EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 2, 2018, 431-443 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On M-Principally Injective and Projective S-acts Javed Hussain1,∗, Muhammad Shabir2 1 Department of Mathematics, Sukkur IBA University, Sukkur, Pakistan 2 Department of Mathematics, Quaid-i-azam University, Islamabad, Pakistan Abstract. In this paper, we have introduced the notions of M-cyclic S-acts, M-principally pro- jective and injective S-acts, semi-projective S-acts and co-cyclic S-acts, whereM is a right S-act. Several interesting properties, characterizations relations between newly defined structures have been investigated. 2010 Mathematics Subject Classifications: 20M30, 08A60, 08B30 Key Words and Phrases: Semigroups, S-acts, Endomorphism Monoids. 1. Introduction Group action has played a very significant role in the development of the theory of groups. Similarly representation of the Semi group S by transformation of a set, i.e. acts plays an essential role in Semigroup theory from the beginning (as can be seen from the title of A. K. Suschkewitsch’s dissertation ”The action of generalized group theory” (in Russian), 1992). A representation of a semigroup S by a transformation of a set defines an S-act just as the representation of a ring R by endomorphisms of an Abelian group defines an R-module. Probably first time, the definition of S-acts appeared in two papers of the H. J. Hoehnke with a different name in connection with the consideration of radicals of a semigroup. Acts over the semigroups appeared and were used in various applications like algebra automata theory, mathematical linguistics, graph theory, system theory, information theory, the theory of communications and electronic circuits, databases and other fields of theoretical computer science. In Ring theory, nowadays it is impossible to imagine the main direc- tions without homological methods using categories of modules. Similarly, it is important for monoids to consider an associated category of S-acts Projective and injective S-acts, injective envelopes, projective covers were mainly developed by P. Berthiaume [11], C. S. Johnson & F. Mc. Morris [3], B.M Schein [10]. The monoids over which all S-acts are (injective) projective were investigated by L.Skronjakov. The most notable difference be- tween the theory of S-acts and the theory of modules was discovered by T .G Mustfin, J. ∗Corresponding author. Email addresses: javed.brohi@iba-suk.edu.pk (J. Hussain), mshabirbhatti@yahoo.co.uk (M. Shabir) http://www.ejpam.com 431 c© 2018 EJPAM All rights reserved. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 432 Fountain and V. Gould, that there exist S-acts which have unstable theories whereas, all complete theories of modules are stable. Principally injective S-acts were first considered by J. Luedeman [8], F. Mc. Morris , and S. K. Sim [13], P. Berthiaume [11], introduced the notion of weak injectivity. M. Shabir and J. Ahsan in [2] have characterized the monoids by P -injective S-acts and a normal system. The novelity of this work is not only that new sturctures has been defined and several of their nontrivial and interesting properties has been studied but also that several of connections of our defined structures has been eastablsihed with existing sturctures like injective, projective, Co-hereditary S-acts and PP-semigroups. For example see Theorems 9, 10, 11, 14 and Corollary 4. In this paper, we have introduced the notions of M-cyclic S-acts, M-principally projec- tive and injective S-acts and semi-projective S-acts, whereM is a right S-act. The paper consists of 4 sections. We are giving a section-wise description of the paper. First sec- tion is running introduction. In section 2, we have defined M-principally injective S-acts and established the connection between the M-cyclic sub-acts and the ideals of endo- morphism monoid of a quasi-principally injective S-act M. Further a relation between the kernel congruence induced by endomorphisms on quasi-principally injective S-act M and the ideals of its endomorphism monoid. After that, some of the properties of M- principally injective S-acts have also been investigated. In section 2, we have defined M-principally projective S-acts and investigated some interesting results and connection with M-principally injective S-acts. A note on Co-hereditary S-acts is included at the end of the section. In section 4, Semi-projective S-acts have been defined and connections with its endomorphism monoid are investigated and significant results are proved. 2. M-principally injective S-acts and their properties in terms of M-cyclic sub-acts Throughout the paper S will denote the semigroup with fixed element θ and E will denote the endomorphism monoid of a right S-act M, i.e. E = EndS (M) . Moreover, by ker γ we mean the usual kernel congruence on M induced by γ ∈ E. Definition 1. Let N be a sub-act of a right S-act M. Then N is called an M-cyclic sub-act of M if N ∼=M/ρ for some congruence ρ on M. Lemma 1. Let M be a right S-act and N be a sub-act of M. Then the following are equivalent. 1) N ∼=M/ρ. 2) N = α (M), for some α ∈ E. Proof. 1)⇒ 2) Let π :M→M/ρ be a natural epimorphism and γ :M/ρ→ N be an isomorphism then it follows that γπ :M→N is an epimorphism and γπ (M) = N . 2)⇒1) Suppose N = α (M) for some α ∈ E. Let K = kerα. Define Φ : N = α (M)→ M/K by Φ(α(m)) = [m]K . Indeed Φ is an S-isomorphism. Thus by Lemma (1) we conclude that α (M), where α ∈ E, can also be used as an alternate notion for M-cyclic sub-act of M. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 433 Definition 2. Let M be a right S-act. A right S-act N is called M-principally in- jective if every S-homomorphism from an M-cyclic sub-act to N can be extended to an S-homomorphism from M to N . Definition 3. A right S-actM is called quasi-principally injective if it isM-principally injective. Definition 4. A semigroup S with a fixed element θ is called self principally injective if S is S-principally injective. Theorem 1. Let M be a quasi-principally injective S-act and α, β ∈ E. 1) If α (M) can be embedded into β (M) then Eα is the E-homomorphic image of Eβ. 2) If β (M) is the E-homomorphic image of α (M) then Eβ can be embedded into Eα. 3) If α (M) ∼= β (M) then Eα ∼= Eβ. Here Eα = {uα : u ∈ E} and Eβ = {uβ : u ∈ E}. Proof. 1) Suppose f : α (M)→ β (M) is an S-monomorphism. Consider the inclusions i : α (M) → M, j : β (M) → M and maps α′ : M → α (M), β′ : M → β (M) such that α = iα ′ and β = jβ ′ . Now since, M is quasi-principally injective so there exists an S-homomorphism g : M → M such that, jf = gi, where g extends f . Moreover, since g(α(m)) = β(m′) ∈ β (M), for some m′ ∈M, therefore gα (M) ⊆ β (M). Define Φ : Eβ → Eα by Φ(uβ) = ugα. We claim that Φ is E-epimorphism. In order to see that Φ is well-defined, take uβ, vβ ∈ Eβ such that uβ = vβ. Suppose contrary that Φ(uβ) 6= Φ(vβ) i.e. ugα 6= vgα, so there exists m ∈ M such that ugα(m) 6= vgα(m), but g(m) = β(m′) for some m′ ∈ M. Thus uβ(m′) 6= vβ(m′) this leads to a contradiction. Now for any uβ ∈ Eβ, Φ(γ(uβ)) = Φ((γu)β) = (γu)gα = γ(ugα) = γΦ(uβ), for all γ ∈ E. So Φ is an E-homomorphism. To see surjectivity of Φ, let uα ∈ Eα, where u ∈ M. Consider ui : α (M) →M, since M is quasi-principally injective so there exists γ :M→M such that γjf = ui. Consider Φ(γβ) = γgα = γgiα′ = γjfα′ = uiα′ = uα. Thus Φ : Eβ → Eα is an E-epimorphism. 2) Let us keep the same notation as in part 1) of proof. Let f : α (M) → β (M) be an S-epimorphism and we also have jf = gi. Define Φ : Eβ → Eα by Φ(uβ) = ugα. The map is well-defined and an E-homomorphism as is in 1). Moreover, β (M) = gα (M) . To show injectivity of Φ, we let uβ, vβ ∈ Eβ such that uβ 6= vβ so uβ(m′) 6= vβ(m′) for some m′ ∈ M. Now gα(m) = β(m′) for some m ∈ M, which implies that ugα(m) 6= vgα(m) and therefore ugα 6= vgα hence Φ(uβ) 6= Φ(vβ). 3) A direct consequence of 1) and 2). An immediate corollary of last theorem is following. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 434 Corollary 1. Let S be a right self-principally injective semigroup and a, b ∈ S then following are equivalent: 1) If bS embeds in aS then Sb is a homomorphic image of Sa. 2) If aS is a homomorphic image of bS then Sb can be embedded in Sa. 3) If bS ∼= aS then Sb ∼= Sa. Theorem 2. If M be a right S-act then, for all α, β ∈ E, following are equivalent. 1) M is quasi-principally injective. 2) If AnnE(kerα) = {β ∈ E : kerα ⊆ kerβ} then AnnE(kerα) = Eα. 3) If kerβ ⊆ kerα then Eα ⊆ Eβ. Proof. 1)⇒2) Let uα ∈ Eα, where u ∈ E. To show that uα ∈ AnnE(kerα), we need to show that kerα ⊆ keruα. Let (x, y) ∈ kerα, we have α(x) = α(y) so uα(x) = uα(y), which implies (x, y) ∈ keruα, it follows that kerα ⊆ keruα and so uα ∈ AnnE(kerα). For reverse inclusion, let β ∈ AnnE(kerα) so kerα ⊆ kerβ. Consider α′ : M → α (M) and β′ : M → β (M)induced by α and β. If i : α (M) → M and j : β (M) → M are inclusions then α = iα ′ and β = jβ′. Since α′ is an epimorphism so there exists an S-homomorphism Φ : α (M) → β (M) such that Φα′ = β′. Since M is quasi-principally injective so there exists ψ ∈ E such that ψi = jΦ. Finally consider, ψα = ψ(iα ′ ) = (ψi)α ′ = (jΦ)α ′ = j(Φα ′ ) = jβ′ = β This shows that β ∈ Eα. Thus AnnE(kerα) = Eα. 2)⇒3) Suppose kerβ ⊆ kerα. By hypothesis it is sufficient to show that AnnE(kerα) ⊆ AnnE(kerβ). Let u ∈ AnnE(kerα) so kerα ⊆ keru then kerβ ⊆ keru and therefore u ∈ AnnE(kerβ). Thus AnnE(kerα) ⊆ AnnE(kerβ). 3)⇒1) Suppose Φ : α (M)→M is an S-homomorphism. Now Φα′ ∈ E. We can easily see kerα ⊆ ker Φα′. So by hypothesis EΦα′ ⊆ Eα. Since Φα′ ∈ EΦα′ ⊆ Eα so Φα′ = uα for some u ∈ E, which implies the desired result. If AnnrS(a) = {s ∈ S : as = θ} is a right annihilator and AnnlS(a) = {s ∈ S : sa = θ} a left annihilator then we can have following corollary from Theorem 2. Corollary 2. The following are equivalent for a semigroup S. a) S is right self-principally injective. b) AnnlS(a) = Sa. c) AnnrS(b) ⊆ AnnrS(a) implies that Sb ⊆ Sa. We denote the set of all homomorphisms from a right S-act M to a right S-act N , by HomS(M,N ). Moreover, HomS(M,N ) is a right E-act by action (u, α) → uα, the usual composition of functions. Lemma 2. Let M and N be two right S-acts then N is M-principally injective iff for all α ∈ E, HomS(M,N )α = {β ∈ HomS(M,N ): kerα ⊆ kerβ}. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 435 Proof. Let K = β ∈ HomS (M,N ) : kerα ⊆ kerβ}. Let β ∈ K then kerα ⊆ kerβ. Define Φ : α (M) → N by Φ(α(m)) = β(m). Clearly Φ is well-defined and an S- homomorphism. Since N is M-principally injective so there exists γ : M → N such that γi = Φ i.e. γ extendsΦ, where i : α (M)→Mis the inclusion. Consider, β(m) = Φ(α(m)) = γi(α(m)) = γ(α(m)) = γα(m) for all m ∈ M, hence β = γα ∈ HomS (M,N )α. For the reverse inclusion let uα ∈ HomS (M,N )α. We claim that kerα ⊆ keruα., which is sufficient to show that uα ∈ K. To do so, let (x, y) ∈ kerα so α(x) = α(y) and therefore uα(x) = uα(y) which shows (x, y) ∈ keruα, hence kerα ⊆ keruα. Hence HomS(M,N )α = K. For the converse, let Φ : α (M) → N be a S-homomorphism. Consider the map Φα ∈ HomS(M,N ). Clearly, kerα ⊆ ker Φα. Therefore Φα ∈ K = HomS(M,N )α so Φα = uα, for some u ∈ HomS(M,N ). Hence N is M-principally injective. Lemma 3. Every X-cyclic sub-act of X is anM-cyclic sub-act ofM, for everyM-cyclic sub-act X of M. Proof. SupposeN is anX-cyclic sub-act ofX soN = α(X) for some α ∈ EndS(X).Now since X is M-cyclic so X = γ (M) for some γ ∈ E, hence N = αγ (M) so N is M-cyclic. Theorem 3. Let N and M be right S-acts then N is M-principally injective iff N is X−principally injective for every M-cyclic sub-act X of M. Proof. Let N be M-principally injective and X be an M-cyclic sub-act of M, so X = γ (M) , for some γ ∈ E. Let Φ : α(X) → N where α ∈ EndS(X) so α(X) = α (γ (M)) = αγ (M) , since N is M-principally injective so Φ̂ :M→N extends Φ. Now Φ̂/X = ψ, clearly ψ : X → N extends Φ, hence N is X-cyclic. Conversely assume as mentioned in the statement. Since i (M) = M, where i is an identity S-homomorphism on M, so M, itself is an M-cyclic sub-act of M. Thus N is M-principally injective. The direct product of S-acts is defined in [4]. If N = ⊕ i∈I Ni is the direct sum of right S-acts Ni, for each i ∈ I. Moreover, we treat λi : Ni → N as a natural S-injection and πj : N → Nj as a natural S-projection such that πjλj = iNj , where iNj is the identity S-homomorphism on Nj . Theorem 4. N = ⊕ i∈I Ni is M-principally injective iff each Ni is M-principally injective, for all i ∈ I. Proof. Suppose Φj : α (M)→ Nj is an S-homomorphism for each j ∈ I. Now sinceN is M-principally injective so we have γ :M→N such that γi = λjΦj , where i : α (M)→M is inclusion. Using πjγ :M→Nj it follows that, (πjγ)i = πj(γi) = πj(λjΦj) = (πjλj)Φj = Φj . J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 436 Conversely assume that Φ : α (M)→ N is an S-homomorphism. Now πjΦ : α (M)→ Nj is also an S-homomorphism. Since each Nj is M-principally injective, so we have γj : M → Nj such that γji = πjΦ. Define γ : M → N by γ(t) = (γj(t)), for t ∈ α (M) . Clearly γ is an S-homomorphism and γ(i(t)) = (γj(i(t))) = (γji(t)) = (πjΦ(t)) = Φ(t), for all t ∈ α (M) . Thus γ extends Φ, so N is M-principally injective. Corollary 3. N = ⊕ i∈I Ni is quasi-principally injective iff each Ni is quasi-principally injective, for all i ∈ I. 3. M-principally projective S-acts In this section we have definedM-principally injective S-acts and characterized them in terms ofM-cyclic sub-acts. Further we have also investigated few basic and important properties of the mentioned structure. 3.1. M-principally and quasi-principally Projective S-act Definition 5. A right S-act N is calledM-principally projective if every S-homomorphism from N to an M-cyclic sub-act of M can be lifted to an S-homomorphism from N to M. Lemma 4. Let M and N be right S-acts then N is M-principally projective iff HomS (N , α (M)) = αHomS(N ,M), for all α ∈ E. Proof. Assume thatN isM-principally projective. Let α ∈ E and αu ∈ αHomS(N ,M) where u ∈ HomS (N ,M) . Since α : M → α (M) so αu : N → α (M) this means αu ∈ HomS (N , α (M)) which gives HomS (N , α (M)) ⊇ αHomS (N ,M) . For the con- verse inclusion, let Φ ∈ HomS (N , α (M)) . Since N isM principally projective so we have an S-homomorphism γ : N →M such that Φ = αγ which implies Φ ∈ αHomS (N ,M) . Thus equality holds. Conversely assume that HomS (N , α (M)) = αHomS (N ,M) for all α ∈ E. Let Φ : N → α (M) so by hypothesis Φ = αu for some u ∈ HomS (N ,M) . Hence N is M-principally projective. Theorem 5. If K ∼= N , M ∼= M′ and N is M-principally projective then K is M- principally projective and N is M′-principally projective. Proof. Straight forward. Lemma 5. Let M and N be two right S-acts then N is M-principally projective iff N is X-principally projective for every M-cyclic sub-act X of M. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 437 Proof. Suppose N is X-principally projective. As X X is a M-cyclic sub-act of M so X = α (M), for some α. Moreover, Φ : N → β(X) be an S-homomorphism, where β ∈ End(X). Now β(X) = βα (M) so β(X) is M-cyclic and since N is M-principally projective so there exists Φ : N →M such that (βα)Φ = Φ. Now consider αΦ : N → X which clearly lifts Φ i.e. β(αΦ) = (βα)Φ = Φ. Hence N is X-principally projective. Converse is trivially follows by taking particular X =M. Theorem 6. Let N be anM-principally projective S-act. If φ is idempotent( i.e. φ2 = φ) then the homomorphic image φ(N ) is also M-principally projective. Proof. Let γ : φ(N ) → α (M) be an S-homomorphism. Consider γφ : N → α (M) , is an S-homomorphism, since N is M-principally projective so there exists θ : N → M lifting γφ i.e. αθ = γφ. Note that θi : φ(N ) →M, where i : φ(N ) → N is inclusion. It follows that for all φ(n) ∈ φ(N ) we have, (α(θi))(φ(n)) = (αθ)i(φ(n)) = αθ(φ(n)) = γφ(φ(n)) = γ(φ(n)), which simply implies that α(θi) = γ. Hence φ(N ) is M-principally projective. Theorem 7. LetM and N be right S-acts, thenM is N -principally projective and every N -cyclic sub-act of N isM-principally injective iff N isM-principally injective and every M-cyclic sub-act of M is N -principally projective. Proof. Assume that M is N -principally projective and each N -cyclic sub-act of N is M-principally injective. Since i(N ) = N , where i is the identity on N , so N is itself N - cyclic sub-act of N and henceM-principally injective. Let α (M) be anM-cyclic sub-act ofM, where α ∈ E. To show α (M) is N -principally projective, we let γ : α (M)→ β(N ) be an S-homomorphism, where β is an endomorphism on N . Since β(N ) isM-principally injective so there exists θ : M → β(N ) such that θi = γ, where i : α (M) → M is inclusion. Now since M is N -principally projective so there exists φ :M→N such that βφ = θ. Now consider φi : α (M) → N which lifts γ, i.e. β(φi) = (βφ)i = θi = γ. Hence α (M) is N -principally projective. Conversely assume that N is M-principally injective and every M-cyclic sub-act of M is N -principally projective. Trivially M is N -principally projective. Let β(N ) be N - cyclic sub-act of N and γ : α (M) → β(N ) be an S-homomorphism. Since α (M) is N -principally projective so there exists θ : α (M) → N such that βθ = γ. Now since N is M-principally injective so there exists φ : M → N such that φi = θ. Consider S-homomorphism βφ : α (M) → N which extends γ i.e. (βφ)i = β(φi) = βθ = γ. Hence β(N ) is M-principally injective. Theorem 8. N = ⊕ i∈I Ni is M-principally projective where each Ni is M-principally projective for all i ∈ I. Proof. Follows on the same lines of Theorem 4. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 438 Definition 6. For right S-acts M and N , M is called N -Projective or projective relative to N , if every right S-act C, every homomorphism f : M → C can be lifted w.r.t every g : N → C i.e. there exists a homomorphism h :M→N such that f = gh. Moreover, M is called N -injective right S-act, if for any Sub-act C of N , and any homor- morphism f : C →M there exists a homormophism g : N →M extending f . Theorem 9. Let M and N be two right S-acts, then M is N -injective and each sub- act of N is M-principally projective iff N is M-principally projective and each M-cyclic sub-act of M is N -injective. Proof. Assume that M is N -injective and each sub-act of N is M-principally projec- tive. SinceN is a sub-act of itself andM-principally projective. Let α (M) be anM-cyclic sub-act of M, where α ∈ E. To show α (M) is N -injective, we let β : N ′ → α (M) be an S-homomorphism, where N ′ is a sub-act of N . Since each sub-act of N is M-principally projective so N ′ is alsoM-principally projective. Therefore there exists γ : N ′ →M such that αγ = β. Since M is N -injective so there exists φ : N → M such that φi = γ. Now consider αφ : N → α (M) and (αφ)i = α(φi) = αγ = β. Hence α (M) is N -injective. Conversely assume that N is M-principally projective and each M-cyclic sub-act of M is N -injective. M itself is an M-cyclic sub-act and N -injective. Let N ′ be a sub-act of N and β : N ′ → α (M) be an S-homomorphism. Since α (M) is N -injective so there exists γ : N → α (M) such that γi = β, where i : N ′ → N is inclusion. Since N is M-principally projective so we have φ : N → M such that αφ = γ. Now keep in view φi : N ′ →M and α(φi) = (αφ)i = γi = β. Hence N ′ is M-principally projective. Theorem 10. The following statements are equivalent for a projective right S-act M. 1) Every M-cyclic sub-act of M is projective. 2) Every factor S-act of an M-principally injective S-act is M-principally injective. 3) Every factor S-act of an injective S-act is M-principally injective. Proof. 1)⇒2) Let N be anM-principally injective S-act and ρ be a congruence on N . Let φ : α (M) → N/ρ be an S-homomorphism, where α ∈ E. Since α (M) is projective so there exists φ : α (M) → N such that πφ = φ, where π : N → N/ρ is a canonical epimorphism. Now as N is M-principally injective so there exists γ : M → N which extends φ. Consider πγ : M → N/ρ which extends φ. Hence N/ρ is M-principally injective. 2)⇒3) LetN be an injective right S-act and ρ be congruence onN . Let φ : α (M)→ N be an S-homomorphism and N be injective, so there exists γ :M→N extending φ. By hypothesis N is M-principally injective. Thus N/ρ is M-principally injective. 3)⇒1) Let α (M) be an M-cyclic sub-act of M, h : A → B be an S-epimorphism and β : α (M) → B be an S-homomorphism. by corollary 2.1.2 of [1] there exists an injective S-act Q in which A can be embedded. Now define φ : B = h(A) → A/K by φ(h(a)) = [a]K , where K = kerh, the kernel congruence on A induced by h. Clearly φ is an S-isomorphism. Let K ′ = K ∪ 4Q be a congruence on Q, where 4Q is a diagonal congruence on Q. Moreover, A/K is a sub-act of Q/K ′. Thus β : α (M)→ B can also be J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 439 viewed as β : α (M) → Q/K ′. By hypothesis Q/K ′ is M-principally injective, so there exists β :M→ Q/K ′ which extends β. Since M is projective so there exists γ :M→ Q such that πγ = β, where π : Q→ Q/K ′ is a canonical epimorphism i.e. γ lifts β. Clearly γ(α (M)) lies in the domain of h, so we must have γ(α (M)) ⊆ A. Thus we have lifted β. This completes the proof of the theorem. Corollary 4. The following statements are equivalent for a semigroup S. 1) S is a right PP-Semigroup. 2) Every factor S-act of PM-injective S-act is PM-injective. 3) Every factor S-act of an injective S-act is PM-injective. Theorem 11. Let N be an M-principally projective S-act. If M0 is either an S- homomorphic image or an S-sub-act of M, then N is M0-principally projective. Proof. If N is a homomorphic image of M0 then the result follows directly from the Lemma 5. Now assume that M0 is S- sub-act of M. To show that N is M0-principally projective, we let φ : N → α(M0) be an S-homomorphism, where α ∈ End(M0). By Lemma 1, it is clear that α(M0) ' M0/ kerα. Consider π0 : M0 → M0/ kerα be the natural epimorphism. Let ρ be the congruence onM defined by ρ = kerα∪4M, where 4M is a diagonal congruence on M. Let π :M→M/ρ ' γ (M) for some endomorphism γ on M. We can take another view of φ, notice that α(M0) 'M0/ kerα ⊂M/ρ ' γ (M) thus we can treat φ as an S-homomorphism φ : N → γ (M) , also surely π extends π0. Since N is M-principally projective so there exists β : N →M such that πβ = φ. But π(β(N )) = πβ(N ) = φ(N ) ⊂ α(M0) 'M0/ kerα, which clearly shows that β(N ) ⊂ M0. Thus we can treat β from N to M0. Since π and π0 agrees at M0, so π0β(n) = π0(β(n)) = π(β(n)) = πβ(n) = φ(n) for all n ∈ N , which implies that π0β = φ. Hence N is M0-principally projective. Theorem 12. Let M be a projective S-act and E = E (M) be an injective hull of M. If E is completely M-principally injective i.e. each factor S-act of E is M-principally injective, then each M-cyclic sub-act of M is M-principally projective. Proof. Let α (M) be an M-cyclic sub-act of M, α ∈ EndS (M) and i : α (M) →M be an inclusion map. Consider π : E→ E/ρ, where ρ is congruence on E. Let γ : α (M)→ β(E), where β is an endomorphism on E. Clearly β(E) ' E/ kerβ. Since E is completely M-principally injective which implies E/ kerβ isM-principally injective. Therefore, there exists φ : M→ E/ kerβ ' β(E), such that φi = γ. Since M is projective so there exists ψ : M → E such that βψ = φ. Let θ := ψi : α (M) → E so βθ = βψi = φi = γ. Thus α (M) is E-principally projective. Now since M can be embedded in E so we can treat M as the sub-act of E and so by Theorem 11, it follows that α (M) is M-principally projective. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 440 Theorem 13. a) Let M be a right S-act, if every M-cyclic sub-act α (M) (where α ∈ EndS (M)) of M is A-principally projective and A is M-principally injective then every A-cyclic sub-act β(A) (where β ∈ EndS(A)) of A is M-principally injective. b) Let i : β(A) → A (where β ∈ EndS(A)) be an inclusion map and M be an S-act. If every M-cyclic sub-act of M is A-principally injective and A is M-principally projective then β(A) is M-principally projective. Proof. a) Let i : α (M) → M be an inclusion map and let γ : α (M) → β(A) be an S-homomorphism, where β ∈ EndS(A). Since α (M) is A-principally projective so there exists h : α (M)→ A such that βh = γ. Since A isM-principally injective so there exists λ :M→ A such that λi = h. Let µ = βλ :M→ β(A), since µi = βλi = βh = γ. Hence β(A) is M-principally injective. b) To show that β(A) is M-principally projective, we let γ : β(A) → α (M) be an S- homomorphism, where α ∈ E. Keeping γ in view, since α (M) is A-principally injective so there exists φ : A → α (M) such that φi = γ. Since A isM-principally projective so there exists θ : A→M such that αθ = φ. Consider θi : β(A)→M and α(θi) = (αθ)i = φi = γ. Hence β(A) is M-principally projective. 3.2. A Note on Co-hereditary S-acts Remark 1. Usually co-hereditary S-acts are defined as those S-acts whose every proper factor S-act is injective. Keeping in view Lemma 1 we can redefine the Co-hereditary S-acts as following. Definition 7. A right S-act M is called Co-hereditary S-act, if each M-cyclic sub-act is injective. Definition 8. A right S-act M is NPI-Co-hereditary if every M-cyclic sub-act of M is N -principally injective. Definition 9. A right S-act is quasi PI-Co-hereditary if every M-cyclic sub-act of M is M-principally injective. Remark 2. Co-hereditary → NPI-co-hereditary →PI- co-hereditary. Theorem 14. If an S-act M is API-co-hereditary, then every A-cyclic sub-act of an M-principally projective S-act A is M-principally projective. Proof. Let α(A) be an A-cyclic sub-act of A. We show it is M-principally projective. By definition every M-cyclic sub-act of M is A-principally injective. Since M is itself M-cyclic sub-act of M so is A-principally injective Thus by Theorem 13 again, α(A) is M-principally projective. Corollary 5. If an S-act M is PI-cohereditary and M-principally projective, then every M-cyclic sub-act of M is M-principally projective. J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 441 4. Semi-projective S-acts Definition 10. A right S-actM is called a Semi-projective S-act if αE = HomS(M, α (M)) for all α ∈ E. Theorem 15. Let M be an S-act then the following conditions are equivalent. 1) M is quasi-principally projective. 2) M is semi-projective. 3) For α, β ∈ E if α (M) ⊆ β (M) then αE ⊆ βE. Proof. 1)⇒2) follows directly from Lemma 4, for N =M. 2)⇒3) let α (M) ⊆ β (M) then for u ∈ HomS(M, α (M)) since α (M) ⊆ β (M) so we may also view u as an S-homomorphism u :M→ β (M) i.e. u ∈ HomS(M, β (M)). So we have HomS(M, α (M)) ⊆ HomS(M, β (M)) and therefore by hypothesis αE ⊆ βE. 3)⇒1) Consider Φ :M→ α (M) an S-homomorphism, so Φ (M) ⊆ α (M) by hypothesis ΦE ⊆ αE. Since Φ ∈ ΦE ⊆ αE so Φ = αu, for some u ∈ E. Hence M is quasi-principally projective. Theorem 16. Let M be a semi-projective right S-act and α, β ∈ E. Then: 1) If α (M) embeds into β (M) then αE can be embedded into βE. 2) If β (M) is a homomorphic image of α (M) then βE is a homomorphic image of αE . 3) If α (M) ∼= β (M) then αE ∼= βE. Proof. 1) Let f : α (M)→ β (M) be an S-homomorphism. SinceM is semi projective so βE = HomS (M, β (M)) . As fα :M→ β (M) , so fα ∈ HomS (M, β (M)) = βE and so fα = βg, for some g ∈ E. Define φ : αE → βE by φ(αu) = βgu for u ∈ E. We can see that φ is well-defined. Indeed, for any u, v ∈ E such that αu = αv we have fαu = fαv implies βgu = βgv and so φ(αu) = φ(αv). Clearly E-homomorphism. To show injectivity we take φ(αu) = φ(αv) then βgu = βgv. As fα = βg therefore fαu = fαv. Since f is 1-1 αu = αv. Hence φ is embbedding. 2) Let f, g and φ be as in part 1). For βu ∈ βE there exists ψ ∈ HomS(M, β (M)) such that βu = ψ. For m ∈ M we have βu(m) = ψ(m) ∈ β(M) so there exist m′ such that βu(m) = ψ(m) = β(m′). As fα (M) = β (M) so there exists m′′ such that βu(m) = ψ(m) = β(m′) = fα(m′′) = βg(m′′). Hence we can define γ : M → M by γ(m) = m′′,for m,m′′ ∈ M, whenever βu(m) = βg(m′′). Let us see that γ is well-defined. Let m1 = m2, where γ(m1) = m′′1 and γ(m2) = m′′2. This holds only when βu(m1) = βg(m′′1) and βu(m2) = βg(m′′2), βu(m2) = βu(m1) = βg(m′′1), which implies γ(m2) = m′′1 = γ(m1), hence γ is well defined. Moreover βu(m) = βg(m′′) = βgγ(m) for all m ∈M. Thus βu = βgγ = φ(αγ) so φ is epimorphism. 3) Follows directly form 1) and 2). J. Hussain, M.Shabir / Eur. J. Pure Appl. Math, 11 (2) (2018), 431-443 442 Theorem 17. Let M be a semi-projective(quasi-principally projective) right S-act then there is a one-one correspondence between M-cyclic sub-acts of M and principal right ideals of E. Proof. Let à be a collection of all M-cyclic sub-acts of M and B̃ be a collection all principal right ideals of E. Define Φ : à → B̃ by Φ(α (M)) = αE. Now Φ is well defined because if α (M) = β (M) for α, β ∈ E, then HomS (M, α (M)) = HomS (M, β (M)) , since M is semi-projective so αE = βE and Φ (α (M)) = Φ(β (M)). To show injectivity we let Φ(α (M)) = Φ(β (M)) so αE = βE and HomS(M, α (M)) = Hom(M, β (M)) which clearly implies α (M) = β (M). Surjection is trivial. Theorem 18. There is a one-one correspondence between f ∈ HomS (α (M) ,M) and F ∈ HomS (αE,E) with kerF (α) ⊃ kerα in such away that F (αu) = fαu, for all u ∈ E and f(α(m)) = F (α)(m), for all α ∈ E and m ∈M. Proof. Let us fix α ∈ E. We claim that for every f ∈ HomS (α (M) ,M) . we can define a unique F : αE → E by F (αu) = fαu, for all u ∈ E. Also we can see that ker F (α) ⊃ kerα. Indeed, for (x, y) ∈ kerα we have α(x) = α(y), which implies fα(x) = fα(y) and therefore F (α)(x) = F (α)(y) i.e.(x, y) ∈ kerF (α). Hence we can define a map Φ : HomS(α (M) ,M)→ HomS(αE,E) as Φ(f) = F. We will show that Φ is the required one-one correspondence. Let us begin by proving that Φ is well defined. Let f, f ′ ∈ HomS(α (M) ,M) such that f = f ′. Then fαu = f ′αu and so F (αu) = F ′(αu),for all αu ∈ αE. Hence F = F ′ i.e. Φ(f) = Φ(f ′). To show that Φ is 1-1. Let f, f ′ ∈ HomS(α (M) ,M) such that Φ(f) = Φ(f ′) i.e. F = F ′. This implies that F (αu) = F ′(αu), for all αu ∈ αE. Therefore fαu = f ′αu, for all u ∈ E. In particular for u = iM, it follows that fα = f ′α and hence f(α(m)) = f ′(α(m)) for all α(m) ∈ α (M) , and f = f ′. Hence Φ is 1-1. To see surjectivity, let F ∈ HomS (αE,E). Define f(αM(m)) = F (α)(m) for all m ∈ M. Clearly f is S-homomorphism and Φ(f) = F. Thus we established the 1-1 correspondence. Theorem 19. For a semi-projective S-act and α ∈ E , α (M) is simple. Converse is true for those S-acts for which mS is M-cyclic for all m ∈M. Proof. Let α (M) be simple. Suppose contrary that αE is not simple, so there exists γ ∈ E such that θ 6= αγE ( αE, and therefore αγ (M) ( α (M) contradiction. Hence αE is simple. conversely assume that αE is simple and M is, as mentioned in the theorem. Let mS = γ (M) , γ ∈ E. Now θ 6= αγ (M) ( α (M) → θ 6= αγE ( αE contradiction. Hence the result. Theorem 20. If N = ⊕ i∈I Ni is quasi-principally projective (respectively semi-projective) then each Ni is quasi-principally projective (respectively semi-projective), for all i ∈ I. REFERENCES 443 References [1] J. Ahsan, L. Zhongkui, A homological approach to the theory of monoids, Science press bejing. [2] J. Ahsan, M. F. Khan, M. Shabir and M. Takahashi, Characterizations of Monoids by P-injective and normal S-system, Kobe J. Math, (1991)8:173-192. [3] C. S. Johnson, Jr. and F.R. McMorris: Completely cyclic injective semilattices, Proc. Amer. Math. Soc 36(1972), 385-388. [4] Fakhruddin S.M., On the category of S-posets. Acta Sci. Math., 1988, 52, 85-92. [5] C.S. Johnson, J.R., McMorris F.R., Injective hulls on certain S-systems over a semi- lattice. Proc. Amer. Math. Soc., 1972, 32, 371-375. [6] J. Fountain, A class of right PP monoids, Quart. J. Math. Oxford (2) 28 (1977), 285-300 [7] J. Fountain: Completely right infective semigroups, Proc. London Math. Soc. 28(1974), 28-44. [8] J.K. Luedeman, F.R. McMorris and S.K. Sim, Semi-groups for which every totally irreducible S-system is injective, Comment. Math. Univ. Carolinae 19 (1978), 27–35. [9] J.J. Rotman. An Introduction to Homological Algebra , Academic Press, New York (1979). [10] Schein B.M. Injectives in certain classes of semigroups. Semigroup Forum, 1974, 9, 159-171. [11] P. Brathiaume, The injective envelope of S-acts, Canad. Math. Bull, (1967)10:261- 273. [12] R. Wisbauer, Foundations of module and ring theory , Gordon and Breach, Philadel- phia, (1991). [13] N.V. Sanh N. V., Shum K. P., Dhompongsa S. and Wongwai S., On quasi-principally injective modules, Algebra Coll. 6(3)(1999),269-276. [14] V. Gould, Divisible S-systems and R-modules , Proc. Edinburgh Math. Soc. 30 (1987), 187–200. [15] S. Wongwai, On the endomorphism ring of a semi-injective module, Acta Math.Univ.Comenianae Vol.LXXI, 1(2002), pp.27-33 [16] Tansee H. and Wongwai S., On the endomorphism ring of a semi-projective module, KYUNGPOOK Math J. 42(2002), 369-38.