EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 2, 2014, 166-178 ISSN 1307-5543 – www.ejpam.com Categorical Properties of Regular Monomorphisms of S-posets Hamid Rasouli Department of Mathematics, College of Basic Sciences, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran Abstract. Recall that monomorphisms in some categories such as the category of posets, and the cat- egory of topological spaces, are not necessarily embeddings. The notion of regular monomorphism, solves this problem in these two categories. We have the same situation in the category S-Pos of S- posets; that is posets with an action of a pomonoid S which preserves the order. In this category, regular monomorphisms exactly determine sub S-posets. In this paper, we study some categorical properties of regular monomorphisms in the category of S- posets with action-preserving monotone maps. 2010 Mathematics Subject Classifications: 06F05, 18A20, 18A30, 20M30 Key Words and Phrases: S-poset, regular monomorphism 1. Introduction and Preliminaries One of the very useful notions in many branches of mathematics as well as in computer science is the action of a semigroup or a monoid on a set. Such acts corresponds to rep- resentation of monoids. Also the action of a pomonoid S on partially ordered sets, namely S-posets, appears as representations of mappings between pomonoids (cf. [6]). Brazegar et al. [5], inspired by the work of Banaschewski [2] on M -injectivity, introduced three differ- ent kinds of essentiality for a subclass M of monomorphisms of a category, and considered some category-theoretic conditions onM to study well-behaviour ofM -injectivity via these essential monomorphisms. In the present paper, considering M to be the class of regular monomorphisms (order-embeddings) in the category S-Pos of S-posets with action-preserving monotone maps, we investigate some categorical properties of M which mostly are useful in the study of well-behaviour of regular injectivity of S-posets (see also [4, 13]). Then we compare some categorical properties of monomorphisms and regular monomorphims in the categories of posets and S-posets. A study of S-posets from a category-theoretic standpoint forms the content of [11], and extends the results found in [8]. For more information on various properties of S-posets, see also [7, 9, 10, 15]. Email address: hrasouli@srbiau.ac.ir http://www.ejpam.com 166 c© 2014 EJPAM All rights reserved. H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 167 In the rest of this section we give some preliminaries about S-acts, posets, and S-posets needed in the sequel. Let S be a monoid with identity 1. Recall that a (left) S-act A is a set equipped with a map λ : S × A→ A, called its action, such that, denoting λ(s, a) by sa, we have 1a = a and (st)a = s(ta), for all a ∈ A, and s, t ∈ S. The category of all S-acts, with action-preserving maps between them, is denoted by S-Act. An S-act congruence θ on A is an equivalence relation with the property that aθa′, a, a′ ∈ A, implies that saθ sa′, for all s ∈ S. A quotient S-act is the set A/θ with the natural action, s[a] = [sa], which makes the canonical map γ : A→ A/θ , a 7→ [a], an S-act map. For more information about S-acts, see [12]. A monoid S is said to be a pomonoid if it is also a poset whose partial order is compatible with the binary operation. For a pomonoid S, a (left) S-poset is a poset A which is also an S-act whose action is mono- tone in both arguments. An S-poset map (morphism) is an action preserving monotone map between S-posets. Note that each poset P can be made into an S-poset with trivial action: sp = p, for every p ∈ P, s ∈ S. Let A be an S-poset. An S-poset congruence on A is an S-act congruence θ with the property that the S-act A/θ can be made into an S-poset in such a way that the canonical S-act map A→ A/θ is an S-poset map. For a binary relation R on A, define the relation≤R on A by a ≤R a′ if and only if a ≤ a1Ra′1 ≤ . . .≤ anRa′n ≤ a′, for some a1, a′1, . . . , an, a′n ∈ A. Then an S-act congruence θ on A is an S-poset congruence if and only if aθa′ whenever a ≤θ a′ ≤θ a. The S-poset quotient is then the S-act quotient A/θ with the partial order given by [a] ≤ [b] if and only if a ≤θ b. Also the S-poset congruence θ (H) on A generated by H ⊆ A×A can be characterized as follows (see [15, Proposition 3.3]): aθ (H)a′ if and only if a = a′, or there exist s1, s2, . . . , sn, t1, t2, . . . , tm ∈ S such that a ≤ s1c1, s1d1 ≤ s2c2, s2d2 ≤ s3c3, . . . , sndn ≤ a′; a′ ≤ t1p1, t1q1 ≤ t2p2, t2q2 ≤ t3p3, . . . , tmqm ≤ a, where (ci , di), (p j , q j) ∈ H ∪H−1 for i = 1, 2, . . . , n and j = 1,2, . . . , m. Moreover, the order relation on A/θ (H) can be defined by: [a]≤ [a′] if and only if a ≤ a′, or there exist s1, s2, . . . , sn ∈ S such that a ≤ s1c1, s1d1 ≤ s2c2, s2d2 ≤ s3c3, . . . , sndn ≤ a′, where (ci , di) ∈ H ∪H−1 for i = 1,2, . . . , n. Recall that the product of a family of S-posets is their cartesian product, with component- wise action and order. The coproduct is their disjoint union, with natural action and componen- twise order. As usual, we use the symbols ∏ and ∐ for product and coproduct, respectively. Also for a family (Aα)α∈I of S-posets each with a unique fixed element 0, the direct sum ⊕ Aα is defined to be the sub S-poset of the product ∏ Aα consisting of all (aα)α∈I such that aα = 0 for all α ∈ I except a finite number of indices. Throughout S denotes a pomonoid unless otherwise stated, andM stands for the class of regular monomorphisms of S-posets. H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 168 2. Categorical Properties of Regular Monomorphisms in S-Pos In this section we investigate the categorical and algebraic properties, regarding composi- tion, limits and colimits of the category S-Pos with respect to the classM of regular monomor- phisms which are exactly order-embeddings. We have divided the section into three subsections as follows: 2.1. Composition Properties of Regular Monomorphisms In this subsection we study the composition properties of regular monomorphisms of S- posets. To find the importance of these properties, the reader is referred to [1, 2]. Lemma 1. The classM is: i) Composition closed; that is, if f : A→ B and g : B→ C belong toM , then g f also belongs toM . ii) Isomorphism closed; that is, it contains all isomorphisms and is closed under composition with isomorphisms. iii) Left cancellable; that is, if g f ∈M , then f ∈M . Proof. SinceM is in fact the class of all order-embeddings, the proof is obvious. Remark 1. For every pomonoid S, the classM is not right cancellable. For example, consider the inclusions 2 f ,→ 2∪̇1 g ,→ 3 with trivial actions of a pomonoid S on 2, 2∪̇1 and 3. Then g f ∈M but g is not inM . Also, for every pomonoid S, there always exists a monomorphism that is not regular. To see this, it suffices to take the inclusion i : 1∪̇1 ,→ 2 with trivial actions of a pomonoid S on 1∪̇1 and 2. Here we investigate the factorization property in S-Pos. Recall that for two morphisms f : A→ B and g : C → D in a category C , f is called vertical on g if for morphisms u : A→ C and v : B → D which v ◦ f = g ◦ u, there exists a unique morphism w : B → C such that w ◦ f = u and g ◦ w = v. Also a factorization diagonalization system for C , is a pair (E ,M ) where E andM are some classes of morphisms, with the following properties: • E andM are composition and isomorphism closed. • For every e ∈ E and m ∈M , e is vertical on m. • Every morphism f ∈ C has a factorization of the form f = m ◦ e in which m ∈M and e ∈ E . In this case, we say that C has (E ,M )-factorization diagonalization property. To find a factorization diagonalization system for S-posets, we use the decomposition and first isomorphism theorems (see [7, Proposition 2.3], [8, Theorem 1]). First recall that for every S-poset homomorphism f : A→ B, the subkernel of f is defined by K f = {(a, a′) ∈ A× A| f (a)≤ f (a′)}. H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 169 Proposition 1. Let E be the class of all S-poset epimorphisms. Then S-Pos has (E ,M )-factorization diagonalization property. Proof. First, notice that in view of Lemma 1, E andM are composition and isomorphism closed. Also [7, Proposition 2.3] implies that each S-poset homomorphism f : A→ B can be decomposed as f = me, where e = f π : A π → A/ker f f ' f (A), and m = i : f (A) ,→ B. Clearly, m ∈M and e ∈ E . Now, we show that every e ∈ E is vertical on every m ∈M . Let u : A→ C and v : B→ D be some S-poset homomorphisms such that the following diagram is commutative: A u �� e // B v �� C m // D We claim that Ke ⊆ Ku. Suppose e(a)≤ e(a′), for a, a′ ∈ A. Then mu(a) = ve(a)≤ ve(a′) = mu(a′). Since m is a regular monomorphism, we get u(a) ≤ u(a′), as claimed. Using [8, Theorem 1], there exists a unique d : B → C such that de = u. Also we have mde = mu = ve and hence md = v because e is an epimorphism. This completes the proof. 2.2. Limits of regular monomorphisms In this subsection some of the categorical properties of regular monomorphisms related to limits such as products and pullbacks are studied. Proposition 2. i) The classM is closed under products. ii) Let { fα : A → Bα|α ∈ I} be a family of regular monomorphisms. Then their product homomorphism f : A→ ∏ Bα is also a regular monomorphism. Proof. It is straightforward. Proposition 3. Let { fα : A → Bα|α ∈ I} be a source of regular monomorphisms. Then the homomorphism f : A → l im←−Bα (existing by the universal property of limits) is also a regular monomorphism. Proof. Let f (a)≤ f (a′) for some a, a′ ∈ A. For every α ∈ I , we have fα(a) = πα f (a)≤ πα f (a′) = fα(a ′), where πα : l im←−Bα → Bα is a limit morphism. Since fα is a regular monomorphism by the assumption, a ≤ a′. Thus f is a regular monomorphism. H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 170 The above result as well as Proposition 2 (ii) are also true whenever for some (not neces- sarily all) α ∈ I , fα is a regular monomorphism. Recall that a class of morphisms of a category is called pullback stable if pullbacks transfer those morphisms. In the next result, we study this property for regular monomorphisms of S-posets. Proposition 4. The classM is pullback stable. Proof. Consider the pullback diagram P pA �� pB // B g �� A f // C where P is the sub S-poset {(a, b) : f (a) = g(b)} of A× B, and pullback maps pA : P → A, pB : P → B are restrictions of the projection maps. Assume that f ∈M . We show that pB ∈M . Let pB(a, b)≤ pB(a′, b′), for a, a′ ∈ A, b, b′ ∈ B. Then b ≤ b′ and hence f (a) = g(b)≤ g(b′) = f (a′). This implies a ≤ a′, because f is a regular monomorphism. Therefore, (a, b) ≤ (a′, b′), as required. 2.3. Colimits of Regular Monomorphisms In this subsection we investigate the colimit properties, such as coproducts, direct sums, pushouts and directed colimits (direct limits) of regular monomorphisms. Proposition 5. The classM is closed under coproducts and direct sums. Proof. Assume that { fα : Aα → Bα|α ∈ I} is a family of regular monomorphisms and ∐ fα : ∐ Aα → ∐ Bα is the coproduct morphism (which uniquely exists by the universal property of coproducts): Aα ια �� fα // Bα ι′α �� ∐ Aα ∐ fα // ∐ Bα We show that ∐ fα is a regular monomorphism. Let ( ∐ fα)(a,α) ≤ ( ∐ fα)(a′,α′), where a ∈ Aα, a′ ∈ Aα′ ,α,α′ ∈ I . It can be written as ( ∐ fα)ια(a)≤ ( ∐ fα)ια′(a′) and by the commutativity of the diagram, we get ( fα(a),α) = ι ′ α fα(a) = ( ∐ fα)ια(a)≤ ( ∐ fα)ια′(a ′) = ι′α′ fα′(a ′) = ( fα′(a ′),α′); H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 171 But this is impossible except α = α′ and then fα(a) ≤ fα(a′). Since fα is order-embedding, a ≤ a′. Consequently, (a,α) = (a,α′)≤ (a′,α′), as claimed. For the second part, let { fα : Aα→ Bα|α ∈ I} be a family of regular monomorphisms such that Aα and Bα have a unique zero element, and f : ⊕ Aα → ⊕ Bα be the homomorphism induced by the product of fα’s. In fact, f = ∏ fα � � � ⊕ Aα . Since ∏ fα is a regular monomorphism by Propoition 2, this clearly implies that so is f . Here we show that pushouts transfer regular monomorphisms in S-Pos. Theorem 1. S-Pos hasM -transferring pushouts. Proof. Consider the pushout diagram A g �� f // B qB �� C qC // Q Recall that Q = (B t C)/θ (H) and θ (H) is the S-poset congruence on B t C generated by H = {((1, f (a)), (2, g(a))) : a ∈ A}, where iB : B → B t C , iC : C → B t C are the coproduct injections given by iB(b) = (1, b) and iC(c) = (2, c), for every b ∈ B, c ∈ C . Also the pushout maps are given as qC = πiC : C → Q, qB = πiB : B → Q, where π : B t C → Q is the canonical epimorphism. Suppose f is a regular monomorphism. To show that qC is a regular monomorphism, let qC(c)≤ qC(c′), for c, c′ ∈ C . Thus we have [(2, c)] = πiC(c) = qC(c)≤ qC(c ′) = πiC(c ′) = [(2, c′)]. Then, by [15, Proposition 3.3], we get (2, c) ≤ (2, c′) (and hence c ≤ c′) or there exist s1, s2, . . . , sn ∈ S such that (2, c)≤ s1c1, s1d1 ≤ s2c2, s2d2 ≤ s3c3, . . . , sndn ≤ (2, c′), where (ci , di) ∈ H ∪H−1 for i = 1,2, . . . , n. It follows that there exist a1, . . . , an ∈ A such that (2, c)≤ s1(2, g(a1)), s1(1, f (a1))≤ s2(1, f (a2)), s2(2, g(a2))≤ s3(2, g(a3)), . . . , sn−1(1, f (an−1))≤ sn(1, f (an)), sn(2, g(an))≤ (2, c′). This gives the following: c ≤ g(s1a1), f (s1a1)≤ f (s2a2), g(s2a2)≤ g(s3a3), . . . , f (sn−1an−1)≤ f (snan), g(snan)≤ c′. Since f is a regular monomorphism, s1a1 ≤ s2a2, s3a3 ≤ s4a4, . . . , sn−1an−1 ≤ snan. H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 172 Hence, we get c ≤ g(s1a1)≤ g(s2a2)≤ g(s3a3)≤ . . .≤ g(sn−1an−1)≤ g(snan)≤ c′. For a class E of morphisms of a category, we say that multiple pushouts transfer E -morphisms if in the multiple pushout (Q, (Aα qα→ Q)α∈I) of a family { fα : A→ Aα|α ∈ I} of E -morphisms, qα ∈ E , for every α ∈ I . Analogously to the pushouts, the following result is obtained. Theorem 2. Multiple pushouts transfer regular monomorphisms. Proof. Assume that (Q, (Aα qα→Q)α∈I) is the multiple pushout of a family { fα : A→ Aα|α ∈ I} of regular monomorphisms. Recall that Q = ( ∐ Aα)/θ (H), where θ (H) is the S-poset congruence on ∐ Aα generated by H = {(iα( fα(a)), iβ( fβ(a)))|a ∈ A,α,β ∈ I}, and qα = πiα, where π : ∐ Aα → Q and iα : Aα → ∐ Aα are the natural map and coproduct injection, respectively. We take α ∈ I and prove that qα is a regular monomorphism. Let qα(aα)≤ qα(a′α), for aα, a′α ∈ Aα. Then we have [(α, aα)] = πiα(aα) = qα(aα)≤ qα(a ′ α) = πiα(a ′ α) = [(α, a′α)]. Using [15, Proposition 3.3], this implies that (α, aα) ≤ (α, a′α) (and then aα ≤ a′α) or there exist s1, s2, . . . , sn ∈ S such that (α, aα)≤ s1c1, s1d1 ≤ s2c2, s2d2 ≤ s3c3, . . . , sndn ≤ (α, a′α), where (ci , di) ∈ H ∪H−1 for i = 1,2, . . . , n. Thus there exist a1, . . . , an ∈ A such that (α, aα)≤ s1(α, fα(a1)), s1(α1, fα1 (a1))≤ s2(α1, fα1 (a2)), s2(α2, fα2 (a2))≤ s3(α2, fα2 (a3)), . . . , sn−1(αn−1, fαn−1 (an−1)) ≤ sn(αn−1, fαn−1 (an)), sn(α, fα(an))≤ (α, a′α). Then we get aα ≤ fα(s1a1), fα1 (s1a1)≤ fα1 (s2a2), fα2 (s2a2)≤ fα2 (s3a3), . . . , fαn−1 (sn−1an−1)≤ fαn−1 (snan), fα(snan)≤ a′α. Since fα1 , . . . , fαn−1 are regular monomorphisms, s1a1 ≤ s2a2, . . . , sn−1an−1 ≤ snan, whence aα ≤ fα(s1a1)≤ fα(s2a2)≤ fα(s3a3)≤ . . .≤ fα(sn−1an−1)≤ fα(snan)≤ a′α, as required. H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 173 Corollary 1. Every multiple pushout of regular monomorphisms (the diagonal maps on the mul- tiple pushout diagram) is a regular monomorphism. Proof. Apply Lemma 1(i) and Theorem 2. Definition 1. Let E be a class of morphisms of a category C . We say that C has: i) E -bounds if for every small and non-empty family {hα : A→ Bα}α∈I of E -morphisms, there is an E -morphism h : A→ B which factorizes through all hα’s. ii) E -amalgamation property if in (i), h factorizes through all hα’s by E -morphisms. In view of Corollary 1, the following is immediate: Proposition 6. S-Pos hasM -amalgamation property and so also hasM -bounds. Finally, we study directed colimit of regular monomorphisms in S-Pos. Recall that a directed system of S-posets and S-poset maps is a family (Ai)i∈I of S-posets indexed by an up-directed set I endowed by a family (ψi j : Ai → A j)i≤ j∈I of S-poset maps such that given i ≤ j ≤ k ∈ I , ψik = ψ jkψi j , and ψii = id. Also the pair (l im−→Ai , {αi : Ai → l im−→Ai}) or in abbreviation, l im−→Ai is called the directed colimit (or direct limit) of the directed system ((Ai)i∈I , (ψi j)i≤ j) if for every i ≤ j ∈ I , α jψi j = αi , and for every (B, fi : Ai → B) with f jψi j = fi , i ≤ j ∈ I , there exists a unique S-poset map ν : l im−→Ai → B such that ναi = fi , for every i ∈ I . Recall from [7] that the directed colimit of a directed system ((Ai)i∈I , (ψi j)i≤ j) of S-posets exists, and may be represented as (A/θ , (ψi : Ai → A/θ )i∈I), where i) A= ∐ Ai; ii) aθa′(a ∈ Ai , a′ ∈ A j) if and only if ∃k ≥ i, j :ψik(a) =ψ jk(a′); iii) [a]θ ≤ [a′]θ (a ∈ Ai , a′ ∈ A j) if and only if ∃k ≥ i, j :ψik(a)≤ψ jk(a′); iv) for each i ∈ I and a ∈ Ai , ψi(a) = [a]θ . Theorem 3. Let I be an up-directed set and {hα : Aα→ Bα | α ∈ I} be a directed family of regular monomorphisms. Then the directed colimit homomorphism induced by h : l im−→Aα → l im−→Bα is a regular monomorphism. Proof. Let (l im−→Aα, fα), (l im−→Bα, gα) be directed colimits of the directed systems ((Aα)α∈I , (ψαβ)α≤β) and ((Bα)α∈I , (ϕαβ)α≤β), respectively. Suppose {hα : Aα → Bα | α ∈ I} is a directed family of regular monomorphisms such that for every α ≤ β , fβψαβ = fα and gβϕαβ = gα. Then gβhβψαβ = gβϕαβhα = gαhα. Thus h = l im−→hα exists by the universal property of colimits. Consider l im−→Aα = ( ∐ α Aα)/ρ and l im−→Bα = ( ∐ α Bα)/ρ′. Let h[aα]ρ ≤ h[aβ]ρ. Then we have [hα(aα)]ρ′ = gαhα(aα) = hfα(aα) = h[aα]ρ ≤ H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 174 h[aβ]ρ = hfβ(aβ) = gβhβ(aβ) = [hβ(aβ)]ρ′ . Therefore, hα(aα)≤ρ′ hβ(aβ) and hence there exists γ ∈ I such that γ≥ α,β and ϕαγhα(aα) ≤ ϕβγhβ(aβ). This implies that hγψαγ(aα) ≤ hγψβγ(aβ). Now, since hγ is a reg- ular monomorphism by hypothesis, we get ψαγ(aα) ≤ ψβγ(aβ) which gives that aα ≤ρ aβ . Consequently, [aα]ρ ≤ [aβ]ρ and hence h is a regular monomorphism. Corollary 2. S-Pos hasM -directed colimits. Proof. Assume that (l im−→Bα, gα) is the directed colimit of the directed system ((Bα)α∈I , (ϕαβ)α≤β), and {hα : A→ Bα | α ∈ I} is a directed family of regular monomorphisms such that gβϕαβ = gα, for every α ≤ β . Let h : A→ l im−→Bα be the directed colimit of regular monomor- phisms hα : A→ Bα, α ∈ I . Recall that h= l im−→hα = gαhα = gβhβ = gγhγ = . . . . Now, take Σ =: {id : Aα→ Aγ | α ∈ I − {γ}, Aα = A= Aγ}. It is clear that Σ is a directed system and l im−→Aα = ∐ Aα. By Theorem 3, the induced directed colimit homomorphism h′ : ∐ Aα → l im−→Bα is a regular monomorphism. On the other hand, obviously the canonical map iα : Aα→ ∐ Aα is also a regular monomorphism. Hence, in view of Lemma 1 (i), h= h′iα : Aα = A→ l im−→Bα is a regular monomorphism. Definition 2. Let E be a class of morphisms of a category C . We say that C fulfills the E -chain condition if for every directed system ((Aα)α∈I , (ϕαβ)α≤β∈I) whose index set I is a well-ordered chain with the least element 0, and ϕ0α ∈ E for all α, there is a (so called “upper bound") family (gα : Aα→ A)α∈I with g0 ∈ E and gβϕαβ = gα. Proposition 7. S-Pos fulfills theM -chain condition. Proof. Take A= l im−→αAα and let gα : Aα→ A be the colimit maps. Then, applying Corollary 2, we get the result. 3. Comparison of Categorical Properties of Monomorphisms and Regular Monomorphisms In this section some category-theoretic notions relative to monomorphisms and regular monomorphisms in Pos and S-Pos is studied. We show that these morphisms have a different behaviour with some categorical properties. First notice that using similar arguments in proofs of results in Section 2 for the class of monomorphisms, some properties regarding composition closed, limits (products and pull- backs), colimits (coproducts and direct sums) and directed colimits hold in Pos and S-Pos. About factorization diagonalization property, note the following: H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 175 Remark 2. S-Pos does not generally have (E ,M ono)-factorization diagonalization property, where E andM ono are the classes of epimorphisms and monomorphisms, respectively. In fact, although each S-poset morphism f has a factorization as f = me, where m ∈M ono, e ∈ E (see the proof of Proposition 1), there exist e ∈ E and m ∈ M ono such that e is not vertical on m. To see this, for a pomonoid S, consider the S-posets A= 1 t 1 = {a, b}, B = 2 = ({a, b}, a≤ b), C = 2 t 1 = ({a, b, c}, a≤ b) and D = 3 = ({a, b, c}, a≤ b≤ c) with trivial actions of S. Take the inclusion maps e : A ,→ B and m : C ,→ D. Clearly, m ∈ M ono and e ∈ E . We show that e is not vertical on m. For this, define the morphisms u : A → C and v : B → D by u(a) = v(a) = b, u(b) = v(b) = c. Consider the following commutative diagram A u �� e // B v �� C m // D One can easily check that there is no morphism d : B→ C such that de = u. The following example shows that, in contrast to some categories such as S-Act, for a monoid S, pushouts do not transfer monomorphisms in S-Pos (see [4, Theorem 3.2(1)]). Example 1. For a pomonoid S, consider the S-posets A= 1t1= {a, b}, B = 2= ({a, b}, a≤ b) and C = 2 t 1 = ({a, b, c}, a≤ c) with trivial actions of S, and also the inclusion i : A ,→ B and the homomorphism j : A→ C given by j(a) = c, j(b) = a. The pushout of i and j is described as � Q = (B t C)/θ (H), qB = πiB, qC = πiC � where θ (H) is the S-poset congruence on B t C generated by H = {((1, a), (2, c)), ((1, b), (2, a))}, π : B t C →Q is the natural map, and iB, iC are the coproduct injections. We claim that [(2, a)] = [(2, c)]. First note that since a ≤ c in C, (2, a) ≤ (2, c) in B t C which implies that [(2, a)]≤ [(2, c)]. On the other hand, (2, c)≤H∪H−1 (2, a) because we have (2, c)≤ (2, c)H−1(1, a)≤ (1, b)H(2, a)≤ (2, a). This gives that [(2, c)]≤ [(2, a)]. Therefore, qC(a) = [(2, a)] = [(2, c)] = qC(c), which shows that qC is not a monomorphism. Injectivity and absolute retractness relative to a class of morphisms of a category are two category-theoretic close notions, which usually coincide under some conditions (cf. [4]). Re- call from [3] that in the category Pos of posets and order-preserving maps, these concepts with respect to regular monomorphisms (order-embeddings) are the same, which are exactly com- plete posets. Here we show that absolute retractness is actually a different notion to injectivity H. Rasouli / Eur. J. Pure Appl. Math, 7 (2014), 166-178 176 of posets relative to monomorphisms (one-one monotone maps). More precisely, there exists no non-trivial injective poset, but we show that absolute retract posets are exactly complete chains. Let us first recall some definitions. A poset P is called (regular) absolute retract if each (regular) monomorphism f : P →Q in Pos is a section. Notice that, using Zorn’s lemma, there exists a total order � on a poset (P,≤) which is compatible with the partial order ≤; this means a ≤ b implies that a � b, for every a, b ∈ P (see [14]). In the following result, all absolute retract posets is characterized. Theorem 4 (Characterization of absolute retract objects in Pos). Let P be a poset. Then P is absolute retract if and only if it is a complete chain. Proof. Assume that (P,≤) is an absolute retract poset. This clearly implies that P is regular absolute retract and then complete by [3, Proposition 1]. To show that P is a chain, consider a compatible total order � on P. Thus the morphism i : (P,≤)→ (P,�), mapping the elements of P identically, is a monomorphism and then has a left inverse by hypothesis. This obviously implies that P is a chain. For the converse, let P be a complete chain and f : P → Q be a monomorphism in Pos. Since P is a chain, f is order-embedding. On the other hand, since P is complete, it is regular absolute retract by [3, Proposition 1]. Consequently, f has a left inverse and hence P is absolute retract. In what follows, we study enough absolute retractness of posets. To this end, we recall some required notions. Let P be a poset. A poset E is said to be an extension of P if P is embedded into E. Also we say that E is a monomorphic extension of P if there exists a monomorphism from P to E. An extension E of P is called join dense if each element of E is the join of its predecessors in P, that is, e = ∨ {p ∈ P : p ≤ e}, for every e ∈ E. Meet density is defined dually. The Dedekind-MacNeille completion of P, de- noted by DM(P), is a complete extension of P which is both join and meet dense. Lemma 2. Let P be a chain. Then so is DM(P). Proof. Suppose P is a chain, and x , y ∈ DM(P), x � y . Then there exists an s ∈ P such that s ≤ x and s � y (join density) and hence also a t ∈ P such that t ≥ y and s � t (meet density). Since P is a chain, t ≤ s. Thus we get y ≤ t ≤ s ≤ x , showing that DM(P) is a chain. Finally, the following result is obtained: Proposition 8. Pos has enough absolute retracts: Each poset can be monomorphically extended to an absolute retract poset. Proof. Let P be a poset. 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