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/B_S of right S-poset maps over B_S, where B_S 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/B_S 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/B_S 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×S→A 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:A→B with f(as) = f(a)s, for s∈S,a∈A), 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., s≤t,s′≤t′ imply ss′≤tt′ (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, a1s1≤a2s2, where a1,a2∈A and s1,s2∈S). We denote it by A_S. 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 {Ai∣i∈I} will be denoted by ∐ i∈IAi. 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 s∈S,t∈T and a∈ A. We denote it by TAS.

We recall the following results from [14]:

(i) For every A_S 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(A_S)-action on A by f⋅a=f(a), for every f ∈ End(A_S), a ∈ A. Note that this action is monotone because if f,g∈End(AS) and a , b ∈ A are such that f ≤ g and a ≤ b then we have f⋅a=f(a)≤f(b)≤g(b)=g⋅b. Thus one has End(AS)AS.

(ii) The following two mappings are pomonoid homomorphisms:

Here, ρs:AS→AS, a↦as and λt: TA→ TA, a↦ta are morphisms in Pos-S and T-Pos, respectively.

(iii) For every T-S-biposet _T A_S recall that if B ∈,Pos-S then the set Pos_S(B , A) of all S-poset maps from B_S to A_S is an object in T-Pos with the action defined by t⋅f=λtf for every t∈T,f∈PosS(B,A). Consequently, we have a functor

by taking

for every B∈ Pos-S.

An S-poset G_S is a generator in the category Pos-S if for any distinct S-poset maps α,β:XS→YS there exists an S-poset map f:GS→XS such that αf≠βf.

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:F→B to another g:E→B are commutative triangles in C:

i.e, gh=f. We write h:f→g.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 A_S 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 S_S.

Proposition 2.2. An S-poset A_S is a cyclic projective generator in Pos-S if and only if AS≅eSS 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 S_S.

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

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

(i) aS_S is projective.

(ii) aS_S ≅ eS_S 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 P_S is projective if and only if PS≅∐ i∈IeiS where ei2=ei∈S,i∈I.

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

• 1. For all XS,YS∈ Pos-S and f,g∈PosS(X,Y),f≤g whenever fk≤gk for all k∈PosS(A;X).

• 2. A_S is a generator.

• 3. For every X_S∈ Pos-S there exists a set I and an epimorphism h:∐IA→X in Pos-S.

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

• 5. S_S is a retract of A_S.

Now we can prove the following result.

Theorem 2.7. Every S-poset P_S is projective generator if and only if PS=∐ i∈IPi where Pi≅eiS for every i ∈ I, and at least one P_j, j ∈ I is a generator with ejJ1.

Proof. On the one hand, let the S-poset P_S be a projective generator. By Theorem 2.5 we have PS≅∐ i∈IeiS where ei2=ei∈S,i∈I. And by Theorem 2.6 there exists a surjective S-poset epimorphism π:PS→SS, so 1=π(a)for some a∈ejS,j∈I. 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 as∈ejS. Hence, e_jS is a generator and by Proposition 2.2, ejJ1.

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

take qj=πj and q_i the composite S-poset map Pi≅eiS↪S for every i∈I, i=j. By the property of the coproduct S-poset PS=∐ i∈IPi, corresponding to the S-poset epimorphisms {qi∣i∈I}, there exists a unique S-poset map π:P→SS such that π|Pi=qi for all i∈I. In particular, π|Pj=πj and π_j is an S-poset epimorphism, so π is also an S-poset epimorphism. Hence, P_S is generator.

Notice that for every pomonoid S and idempotent e∈ S, the sub S-poset eS_S of S_S 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 1≠e∈S, 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 eS_S is cyclic, hence it is a genreator by assumption. The result thus follows by Proposition 2.2. .

For the implication (iii) ⇒ (i), let P_S be an S-poset. By Theorem 2.5 we have PS≅∐ i∈IeiS where ei2=ei∈S,i∈I. By the assumption we have eiJ1 for every i ∈ I and so P_S 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 (IS⊆I) and a down set (a≤b,b∈I imply that a∈I). It is principal if it is generated (as a monoid right ideal of S) by a single element. For example

is a principal poideal of S, for every r∈S.

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 eS_S of S_S is a generator in Pos-S, then ↓eS is also a generator.

Proof. By assumption there exists an S-poset epimorphism f:eSS→SS. Define the mapping g: ↓S→SS by g(x):=f(ex) for every x∈↓eS. 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

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,y∈S such that 1≤yx, and za≤zb, a, b∈ S implies ya≤yb.

Proof. Since ↓zS is a generator in Pos-S, by Theorem 2.6, there exists an epimorphism g: ↓zS→SS. Hence, there are elements u∈ ↓zS and t ∈ S such that u≤zt and g(u)=1. Let y=g(z) and x=t. Then yx=g(z)x=g(zx). Since u≤zx, 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 eS_S of S_S 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 1≤yx, and ea≤eb, a, b ∈ S, always implies ya≤yb. In particular, since e1≤ee we have y ≤ ye, so 1≤yx≤yex. As we have yex≤1 by the hypothesis, we get yex=1, which means that eJ1. So eS_S 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 eS_S 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 eS_S of S_S 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:P→F with the universal property that given any S-poset A and any poset map f:P→A there exists a unique S-poset map f¯:F→A 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 eS_S is a cyclic projective generator in Pos-S. But eS_S is not free (see Lemma 2.14 below).

Now, we present some condition under which the sub S-posets eS_S of S_S 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 eS_S of S_S 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 eS_S for some idempotent e ∈ S. Let A=SS∐eSS. By Proposition 2.7, A_S is a projective generator in Pos-S. By hypothesis A_S is free which implies that eS_S 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:U→V and morphism u:U→I there exists a morphism s:V→I 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:X→B is M-injective if, for any commutative diagram in C

with h∈M, there exists an arrow s:V→X such that sh=u and fs=υ.

Recall that regular monomorphisms (morphisms which are equalizers) in Pos-S (and also in Pos-S/B_S) 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/B_S, 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 A_S 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 A_S is Emb-injective in Pos-S.

Note that the class of all embeddings of right poideals into S_S is a subclass of all down-closed embeddings in Pos-S, i.e. all embeddings g:BS→CS 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 A_S is called (principally) weakly regularly d-injective if it is injective with respect to all embeddings of (principal) right poideals into S_S.

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

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

Now by composition v with the projection πA:AS×SS→AS, we get A_S 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: ↓sS→SS,x↦x. 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

On the other hand, sts≤s, 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 A_S 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 T⊆AS×BS we have ∨T=(∨πA(T),∨πB(T)) where π_A and π_B are canonical projections on A_S and B_S, respectively. Therefore, for any subset B⊆A, ∨B exists and so A_S 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:AS→BS is an Emb-injective object in Pos-S/B_S. 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∪˙B→B such that the following diagram commutes where i_A and i_B are injection S-poset maps.

In fact,

Now, let us consider the following commutative square

Since f is an Emb-injective object in Pos-S/B_S, there exists a unique S-poset map h:A∪˙B→A 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/B_S. 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:XS→BS, 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/B_S. Since f−1(0)={⊥,a,b} is not a complete lattice, the authors in [6] showed that it is not Emb-injective in Pos-S/B_S.

On theother hands, define the S-poset map g:BS→XS 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 f∈P(S) (see also [3, Theorem 13]).

Corollary 3.9. Suppose f:AS→BS is an Emb-injective object in Pos-S/B_S. If A is a complete lattice which is also a retract of the cofree S-poset A(S), then A_S and B_S 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 A_S is an Emb-injective S-poset. Also, by Proposition 3.7 the S-poset map f is a split epimorphism. Consequently B_S 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/B_S and generators and cyclic projectives in Pos-S.

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

Proof. Since f:AS→BS is Emb-injective object in Pos-S/B_S, by Proposition 3.7, there exists g:BS→AS in Pos-S such that fg=idB. As B_S is a generator in Pos-S and f is an epimorphism, A_S 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:AS→BS is an Emb-injective object in Pos-S/B_S where A_S is a cyclic projective S-poset. Then B_S is a cyclic projective S-poset. Moreover, End(BS)B is a generator in End(BS)-Pos.

Proof. Since f:AS→BS is Emb-injective object in Pos-S/B_S, by Proposition 3.7, there exists g:BS→AS in Pos-S such that fg=idB. Also, A_S is a cyclic projective in Pos-S hence by Proposition 2.1, there exist two S-poset maps SS⇄γπAS such that πγ=idA. This yields fπγg=idB which shows that B_S is a retract of S_S. We get B_S 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:AS→BS is an Emb-injective object in Pos-S/B_S. 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(A_S).

Proof. Since f:AS→BS is Emb-injective object in Pos-S/B_S, in view of Proposition 3.7, there exists g:BS→AS such that fg=idB. Applying the functors Pos_S(B_S, -) and Pos_S(-, B_S) to the identity map idBS we can easily get the assertions (i)} and (ii), respectively. Again by applying the functors PosS(−,AS) and Pos_S(A_S, -) 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 A_S be an S-poset. Then in any of the following cases Pos_S(A_S× B_S, B_S) is a generator in End(B_S)-Pos, for every B_S∈Pos-S:

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

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

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

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

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

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

Proposition 3.13. Suppose that B_S is in Pos-S, _TA_S is a T-S-biposet, and A×B is a cyclic projective S-poset. If f:AS→BS is an Emb-injective object in Pos-S/B_S and λ:T→End(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×B→AS and the unique S-poset map φA:AS→A×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×B⇄πγSS such that πγ= idA×B. Applying the functor PosS(−,AS) to the former identity and knowing that the composition πAπγφA= idA, we obtain

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.