EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No. 4, 2008, (41-55) ISSN 1307-5543 – www.ejpam.com Sequentially pure monomorphisms of acts over semigroups H. Barzegar, Mohammad Mehdi Ebrahim∗,† Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran Abstract. Any notion of purity is normally defined in terms of solvability of some set of equations. In this paper we first take this point of view to introduce a kind of purity, called sequential purity, for acts over semigroups (which is of some interest to computer scientists, too), and then show that it is actually equivalent to C p-purity resulting from a closure operator. The main objective of the paper is to study properties of the category of all acts over a semigroup with respect to sequentially pure monomorphisms. These properties are usually needed to study the homological notions, such as injectivity, of acts. AMS subject classifications: 18A20, 18A30, 20M30, 20M50 Key words: Action of a semigroup, Purity 1. Introduction and Preliminaries To study mathematical notions in a categoryA with respect to a classM of its morphisms, one should know some of the categorical properties of the pair (A ,M ). One of the very useful categories in many branches of mathematics, as well as in computer sciences, is the category Act-S of sets with a right action of a semigroup S on them. In this paper we take A to be this category and Mp to be a particularly interesting class of monomorphisms, to be called sequentially pure, and investigate its categorical properties. First we give the following preliminaries needed in the sequel. 1.1. The category of acts over semigroups First recall the following, for example from [13] or [5]. Let S be a semigroup and A be a set. If we have a mapping µ : A× S→ A (a, s) 7−→ as := µ(a, s) such that a(st) = (as)t for a ∈ A, s, t ∈ S, we call A a (right) S-act or a (right) act over S. If S is a monoid with an identity 1, we usually also require that a1= a for a ∈ A. ∗Corresponding author. Email addresses: m-ebrahimi@sbu.ac.ir (M.M. Ebrahimi) †M.M. Ebrahimi would like to acknowledge the financial support from Iran National Science Foundation (INSF) with thanks. http://www.ejpam.com 41 c© 2008 EJPAM All rights reserved. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 42 In fact, an S-act is a universal algebra (A, (µs)s∈S) where each µs : A → A is a unary operation on A such that µs ◦µt = µst for each s, t ∈ S, and µ1 = idA if S has an identity 1. An element a of an S-act A is called a fixed or a zero element if as = a for all s ∈ S. Note that one can always adjoin a zero element to A and get an act A0 = A∪{0} with a zero element. Let A be an S-act and A′ ⊆ A. Then A′ is called a subact of A if a′s ∈ A′ for all s ∈ S and a′ ∈ A′. Note that the semigroup S can itself be regarded as an S-act with its multiplication as its action. A subact of the S-act S is called a right ideal of the semigroup S. A homomorphism (also called an equivariant map, or an S-map) from an S-act A to an S-act B is a function from A to B such that for each a ∈ A and s ∈ S, f (as) = f (a)s. Since idA and the composition of two equivariant maps are equivariant, we have the cate- gory Act-S of all (right) S-sets and S-maps between them. Recall that to every semigroup S without an identity one can adjoin an identity 1 by setting 1s = s = s1 for all s ∈ S and get an S-act denoted by S1. As a very interesting example, used in computer sciences as a convenient means of alge- braic specification of process algebras (see [7, 8]), consider the monoid (N∞, ·), where N is the set of natural numbers and N∞ = N ∪ {∞} with n < ∞,∀n ∈ N and m · n = min{m, n} for m, n ∈ N∞. Then an N∞-act is called a projection algebra or a projection space (see also [11,14]). 1.2. Some ingredients of the category Act-S In this subsection we give some categorical and algebraic ingredients of Act-S needed in the sequel. Since the class of S-acts is an equational class, the category Act-S is complete (has all products and equalizers). In fact, limits in this category are computed as in the category Set of sets and equipped with a natural action. In particular, the terminal object of Act-S is the singleton {0}, with the obvious S-action. Also, for S-acts A, B, their cartesian product A× B with the S-action defined by (a, b)s = (as, bs) is the product of A and B in Act-S. Recall that for a family {Aα : α ∈ I} of S-acts each with a unique fixed element 0, the direct sum ⊕ α∈I Aα is defined to be the subact of the product ∏ α∈I Aα consisting of all (aα)α∈I such that aα = 0 for all α ∈ I except a finite number of indices. The pullback of a given diagram A ↓ f C g → B in Act-S is the subact P = {(c, a) : c ∈ C , a ∈ A, g(c) = f (a)} of C × A, and pullback maps pC : P → C , pA : P → A are restrictions of projection maps. Notice that for the case where g is a monomorphism, P can be taken as (isomorphic to) f −1(C). All colimits in Act-S exist and are calculated as in Set with a natural action of S on them. In particular, ;with the empty action of S on it is the initial object of Act-S. Also, the coproduct of two S-acts A, B is their disjoint union At B = (A× {1})∪ (B× {2}) with the action of S on H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 43 At B defined by (a, 1)s = (as, 1), (b, 2)s = (bs, 2) for s ∈ S, a ∈ A, b ∈ B. The coproduct At {0} is denoted by A0. Definition 1.1. Let A be an S-act. An equivalence relation ρ on A is called an S-act congruence on A if aρa′ implies asρa′s for a, a′ ∈ A, s ∈ S. The right action making A/ρ an S-act is defined by [a]s = [as]. For H ⊆ A× A, the congruence generated by H, that is the smallest congruence on A con- taining H, is denoted by ρ(H). For a subset H of A× A let H e be the equivalence relation generated by H and H c = {(as, bs) : (a, b) ∈ H, s ∈ S} Then we have: Lemma 1.1. Let A be an act over a semigroup S and H ⊆ A× A. Then ρ(H) = (H ∪H c)e. Proof. It is enough to show that the equivalence relation (H ∪ H c)e is a congruence. Sup- pose that x(H ∪H c)e y and s ∈ S. Then x = y , and so xs = ys, or there exist z1, ..., zn ∈ A such that x = z1, y = zn and (zi , zi+1) ∈ (H ∪ H c) ∪ (H ∪ H c)−1. Then we also have (zis, zi+1s) ∈ (H ∪H c)∪ (H ∪H c)−1. Corollary 1.1. Let H ⊆ A × A and ρ = ρ(H). Then, for a, b ∈ A, one has aρb if and only if either a = b or there exist p1, p2, ..., pn, q1, q2, ..., qn ∈ A, s1, s2, ..., sn ∈ S1 where for i = 1, ..., n, (pi , qi) ∈ H ∪H−1, such that a = p1s1, q1s1 = p2s2, q2s2 = p3s3, ...,qnsn = b. The pushout of a given diagram A f → B g ↓ C in Act-S is the factor act Q = (BtC)/θ where θ = ρ(H) and H consists of all pairs (uB f (a), uC g(a)), a ∈ A, where uB : B→ BtC , uC : C → BtC are coproduct injections. Also, the pushout maps are given as q1 = πuC : C → (BtC)/θ , q2 = πuB : B→ (BtC)/θ , where π : BtC → (BtC)/θ is the canonical epimorphism. Multiple pushouts in Act-S are constructed analogously. Recall that a directed system of S-acts and S-maps is a family (Bα)α∈I of S-acts indexed by an updirected set I endowed by a family (gαβ : Bα → Bβ)α≤β∈I of S-maps such that given α ≤ β ≤ γ ∈ I we have gβγgαβ = gαγ, also gαα = id. Note that the direct limit (directed colimit) of a directed system ((Bα)α∈I , (gαβ)α≤β∈I) in Act-S is given as l im−→αBα = ∐ α Bα/ρ where the congruence ρ is given by bαρbβ if and only if there exists γ ≥ α,β such that uγgαγ(bα) = uγgβγ(bβ), in which each uα : Bα→ ∐ α Bα is an injection map of the coproduct. Notice that the family gα = πuα : Bα → l im−→αBα of S-maps satisfies gβ gαβ = gα for α ≤ β , where π : ∐ α Bα→ l im−→αBα is the natural S-map. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 44 The left and the right adjoints F and H, respectively, of the forgetful functor U: Act-S → Set exist and are defined as follows: The free functor F :Set → Act-S is defined by: F(X ) = X × S1 with the S-action given by (x , t)s = (x , ts), for t ∈ S1, s ∈ S, x ∈ X , and for every map f : X → Y in Set, F( f ) = f × id : X × S1→ Y × S1. The existence of free S-acts, in particular on the singleton set, shows that an S-map is a monomorphism if and only if it is one-one. Therefore, we do not distinguish between monomorphisms of acts and inclusions. The cofree functor H : Set → Act-S is defined by HX = X S1 , the set of all functions from S1 to the set X , with the action of S on X S1 given by ( f s)(t) = f (st) for f ∈ X S1 , s ∈ S, and t ∈ S1. Also, for a function h : X → Y , H(h) : X S1 → Y S1 is defined by (Hh)( f ) = hf for f ∈ X S1 . Since a left adjoint preserves colimits, the functor U preserves epimorphisms. So, epimor- phisms in Act-S are exactly onto S-maps. 2. Sequentially Pure Monomorphisms Any notion of pure monomorphisms is normally defined in terms of solvability of some set of equations. In the following we first consider this point of view to define a kind of pure monomorphisms, which is also of interest to computer scientists, which we are going to study their behaviour in this paper, and then show that they are actually equivalent to C p-pure monomorphisms resulting from a closure operator on the category Act-S. 2.1. Sequentially pure monomorphisms In [10, 14, 16], it is shown that the equations with constants from an S-act A are of one the following three types: xs = y t, xs = x t, xs = a where s, t ∈ S, a ∈ A. Here we are concerned with the equations of the type xs = a only. Gould in [10] defines an α-system of equations on an S-act A to be Σ = {xs j = a j : j ∈ J , |J |< α, s j ∈ S, a j ∈ A} in which si = s j need not imply ai = a j . But, note that if for any s ∈ S there exist two equations of the form xs = a1, xs = a2 in Σ and Σ has a solution b in some extension B of A, then a1 = a2. So, for a system Σ of equations to have a solution there can only be at most one equation xs = a in Σ for each s ∈ S. Therefore, Σ should actually be taken of the form ΣT = {xs = at : t ∈ T, at ∈ A} for some T ⊆ S. Hence, for any fixed T ⊆ S, there is a one to one correspondence between the set of all systems of equations of the above form on an S-act A and the set of all functions k : T → A. In fact, to each system of equations ΣT we get the function kΣ : T → A given by k(t) = at and conversely, for any function k : T → A one has the system of equations Σk = {x t = k(t) | t ∈ T} on A. Thus we have the following definition. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 45 Definition 2.1. (1) Let T be a subset of S and A be an S-act. Any ΣT = {xs = at : t ∈ T, at ∈ A} (or, equivalently, a map k : T → A ) will be called a T -system of equations (or a T -sequence) on A. (2) We say that a system ΣT (or k : T → A) is solvable in an extension B of A if there is some b ∈ B such that bt = at (or k(t) = bt) for all t ∈ T ; that is, k = λb, where λb(t) = bt. (3) We say that a system ΣT (or k : T → A) is consistent if it has a solution in some extension B of A. Now, we are going to show that we should actually only consider just I -sequences for an ideal I of S rather than any T -sequence for any subset T of S. Note that, although an I -sequence k : I → A is just a function and not necessarily a homomorphism, we have the following: Theorem 2.1. Let I be an ideal of S and A be an S-act. Then the following are equivalent for an I-sequence k : I → A: (1) k : I → A is a homomorphism. (2) k : I → A is a consistent map. (3) The system Σ = {xs = k(s)| s ∈ I} is a consistent system. Proof. (1)⇒(2) Let k : I → A be a homomorphism. Let E be an injective S-act containing A (see [13]). Considering the extension I1 of I , k can be lifted to k̄ : I1→ E. Now it is easily seen that k = λx where x = k(1) ∈ E, and so k is consistent. (2)⇒(1) Let k be a consistent map. So, there exist an extension B of A and b ∈ B such that k = λb which is in fact a homomorphism. The equivalence of (3) and (2) follows just from the definition. Corollary 2.1. A T-system ΣT = {x t = at | t ∈ T} (or a T-sequence k : T → A) is consistent if and only if k̂ : TS1 → A defined by k̂(ts) = ats for t ∈ T, s ∈ S1 is a “well defined" equivariant map. Proof. Clearly ΣT = {x t = at | t ∈ T} is consistent if and only if Σ1 = {x(ts) = ats| t ∈ T, s ∈ S1} is consistent. Now, since TS1 is a right ideal of S, the latter is true by the above theorem if and only if k̂ : TS1→ A with k̂(ts) = ats is a homomorphism. The above corollary shows that, as long as consistent systems of equations are concerned, we may as well consider only equivariant maps k : I → A from a right ideal I of S to A rather than functions from an arbitrary set T to A. In particular, we take I to be S itself and have the following: Definition 2.2. Let A be a subact of B. We say that A is sequentially pure or s-pure in B if any one of the following equivalent conditions hold: (1) Every Σ = {xs = as : s ∈ S, as ∈ A} is solvable in A whenever it is solvable in B. (2) For every b ∈ B with bS ⊆ A there is an element a ∈ A such that λb = λa; in the sense that bs = as for each s ∈ S. (3) Every homomorphism k : S→ A is of the form λa for some a ∈ A whenever it is of the form λb for some b ∈ B. A monomorphism f : A→ B is said to be s-pure if f (A) is s-pure in B. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 46 The following is a simple fact which will be used later: Lemma 2.1. For every S-acts A and B, we have: (1) A is s-pure in At B. (2) A is s-pure in A0. Remark 2.1. Note that one may say that a subact A of an act B is finitely pure in B if for every finite subset T ⊆ S, the system ΣT with constants from A has a solution in A whenever it has a solution in B. Clearly, if S is finitely generated, say by T , then by Corollary 2.1, finite (or even T -) purity implies s-purity. But, the converse is not true: Consider the semigroup S = {1, a, b} in which 1 is a left identity and a, b are zero elements. Then A = {a, b} is an s-pure subact of S, but the finite system Σ = {xa = a, x b = b} having solution 1 in S does not have any solution in A. 2.2. Sequential purity versus C p-purity In this subsection we introduce a closure operator which is closely related to sequential purity defined above. We are not going to fully investigate the properties of this closure operator as in [4]. First recall the following definition of a categorical closure operator from [3]. Denoting the lattice of all subacts of an S-act B by Sub(B), we have: Definition 2.3. A family C = (CB)B∈Act−S, with CB : Sub(B) → Sub(B), taking any subact A≤ B to a subact CB(A) (or C(A), if no confusion arises) is called a closure operator on Act-S if it satisfies the following: (c1) (Extension) A≤ C(A), (c2) (Monotonicity) A1 ≤ A2 ≤ B implies C(A1)≤ C(A2), (c3) (Continuity) f (CB(A))≤ CC( f (A)) for all morphisms f : B→ C . Now, one has the usual two classes of monomorphisms related to any closure operator as follows: Definition 2.4. Let A ≤ B be in Act-S. We say that A is C-closed in B if C(A) = A, and it is C-dense in B if C(A) = B. Also, an S-map f : A→ B is said to be C-dense (C-closed) if f (A) is a C-dense (C-closed) subact of B. Now we recall the following closure operator needed in the sequel and has been studied in [4] and used in [6,9,14] to study a kind of injectivity. Definition 2.5. For any subact A of an S-act B, define a closure operatorCd by Cd(A) = {b ∈ B : bS ⊆ A} Now, note that A is Cd -dense (or simply s-dense) in an extension B of A if Cd(A) = B, that is, for every b ∈ B, bS ⊆ A. Notice that in the case where S is a monoid, Cd(A) = A for every A ≤ B. So, it is more interesting to consider the closure operator Cd only for semigroups, or for semigroup part S of monoids of the form T = S1. We now introduce and study another closure operator on Act-S which will be shown to be closely related to sequential purity. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 47 Definition 2.6. The sequential pure closure operator C p on Act-S is defined as C p(A) = {b ∈ B : ∃a ∈ A,λb = λa} where λx : S→ A is defined by λx(s) = xs. Now, note that A is C p-dense in an extension B of A if C p(A) = B (this means that for every b ∈ B there is an a ∈ A with λb = λa; that is, bs = as for every s ∈ S). And A is C p-closed in B if C p(A) = A (that is, for every b ∈ B− A and a ∈ A there is an s ∈ S with bs 6= as). Notice that C p-closedness is preserved by inverse image of S-maps and C p-denseness is preserved by images of onto S-maps. Some easily proved properties of this last closure operator is stated in the following: Lemma 2.2. C p is: (1) a closure operator, (2) idempotent, (3) hereditary; for C ≤ A ≤ B, C p B (A) = C p B (C) ∩ A, (4) weakly hereditary; every A ≤ B is C p-dense in C p B (A), (5) grounded; C p(;) = ;, (6) additive; C p(A∪C) = C p(A)∪C p(C), (7) fully additive; C p( ⋃ i∈I Ai) = ⋃ i∈I C p(Ai), (8) C p A ( ⋂ Ai) ⊆ ⋂ C p A (Ai), (9) productive; for every family of subacts Ai of Bi , taking A= ∏ i Ai and B = ∏ i Bi , C p B (A) = ∏ i C p Bi (Ai). And, some of the properties that C p does not satisfy in general are: Lemma 2.3. For any semigroup S, C p is not: (1) discrete; C p B (A) = A for every S-act B and every A ≤ A, (2) trivial; C p B (A) = B for every B and every A ≤ B, (3) minimal; for C ≤ A ≤ B, C p B (A) = A∪ C p B (C). Proof. Let 0 ∈ A be a fixed element of A, and adjoin two elements θ ,ω to A with actions ωs =ω and θ s = 0. Then C p B (A) = A∪{θ} where B = A∪ {θ ,ω}. Hence C p is neither discrete nor trivial. Also, it is not minimal. Because, adjoining two elements θ ,ω to an S-act C with actions ωs = θ and θ s = θ , and taking A= C ∪ {θ}, B = C ∪ {θ ,ω}, we get C ⊂ A⊂ B, and C p B (A) = B while C p B (C) = C . Another monomorphism which corresponds to this closure operator, and is of main interest in this paper, is defined as follows: Definition 2.7. An S-act A is said to be C p-pure in an extension B of A if C p(A) = Cd(A). Remark 2.2. Let Ai be a family of subacts of A, (1) If ⋂ Ai is s-pure in A then C p A ( ⋂ Ai) = ⋂ C P A (Ai). (2) If for every i ∈ I , Ai is s-pure in A and C p A ( ⋂ Ai) = ⋂ C p A (Ai) then ⋂ Ai is s-pure in A. Note 2.2. For A ≤ B, we have A ≤ C p(A) ≤ Cd(A) ≤ B. So, if A is C p-dense in B, then C p(A) = Cd(A) = B and so A is Cd -dense as well as C p-pure. Similarly, if A is Cd -closed in B, then A= C p(A) = Cd(A) and hence A is C p-closed as well as C p-pure. The following result, the proof of which is straightforward, is what we promised in the beginning of this section. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 48 Theorem 2.3. The following are equivalent: (1) A is s-pure in B. (2) A is C p-pure in B. Lemma 2.4. (1) Any retraction is s-pure. (2) Any C p-dense monomorphism is a retraction. Proof. : (1) Let A ,→ B π→ A= idA be a retraction and S k→ A ,→ B = λb for some b ∈ B. It is clear that k = λπ(b). (2) Let A ,→ B be a C p-dense subact. Then, by ?? it is C p-pure as well as Cd -dense subact. So, for every b ∈ B there exists ab ∈ A such that λb = λab . Now, for every b ∈ B − A choose and fix such an ab ∈ A. Define π : B→ A by π(x) = ¨ x , if x ∈ A ax , if x 6∈ A Then, clearly π is a retraction. It is a homomorphism because it is a homomorphism on A, and for x ∈ B− A, s ∈ S, we have xs ∈ A and so π(xs) = xs = ax s = π(x)s. 3. Categorical Properties of s-Pure Monomorphisms In this section we investigate the categorical and algebraic properties, regarding compo- sition, limits, and colimits, of the category Act-S with respect to the classMp of sequentially pure monomorphisms. We have divided the section into three subsections as follows: 3.1. Composition properties of s-pure monomorphisms In this subsection we investigate some properties of the classMp which are mostly related to the composition of pure monomorphisms. These properties and the ones given in the next two subsections are what normally used to study injectivity with respect to a class of monomorphisms (see [1,17]) Lemma 3.1. The classMp is: (1) Isomorphism closed; that is, contains all isomorphisms and is closed under composition with isomorphisms. (2) Composition closed; that is, if f : A→ B and g : B → C belong to Mp, then g f also belongs toMp. (3) Left cancellable; that is, if g f ∈Mp then f ∈Mp. Proof. We just prove (2), which may be less clear. For convenience and without loss of generality, we consider f and g to be s-pure inclusions. Let S k→ A f ,→ B g ,→ C = λc , for some c ∈ C . Since g is s-pure, there is an element b ∈ B such that f k = λb. Now, the s-purity of f provides an element a ∈ A with k = λa. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 49 Theorem 3.1. The following are equivalent: (1) S has a left identity. (2) Every monomorphism is s-pure. (3) S is s-pure in S1. (4) Mp is right cancellable; that is, for monomorphisms f and g, if g f is s-pure then g is s-pure. (5) For morphisms f and g, if f is an s-pure monomorphism, g is an epimorphism, and g f is a monomorphism, then g f is s-pure. CATEGORICAL PROPERTIES OF Proof. (1)⇒(2,3,4,5): Clearly if S has a left identity, then every monomorphism is s-pure. So, (1) implies (2), (3), (4), and (5). (2)⇒(3) is clear. (3)⇒(1) Since 1 ∈ Cd S1(S), there exists e ∈ S with λe = λ1. This shows that e is a left identity of S. (4)⇒(3) Use the fact that the empty set is s-pure in every right S-act, and apply (3) to ; f ,→ S g ,→ S1. (5)⇒(3) Consider the natural homomorphisms S τ→ S t S1 π→ S1. By Lemma 2.1, τ is s-pure. Since π is an epimorphism and πτ (the inclusion map) is one-one, by (5), πτ is s-pure. As the above theorem shows, Mp is not generally right cancellable. But for some semi- groups, regardless of having a left identity, some special monomorphisms may be cancelled from the right. See the following: Lemma 3.2. If S2 = S, f : A ,→ B and g : B ,→ C are monomorphisms, f is s-dense, and g f is s-pure, then g is s-pure. Proof. Without loss of generality, we again assume that f and g are inclusions. Let c ∈ C be such that cS ⊆ B. So, since f is s-dense, we get (cS)S ⊆ A. Now, since S2 = S and g f is s-pure, we get an a ∈ A⊆ B with λc = λa, which proves that g is s-pure. Definition 3.1. Let E be a class of homomorphisms. We say that Act-S has (E ,Mp) diagonal- ization property if for any commutative diagram A e−→ B f ↓ ↓ g C m−→ D with e ∈ E and m ∈Mp there exists a unique diagonal S-map d : B→ C such that de = f and md = g. Proposition 3.1. Act-S has (E ,Mp) diagonalization property, for E the class of all epimor- phisms. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 50 Proof. Consider the diagram given in the above definition. First, we see that Ker e ⊆ Ker f . Let e(a) = e(a′) and so ge(a) = ge(a′). Thus, mf (a) = mf (a′), and so f (a) = f (a′), since m is a monomorphism. Then, by The Decomposition Theorem (which holds since S-acts form an equational class), there exists a unique S-map d : B→ C with de = f (given by d(b) = f (a), where e(a) = b). It is also easily seen that md = g. Recall that a category is said to have unique (E ,M ) factorization property if every mor- phism f can be uniquely represented as f = me with e ∈ E and m ∈ M , where E , M are some classes of morphisms. Remark 3.1. Act-S does not generally have unique (E ,Mp) factorization property, where E is the class of all epimorphisms. To see this, let S be a semigroup that does not have a left iden- tity. Then, by Theorem 3.1, S is not s-pure in S1. On the contrary, let the inclusion morphism τ : S → S1 have an (E ,Mp)-factorization S e→ A m→ S1. Since me is a monomorphism, e is a monomorphism and hence an isomorphism. Thus, τ= m is s-pure which is a contradiction. 3.2. Limits of s-pure monomorphisms In this subsection some of the categorical properties of s-pure monomorphisms related to limits are studied. The proof of the following is straightforward. Proposition 3.2. (1)Mp is closed under products. (2) Let { fα : A → Bα|α ∈ I} be a family of s-pure monomorphisms. Then their product homomorphism h : A→ ∏ α∈I Bα is also an s-pure monomorphism. Note that the above result (2) is also true whenever for some (not necessarily all) α ∈ I , fα is an s-pure monomorphism. Lemma 3.3. In Act-S, pullbacks transfer s-pure monomorphisms if and only if S has a left identity. Proof. Necessity: By Theorem 3.1, it is enough to show that S is s-pure in S1. Let E be an injective S-act and 0 be a zero element of E (see [13], Lemma III.1.7). Adjoin an element θ to E and define θ s = 0 for all s ∈ S. Then, E is clearly s-pure in Eθ = E ∪ {θ}. Taking a homomorphism f : S1 −→ Eθ given by f (s) = θ s (s ∈ S1) we have the pullback diagram: S τ−→ S1 f ↓ ↓ f E τ′−→ Eθ where τ, τ′ are inclusion morphisms. Then, since τ′ is s-pure, we get that S is s-pure in S1, by the hypothesis. Sufficiency: Let S have a left identity. In this case, by Theorem 3.1, every monomorphism is s-pure and pullbacks clearly preserve monomorphisms. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 51 Proposition 3.3. Let { fα : A→ Bα|α ∈ I} be a source of s-pure monomorphisms. Then the ho- momorphism f : A→ l im←−Bα (existing by the universal property of limits) is an s-pure monomor- phism. Proof. It is clear that f is one-one, because so is every fα. Also, if k : S→ A is an S-map with f k = λx for some x ∈ l im←−Bα, then for every α fαk = πα f k = λπα(x), where πα : l im←−Bα→ Bα is a limit morphism. So, k = λa for some a ∈ A, since fα is s-pure. Note that, the above result is also true whenever for some (not necessarily all) α ∈ I , fα is an s-pure monomorphism. 3.3. Colimits of s-pure monomorphisms In this subsection we investigate the colimit properties of s-pure monomorphisms. Proposition 3.4. The classMp is closed under coproducts. Proof. Let { fα : Aα → Bα|α ∈ I} be a family of s-pure monomorphisms and f : ∐ Aα → ∐ Bα be the coproduct (mono)morphism induced by all fα. Let k : S→ ∐ Aα be a homomor- phism such that f k = λb for some b ∈ ∐ Bα. Since b ∈ Bα for some α ∈ I , k(S) ⊆ Bα, and so k(S)⊆ Aα. Since Aα is s-pure in Bα, k = λaα for some aα ∈ Aα, which proves the result. In the following proposition, suppose that every Aα has a fixed element 0. Proposition 3.5. (1) The classMp is closed under direct sums. (2) If S is a finitely generated semigroup, then ⊕α∈IAα is s-pure in ∏ α∈I Aα. Proof. (1) Let { fα : Aα → Bα|α ∈ I} be a family of s-pure monomorphisms, and f : ⊕Aα → ⊕Bα be the monomorphism induced by the product of fα, s. Let k : S → ⊕Aα be a homomorphism such that k = λb for some b = (bα)α∈I ∈ ⊕Bα. Let J be a finite subset of I such that for all α 6∈ J , bα = 0. So, for every β ∈ J , fβ pβk = λbβ , where pβ : ∏ Aβ → Aβ is the projection map. Since each fβ is s-pure, there exists aβ ∈ Aβ such that pβk = λaβ . Thus, k = λ(aα)α∈I , where for all α /∈ J , aα = 0. (2) Let k : S →⊕Aα be a homomorphism with k = λa for some a = (aα)α∈I ∈ ∏ Aα, and S = ∪n i=1 t iS 1. So, since k(t i) ∈ ⊕Aα, at i = (aα t i)α∈I ∈ ⊕Aα. Thus, for every i there exists a finite subset Ji of I such that for every α 6∈ Ji , aα t i = 0. Now considering the finite subset J = ⋃n i=1 Ji ⊆ I and bα = ¨ aα, if α ∈ J 0, if α 6∈ J , it is clear that k = λb for b = (bα)α∈I . Theorem 3.2. For the following pushout diagram in Act-S, we have: (1) If f is a monomorphism then h is a monomorphism. H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 52 (2) If f is s-pure then h is s-pure. A f −→ B g ↓ ↓ h′ C h−→ Q Proof. (1) Recall that Q = (B t C)/θ where θ = ρ(H) and H consists of all pairs (uB f (a), uC g(a)), a ∈ A, where uB : B → B t C , uC : C → B t C are coproduct injections. And h = πuC : C → (B t C)/θ , h′ = πuB : B → (B t C)/θ , where π : B t C → (B t C)/θ is the canonical epimorphism. Let h(c) = h(c′), c, c′ ∈ C , and so uC(c)ρ(H)uC(c′). Thus c = c′, and the result is proved, or there exist a1, a2, ...an ∈ A, s1, s2, ..., sn ∈ S1 such that c = g(a1s1), g(ansn) = c′, and f (a1s1) = f (a2s2) · · · f (an−1sn−1) = f (ansn) g(a2s2) = g(a3s3) · · · and then, the fact that f is a monomorphism gives a1s1 = a2s2, a3s3 = a4s4, ..., an−1sn−1 = ansn. Thus, we get g(a1s1) = g(a2s2) = g(a3s3) = · · ·= g(ansn) and hence c = c′. (2) Let k : S → C be a homomorphism such that hk = λx for some x ∈ Q. Two cases may occur: (i) there exists c ∈ C such that x = [uC(c)]ρ(H). Therefore, hk = λx = λh(c) and hence k = λc , since h is one-one. (ii) there exists b ∈ B such that x = [uB(b)]ρ(H). For every s ∈ S, [uC(k(s))]ρ(H) = [uB(bs)]ρ(H), and so there exist elements as 1, · · · , as n in A such that uB(bs) = uB f (as 1), uC g(as 1) = uC g(as 2), uB f (as 2) = uB f (as 3), · · · , uC g(as n) = uC(k(s)). Then, since f is a monomorphism, as 2 = as 3, as 4 = as 5, · · · , and hence uC g(as 1) = uC g(as 2) = uC g(as 3) = · · · = uC g(sn) = uC(k(s)). Now, for all s ∈ S, bs = f (as 1) ∈ f (A), and thus there exists k1 : S→ A with k1(s) = as 1, and so f k1 = λb. Then, since f is s-pure, there is an element a ∈ A such that k1 = λa. Therefore, f (as 1) = f (k1(s)) = f (as) and so hg(as 1) = h′ f (as 1) = h′ f (as) = hg(as) which, since h is a monomorphism, yields k(s) = g(as 1) = g(as) = g(a)s, that is k = λg(a), and h is s-pure. Theorem 3.3. Let I be a directed set which has a maximal element γ and {hα : Aα→ Bα| α ∈ I} be a directed family of s-pure monomorphisms. Then, the directed colimit homomorphism induced by h : l im−→Aα→ l im−→Bα is s-pure . Proof. Let (l im−→Aα, fα), (l im−→Bα, gα) be direct limits of the directed systems ((Aα), (ψαβ))α≤β∈I and ((Bα), (ϕαβ))α≤β∈I and suppose {hα : Aα → Bα| α ∈ I} is a directed family of s-pure monomorphisms such that for every α ≤ β , fβψαβ = fα and gβϕαβ = gα. Then, for every α ≤ β , gβhβψαβ = gβϕαβhα = gαhα, so h = l im−→hα exists by the universal prop- erty of colimits. Consider l im−→Aα = Aα/ρ and l im−→Bα = Bα/ρ ′ as defined in section 1. Let H. Barzegar and M.M. Ebrahimi / Eur. J. Pure Appl. Math, 1 (2008), (41-55) 53 h[aα]ρ = h[aβ]ρ. Then, [hα(aα)]ρ′ = gαhα(aα) = gβhβ(aβ) = [hβ(aβ)]ρ′ , and so there ex- ists γ ∈ I with γ ≥ α,β and ϕαγhα(aα) = ϕβγhβ(aβ) which implies that [aα]ρ = [aβ]ρ, and so h is a monomorphism. Now, let k : S → l im−→Aα be a homomorphism such that hk = λ[bα]ρ′ and for s ∈ S, k(s) = [aαs ]ρ, aαs ∈ A. Notice that γ ≥ α,αs, for s ∈ S, and define k1 : S → Aγ by k1(s) = ψαsγ (aαs ). Then, since fγk1(s) = fγψαsγ (aαs ) = fαs (aαs ) = [aαs ]ρ = k(s) and fγk1(st) = k(st) = k(s)t = fγk1(s)t, it follows that k1 is a homomorphism, since fγ is a monomorphism. Now, for every s ∈ S, gγhγk1(s) = hfγk1(s) = hk(s) = [bα]s = gα(bα)s = gγϕαγ(bαs) and then, since gγ is a monomorphism, we have hγk1 = λϕαγ(bα). But, hγ is s-pure and so k1 = λa for some a ∈ Aγ. Hence k = λ fγ(a), because fγk1 = k. Corollary 3.1. Let I be a directed set which has a maximal element γ and {hα : A→ Bα|α ∈ I} be a directed family of s-pure monomorphisms. Then, the directed limit (colimit) of hα,s is s-pure. Proof. Let h : A→ l im−→αBα be a direct limit in Act-S of s-pure monomorphisms hα : A→ Bα, α ∈ I , and consider gα : Bα → l im−→αBα as in the definition. Recall that h = l im−→αhα = gγhγ = gαhα = gβhβ = . . . . It is clear that h : A→ l im−→αBα is a directed colimit of the directed family {hα : id : Aα → Aγ | α ∈ I − {γ}, Aα = A}. Then, apply the above theorem to complete the proof. Another condition which gives the above result is finitely generatedness of semigroup: Theorem 3.4. Let I be a directed set and S be a finitely generated semigroup. Then, the category Act-S hasMp-directed colimits. Proof. Let h : A→ l im−→αBα be a direct limit in Act-S of s-pure monomorphisms hα : A→ Bα, α ∈ I , and consider gα : Bα → l im−→αBα as in the definition. Recall that h = l im−→αhα = gγhγ = gαhα = gβhβ = . . . . Let S = ∪n i=1 t iS 1 and k : S→ A be an S-map such that hk = λ[bα]. Then, for every 1 ≤ i ≤ n, [hαk(t i)] = [bα t i], and so there exist γi ∈ I such that ψαγi (hαk(t i)) = ψαγi (bα t i). Now, let γ≥ max{γ1, · · · ,γn} and so for every s ∈ S, hγ(k(s)) =ψαγ(bα)s. Then, since hγ is s-pure, k = λa for some a ∈ A. We say that mul tiple pushouts transfer s-pure monomorphisms if in multiple pushout (P, Aα hα→ P) of a family of s-pure monomorphisms { fα : A→ Aα|α ∈ I}, every hα, α ∈ I , is an s-pure monomorphism. Theorem 3.5. Multiple pushouts transfer s-pure monomorphisms. Proof. Let (P, Aα hα→ P) be the multiple pushout of the family { fα : A→ Aα|α ∈ I} of s-pure monomorphisms. We know that P = ∐ Aα/ρ(H) where H = {( fα(a), fβ(a)) | a ∈ A,α,β ∈ I} (we have taken the image of each element Aα under coproduct morphisms equal to itself). Let hα(a) = hα(a′), a, a′ ∈ Aα, and so there exist p1, p2, ..., pn, q1, q2, ..., qn ∈ A, s1, s2, ..., sn ∈ REFERENCES 54 S1 where for i = 1, ..., n, (pi , qi) ∈ H ∪ H−1 and such that a = p1s1, q1s1 = p2s2, q2s2 = p3s3, ...,qnsn = a′. Then, a = fα(a1)s1 and there exists β ∈ I such that fβ(a1)s1 = fβ(a2)s2. Then, since fβ is a monomorphism, a1s1 = a2s2. Continuing this process we get that a1s1 = a2s2 = ... = ansn, and therefore a = a′. 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