.

(ii) All cyclic projective right S-posets are generators in Pos-S.

(iii) eJ1 for every idempotent e ∈ S.

(iv) All principal right poideals of S which are generated by an idempotent, are generators in Pos-S.

Proof. The equivalence of the first three statements is Theorem 2.8.

That (iii) implies (iv) is easy.Indeed, by Proposition 2.2 we get that eSS is a cyclic projective generator, and Proposition 2.9 shows that eS is a generator in Pos-S.

To finish off, we show that (iv) implies (iii). Consider the principal right poideal eS for every idempotent e ∈ S which is a generator in Pos-S. By a proof similar to that of Proposition 2.11, the cyclic projective sub S-poset eSS of SS is a generator. Using Proposition 2.2, we conclude that eJ1.

By a free S-poset on a poset P we mean an S-poset F together with a poset map τ:PF with the universal property that given any S-poset A and any poset map f:PA there exists a unique S-poset map f¯:FA such that f¯τ=f, i.e, the diagram

commutes. The S-poset F (up to isomorphism) is given by F=P×S with componentwise order and the action (x, s)t=(x, st), for x ∈ P and s, t ∈ S (see [3] for example). Furthermore, by a free S-poset we mean an S-poset which is free on some poset.

Example 2.13. Let S be a pomonoid generated by the elements e,k,k and with discrete order such that kk=1,e2=e and ek=k. Then eSS is a cyclic projective generator in Pos-S. But eSS is not free (see Lemma 2.14 below).

Now, we present some condition under which the sub S-posets eSS of SS are free for idempotent elements e∈ S. The proof of the following result is similar to the proof for the unordered case in [13, Proposition 3.17.17], so we omit it. Moreover, we conclude when projectivity (or cyclic projectivity) implies freeness in Pos-S.

Lemma 2.14. Let e be an idempotent of a pomonoid S. Then the sub S-poset eSS of SS is a free right S-poset if and only if eD1.

This allows us to prove the following.

Theorem 2.15. For any pomonoid S the following statements are equivalent:

(i) All projective right S-posets are free.

(ii) All projective generators in Pos-S are free.

(iii) All cyclic projective right S-posets are free.

(iv) eD1 for every idempotent e ∈ S.

Proof. The implication (i)(ii) is trivial.

To see (ii)(iii), observe that by Proposition 2.4, all cyclic projective S-posets are isomorphic to eSS for some idempotent e ∈ S. Let A=SSeSS. By Proposition 2.7, AS is a projective generator in Pos-S. By hypothesis AS is free which implies that eSS is free.

Now we show that (iii)(i). By decomposition theorem in [19], every projective S-poset is isomorphic to a coproduct of cyclic projective S-posets which are free by assumption. Now since the coproducts of free S-posets being free we get the result.

By the characterization of cyclic projective S-posets in Proposition 2.4 and Lemma 2.14 we get the equivalence of (iii) and (iv), which completes the proof.

Let C be a category and M a class of its morphisms. An object I of C is called M-injective if for each M-morphism h:UV and morphism u:UI there exists a morphism s:VI such that sh = u. That is, the following diagram is commutative:

In particular, this means that, in the slice category C/B, an object f:XB is M-injective if, for any commutative diagram in C

with hM, there exists an arrow s:VX such that sh=u and fs=υ.

Recall that regular monomorphisms (morphisms which are equalizers) in Pos-S (and also in Pos-S/BS) are exactly order-embeddings (see [3] and [6]). By Emb-injectivity in Pos-S we mean M-injectivity in Pos-S, where M=Emb is the class of all order-embeddings of S-posets. In the following we shall deal with Emb-injectivity in Pos-S and Pos-S/BS, where Emb is the class of all order-embeddings of S-posets.

Theorem 3.1. All generators in Pos-S are Emb-injective if and only if all S-posets are Emb-injective.

Proof. Clearly it is enough to show the forward implication. Let AS be an S-poset. Consider the product S-poset AS×SS which is a generator in Pos-S by Theorem 2.6 and so is Emb-injective. By a general category-theoretic result which states that a product of a family of injective objects in a category is injective if and only if each component of the product is injective, we get that AS is Emb-injective in Pos-S.

Note that the class of all embeddings of right poideals into SS is a subclass of all down-closed embeddings in Pos-S, i.e. all embeddings g:BSCS with the property that g(B) is down-closed in C, and hence is a subclass of all embeddings.

Definition 3.2. An S-poset AS is called (principally) weakly regularly d-injective if it is injective with respect to all embeddings of (principal) right poideals into SS.

Proposition 3.3. If all generators in Pos-S are weakly regularly d-injective then all S-posets are weakly regularly d-injective.

Proof. Let AS be an S-poset. Since AS×SS is a generator in Pos-S it is a weakly regularly d-injective. To show that AS is weakly regularly d-injective consider the following diagram

where I is a poideal of S. Define S-poset map u¯:ISAS×SS by u¯(s)=(u(s),s) for each sIS. By the assumption, there exists an S-poset map v:SSAS×SS such that vi=u¯.

Now by composition v with the projection πA:AS×SSAS, we get AS is a weakly regularly d-injective.

For a pomonoid S recall that an element s∈ S is called regular if there exists t∈ S such that sts=s. One calls S a regular pomonoid if all its elements are regular.

Theorem 3.4. Let S be a pomonoid whose identity element is the top element.

Then the following statements are equivalent:

(i) All S-posets are principally weakly regularly d-injective.

(ii) All principal right poideals of S are principally weakly regularly d-injective.

(iii) All generators in Pos-S are principally weakly regularly d-injective.

(iv) S is a regular pomonoid.

Proof. The equivalence of (i) and (iii) comes from (the proof of) Proposition 3.3. The implication (iv) (i) is in [18, Theorem 3.6] and the implication (i) (ii) is trivial, so it is enough for us to show the implication (ii) (iv).

So assume (ii). For every s∈ S, consider the down-closed embedding i: sSSS,xx. It has a left inverse f, as sS is principally weakly regularly d-injective. Taking f(1)=z, we have z ≤ st for some t ∈ S and

s=f(s)=f(1)s=zssts.

On the other hand, stss, as 1 is the top element of S. Therefore sts=s, showing that s is a regular element. As this was for any s, S is a regular pomonoid.

Recall from [4] that a pomonoid S which has no proper non-empty left (right) poideal is said to be left (right) simple.

Corollary 3.5. If all generators in Pos-S are Emb-injective then S is left simple.

Proof. From the hypothesis and Theorem 3.1, we conclude that all complete S-posets are Emb-injective. It follows then from [4, Theorem 3.9] that S is left simple.

Proposition 3.6. For any pomonoid S the following statements are equivalent:

(i) All generators in Pos-S are complete S-posets.

(ii) All S-posets are complete.

Proof. First assume (i). Let AS be an S-poset. Consider the generator AS×SS, which is a complete S-poset by assumption. Since the order on the product AS×SS is the componentwise order, joins are computed componentwise in the product as well. That is, for a subset TAS×BS we have T=(πA(T),πB(T)) where πA and πB are canonical projections on AS and BS, respectively. Therefore, for any subset BA, B exists and so AS is complete, giving (ii).

The converse implication is trivial.

We state the following result from [7, Proposition 3.17] that will be used later on. We give a direct proof of it here, for the convenience of the reader.

Proposition 3.7. Let S be a pomonoid and BS Pos-S. Suppose f:ASBS is an Emb-injective object in Pos-S/BS. Then f is a split epimorphism in Pos-S.

Proof. By the universal property of the coproduct S-poset A˙B (the disjoint union of A and B) there exists a unique S-poset map f¯:A˙BB such that the following diagram commutes where iA and iB are injection S-poset maps.

In fact,

f¯(x)=f(x)ifxAxifxB.

Now, let us consider the following commutative square

Since f is an Emb-injective object in Pos-S/BS, there exists a unique S-poset map h:A˙BA such that fh=f¯ and hiA=idA. So fhiB=f¯iB=idB, which shows that f is a split epimorphism in Pos-S.

Remark 3.8. There exist split epimorphisms in Pos-S which are not Emb-injective in Pos-S/BS. To present an example, take an arbitrary pomonoid S and let X and B be, respectively, the first and second lattices shown in the following diagram:

Evidently, X is an S-poset with the action defined by s= and as=bs=s=a for all s∈ S, also we consider B with the trivial action as an S-poset. Define the S-poset map f:XSBS, by f(a)=f(b)=f()=0 and f()=1. Then f is a convex map. We show that it is not a regular injective object in Pos-S/BS. Since f1(0)={,a,b} is not a complete lattice, the authors in [6] showed that it is not Emb-injective in Pos-S/BS.

On theother hands, define the S-poset map g:BSXS by g(0)=,g(1)=. Then we have fg=idB, so f is a split epimorphism. Therefore, the converse of the above proposition is not true generally.

Next recall that for a given poset P and a pomonoid S, the cofree S-poset on P is the set P(S) of all monotone maps from S to P, with pointwise order and action given by (fs)(t)=f(st) for s, t ∈ S and fP(S) (see also [3, Theorem 13]).

Corollary 3.9. Suppose f:ASBS is an Emb-injective object in Pos-S/BS. If A is a complete lattice which is also a retract of the cofree S-poset A(S), then AS and BS are Emb-injective object in Pos-S.

Proof. By hypothesis we conclude that A(S) is an Emb-injective S-poset (see [4, Theorem 3.3]). Also it is straightforward to see that a retract of a Emb-injective S-poset is Emb-injective and so we get AS is an Emb-injective S-poset. Also, by Proposition 3.7 the S-poset map f is a split epimorphism. Consequently BS being a retract of an Emb-injective S-poset is an Emb-injective S-poset.

At the rest of this section, we investigate some connections between Emb-injectivity in Pos-S/BS and generators and cyclic projectives in Pos-S.

Theorem 3.10. If f:ASBS is an Emb-injective object in Pos-S/BS and BS is a generator in Pos-S then AS is a generator. Further, End(AS)A is a cyclic projective in End(AS)-Pos.

Proof. Since f:ASBS is Emb-injective object in Pos-S/BS, by Proposition 3.7, there exists g:BSAS in Pos-S such that fg=idB. As BS is a generator in Pos-S and f is an epimorphism, AS is also a generator (see [14]). Now, applying this fact and [14, Theorem 2.2], we get that End(AS)A is a cyclic projective.

Theorem 3.11. Suppose f:ASBS is an Emb-injective object in Pos-S/BS where AS is a cyclic projective S-poset. Then BS is a cyclic projective S-poset. Moreover, End(BS)B is a generator in End(BS)-Pos.

Proof. Since f:ASBS is Emb-injective object in Pos-S/BS, by Proposition 3.7, there exists g:BSAS in Pos-S such that fg=idB. Also, AS is a cyclic projective in Pos-S hence by Proposition 2.1, there exist two S-poset maps SSγπAS such that πγ=idA. This yields fπγg=idB which shows that BS is a retract of SS. We get BS is a cyclic projective S-poset by Proposition 2.1, so by [14, Proposition 3.1], we conclude that End(BS)B is a generator in End(BS)-Pos.

Theorem 3.12. Suppose f:ASBS is an Emb-injective object in Pos-S/BS. Then all of the following hold.

(i) PosS(BS,AS) is a generator in Pos-End(BS).

(ii) PosS(AS,BS) is a generator in End(BS)-Pos.

(iii) PosS(BS,AS) is a cyclic projective in End(AS)-Pos.

(iv) PosS(AS,BS) is a cyclic projective in Pos-End(AS).

Proof. Since f:ASBS is Emb-injective object in Pos-S/BS, in view of Proposition 3.7, there exists g:BSAS such that fg=idB. Applying the functors PosS(BS, -) and PosS(-, BS) to the identity map idBS we can easily get the assertions (i)} and (ii), respectively. Again by applying the functors PosS(,AS) and PosS(AS, -) to the above identity, in light of Proposition 2.1, we can deduce that the statements (iii) and (iv) are true.

Proposition 3.13. Let AS be an S-poset. Then in any of the following cases PosS(AS× BS, BS) is a generator in End(BS)-Pos, for every BSPos-S:

(i) AS is an Emb-injective S-poset.

(ii) f: AS→ BS is an Emb-injective object in Pos-S/BS.

Proof. (i) Consider the second projection S-poset map πB:AS×BSBS. The authors in [6] have showed that it is an Emb-injective object in Pos-S/BS. Consequently, by Theorem 3.12(ii), we get the result.

(ii) By Proposition 3.7, there exists an S-poset map g:BSAS such that fg=idB. By the universal property of the product S-poset A×B there exists a unique S-poset map φB:BSA×B (indeed b(g(b),b)) such that the following diagram commutes:

i.e., πBφB=idB and πAφB=g. Applying the functor PosS(-, BS) to the first identity above we obtain

End(BS)=PosS(B,B)φ¯Bπ¯BPosS(A×B,B)

such that φ¯Bπ¯B=idEnd(BS). This means that End(BS) is a retract of PosS(A×B,B) as we needed (see Theorem 2.6 again).

Proposition 3.13. Suppose that BS is in Pos-S, TAS is a T-S-biposet, and A×B is a cyclic projective S-poset. If f:ASBS is an Emb-injective object in Pos-S/BS and λ:TEnd(AS), defined as in (1.1), is an isomorphism then T A is a generator in T-Pos.

Proof. Consider the second projection S-poset map πA:A×BAS and the unique S-poset map φA:ASA×B for which πAφA=idA. That is, let φA(a)=(a,f(a)). Since A×B is a cyclic projective S-poset by assumption, there exist S-poset maps A×BπγSS such that πγ= idA×B. Applying the functor PosS(,AS) to the former identity and knowing that the composition πAπγφA= idA, we obtain

TPosS(A,A)φ¯Aπ¯APosS(A×B,A)γ¯π¯PosS(S,A)TA

in which φ¯Aπ¯A=idPosS(A,A) and γ¯π¯=idPosS(S,A). Thus, T is a retract of T A and hence T A is a generator in Pos-S.

The authors would like to express their sincere thanks to the anonymous referee for a careful reading of the manuscript and for invaluable comments which improved the exposition of the article. Parts of this research were completed while the second author was on sabbatical leave at the Department of Mathematics, Vanderbilt University (VU), Nashvile, TN, USA. This author expresses his thanks for the warm hospitality and facilities provided by Prof. Constantine Tsinakis and the Department of Mathematics of VU. He is greatly indebted to Semnan University for its financial support during the sabbatical.

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