/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 1, 2016, 19-26 ISSN 1307-5543 – www.ejpam.com Essentiality in the Category of S-acts Hasan Barzegar Department of Mathematics, Tafresh University, Tafresh, Iran. Abstract. Essentiality is an important notion closely related to injectivity. In this paper, we study es- sentiality with respect to monomorphisms of acts. We give some criterion to characterize and describe essentiality explicitly. 2010 Mathematics Subject Classifications: Primary 20M30, 08A60; Secondary 08B30. Key Words and Phrases: Essential extension, S-acts 1. Introduction and Preliminaries An important notion related to injectivity with respect to monomorphisms or any other classM of morphisms in a categoryA , is essentiality. In fact, injectivity is characterized and injective hulls are defined using essentiality (see, for example, [1, 10] and [5] ). Throughout this paper S will be denoted by the semigroup with or without identity. We take A = Act-S to be the category of right acts over a semigroup S and M ono to be the class of all monomorphisms of right S-acts, and then, we study the notion of essentiality with respect to this class. Essentiality with respect to the subclass M of monomorphisms have studied and some equivalent conditions, so called “essential test lemma for essentiality", have introduced(see, [7, 9]). We substantially improve the usual characterization of essentiality in terms of congruences, Lemma 1, of an extension B of A by giving a characterization in terms of the elements of B, Theorems 5 and 6, which are essential test lemmas. Also, similar to the case of modules, which essentiality has an expression by submodules, in Theorem 1 an equivalent condition in terms of Ress congruences is obtained for essentiality. Although the Baer Criterion for injectivity (weak injectivity implies injectivity) is true for modules over a ring (with an identity), it is an open problem for acts over a semigroup S (with or without identity). In fact, we are not aware of any type of weak injectivity implying injec- tivity of S-acts, in general, other than Skornjakov-Baer Criterion, which says that injectivity with respect to subacts of cyclic acts implies injectivity with respect to all monomorphisms. Email address: h56bar@tafreshu.ac.ir http://www.ejpam.com 19 c© 2016 EJPAM All rights reserved. H. Barzegar / Eur. J. Pure Appl. Math, 9 (2016), 19-26 20 One of the well known theorem about the injectivity says that, an S-act A is injective if and only if it has no proper essential extension(see, [8] or [4]). So essentiality play an important role in the study of the Baer problem. In section 2 some necessity conditions on essentiality are obtained and Theorem 5, “which is the main result of this article", is in fact an essential test lemma which introduces an equiv- alent condition to essentiality. Let us first recall the definition and some ingredients of the category Act− S of acts over a semigroup S needed in the sequel. For more information and the notions not mentioned here see, for example, [6] and [8]. Recall that, for a semigroup S, a set A is an S-act (or an S-set) if there is a, so called, action µ : A× S → A such that, denoting µ(a, s) := as, a(st) = (as)t and if S is a monoid with 1, a1= a. Each semigroup S can be considered as an S-act with the action given by its multiplication. Notice that, adjoining an external left identity 1 to a semigroup S an S-act S1 := S ∪ {1} is obtained. Also, recall that an element a ∈ A is said to be fixed if as = a for all s ∈ S. The S-act A∪{0} with a fixed adjoined to A is denoted by A0. All fixed elements of as an S-act A is a subact of A and denoted by F ix(A). A fixed element of a Semigroup S is called a left zero element. All left zero elements of a Semigroup S is a right ideal of S and denoted by Z(S). The definitions of a homomorphism of S-acts or S-maps, subact A of B, written as A≤ B, an extension of A, a congruence ρ on A and a quotient A/ρ of A are all clear. For H ⊆ A× A, the congruence generated by H, that is the smallest congruence on A containing H, is denoted by ρ(H). 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. 2. Essentiality of Acts Here, some characterizations and some properties of essentiality are given. Many of results of this section are similar to the work done in [3]. Definition 1. A monomorphism f : A→ B of S-acts is said to be essential if for each homomor- phism g : B→ C which g f is a monomorphism, then g is so. If f is an inclusion map, then B is said to be an essential extension of A. The following two theorems give the usual (external) characterizations for the essentiality (mainly in terms of congruences). More (internal) characterizations (in terms of elements) will be given later in this section. The set of all congruences on an S-act B is denoted by Con(B) and ∆ is the trivial congru- ence(i.e. a∆b if and only if a = b.) Lemma 1. For a monomorphism f : A→ B, the following are equivalent: (i) f is an essential monomorphism. H. Barzegar / Eur. J. Pure Appl. Math, 9 (2016), 19-26 21 (ii) For every epimorphism g : B→ C such that g f is a monomorphism, g itself is a monomor- phism. (iii) For every congruence ρ on B such that for the canonical epimorphism π : B→ B/ρ, π f is a monomorphism, we get ρ =∆. (iv) For every monogenic congruence ρ on B such that for the canonical epimorphism π : B→ B/ρ, π f is a monomorphism, we get ρ =∆. Proof. We just prove (iv)⇒ (i). Let g : B→ C be a homomorphism with g f a monomor- phism, and g(b) = g(b′). Then, sinceρ(b, b′) ⊆ ker(g), we can factorize g through B/ρ(b, b′), and hence π f is a monomorphism, where π : B→ B/ρ(b, b′). So, by (iv), ρ(b, b′) = ∆, and thus b = b′. Corollary 1. An S-act B is an essential extension of A if and only if for each congruence ρ on B, if ρ | A =∆, then ρ =∆. Theorem 1. An extension B of A is an essential extension if and only if for every non trivial θ ∈ Con(B), θ ∩ρ A 6=∆, where ρA is the Rees congruence on B. Proof. (⇒) Let θ 6= ∆ and θ ∩ ρ A = ∆. Then, considering the canonical epimorphism π : B → B/θ , we see that π|A is a monomorphism, and so by hypothesis θ = ∆ which is a contradiction. (⇐) Let g : B→ C be a homomorphism such that g|A is a monomorphism. It is clear that ker(g)∩ρA =∆. So by hypothesis, ker(g) = ∆ and hence g is a monomorphism. Theorem 2. The monomorphisms f : A→ B and g : B→ C are essential monomorphisms if and only if g f is so. Proof. We just prove the case that if g f is an essential monomorphism, then f is so. Let h : B→ D be a homomorphism such that hf is a monomorphism. Then there exists an exten- sion h̄ : C → E(D) of h to the injective hull of D, and since g f is essential, h̄ is a monomorphism, and hence so is h. Proposition 1. Let A and C be subacts of B such that |C | ≥ 2 and B is an essential extension of A. Then |C ∩ A| ≥ 2. In particular, B \ A does not have two fixed elements. Proof. Let |C∩A| ≤ 1. It is clear thatπ|A : A→ B/ρ C , in which ρ C is a Ress congruence on C , is a monomorphism. Hence,π is a monomorphism and so |C |= 1, which is a contradiction. Corollary 2. If a0 ∈ A and b0 ∈ B \ A are fixed elements, then B is not an essential extension of A. For a subact A of an S-act B and b ∈ B, we use the notation Ib = {s ∈ S | bs ∈ A}. Corollary 3. Let A have at least one fixed element and B be an essential extension of A. Then: H. Barzegar / Eur. J. Pure Appl. Math, 9 (2016), 19-26 22 (i) F ix(B) ⊆ A. (ii) For every b ∈ B, Ib 6= ;. Corollary 4. If S has at least one left zero element and S is an essential extension of a right ideal I , then Z(S) ⊆ I . If S is a left zero semigroup, then I = S. Lemma 2. If A has no fixed element, then A0 is an essential extension of A. Proof. Let g : A0→ B be a homomorphism such that g|A is a monomorphism. Then g itself is one-one. In fact, if g(a) = g(0) for some a ∈ A, then for every s ∈ S, g(as) = g(a)s = g(0)s = g(0s) = g(0) = g(a) and so as = a. This means that a is a fixed element, which is a contradiction. Thus g is an injection. Lemma 3. Pushouts do not necessarily transfer essential monomorphisms. Proof. Let A have no fixed element. By Lemma 2, the inclusion τ : A→ A0 is an essential extension. Consider the pushout diagram A τ → A0 τ ↓ ↓ q A0 p → Az1,z2 where z1, z2 are two fixed elements adjoint to A and p(a) = q(a) = a (a ∈ A) and p(0) = z1, q(0) = z2. By [2, Theorem 3.2(1)], p and q are monomorphisms. Define a homomorphism h : Az1,z2 → A0 by h(a) = a and h(z1) = h(z2) = 0. Then hp = idA0 . But, h is not one-one, and hence p is not an essential extension. 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. 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 −→α Bα = ∐ 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 −→α Bα of S-maps satisfies gβ gαβ = gα for α≤ β , where π : ∐ α Bα→ l im −→α Bα is the natural S-map. Theorem 3. Any direct limit of essential monomorphisms is an essential monomorphism. Proof. Let f : A→ l im −→α Bα be a direct limit in Act-S of essential monomorphisms fα : A→ Bα, α ∈ I , and directed S-maps gαβ : Bα→ Bβ (α≤ β), and consider gα : Bα→ l im −→α Bα as before. To show that f : A→ l im −→ Bα is essential, let hf , for h : l im −→α Bα → C , be a monomorphism. Then, for every α ∈ I , hgα fα ∈ M . Since each fα is essential, each hgα is a monomorphism. Now if h([bα]) = h([bβ]), then hgγgαγ(bα) = hgγgβγ(bβ ), for some γ ≥ α,β . Thus gαγ(bα) = gβγ(bβ ) which means that bαρbβ , and hence h is a monomorphism and f is essential. H. Barzegar / Eur. J. Pure Appl. Math, 9 (2016), 19-26 23 Definition 2. The category A is called essentially bounded, if every A ∈ A has only a set of essential extensions. Proposition 2. The category Act-S is essentially bounded. Proof. Any essential extension B of A can be clearly embedded into the injective hull E(A) of A. So, we get the result. The following theorem is an other form of [4, Theorem 8]. Theorem 4. Act-S fulfills Banaschewski’s condition, that is, for every monomorphism f : A→ B there exists a homomorphism g : B→ C such that g f is an essential monomorphism. For an S-act A and a ∈ A we denote the homomorphism f : S→ A, given by f (s) = as for all s ∈ S, by λa. Lemma 4. Let B be an essential extension of A and C p(A) = {b ∈ B | ∃a ∈ A,λb = λa}. Then C p(A) = A. Proof. Since B is an essential extension of A, C p(A) is an essential extension of A too. Let C p(A) 6= A and b ∈ C p(A) \ A. By the Axiom of choice, choose and fix an element ab ∈ A such that λb = λab . Consider the homomorphism g : C p(A)→ A defined by g(b) = ¨ b, if b ∈ A ab, if b 6∈ A . It is clear that g | A = id A and hence g is an isomorphism. So b = ab which is a contradiction. Thus C p(A) = A. Proposition 3. Suppose that B is an essential extension of A and b ∈ B \ A, b′ ∈ B such that Ib = Ib′ . If for each s ∈ Ib = Ib′ , bs = b′s, then b = b′. Proof. If b′ ∈ A, then Ib = Ib′ = S and b ∈ C p(A). By Lemma 4, b ∈ A, which is impos- sible. So b′ /∈ A. Consider the canonical epimorphism π : B → B/ρ(b, b′). Let a, a′ ∈ A and aρ(b, b′)a′. then a = a′ or there exist pi ,qi ∈ {b, b′}, si ∈ S1 and n ∈ N such that a = p1s1, q1s1 = p2s2,q2s2 = p3s3, . . . ,qnsn = a′. If s1 = 1, then a = p1s1 = p1 ∈ {b, b′}, which is impos- sible. Let for i ≥ 2, si = 1 and for each j < i, s j 6= 1. So s1 ∈ Ip1 = Iq1 and a = p1s1 = q1s1 = p2s2, which implies s2 ∈ Ip2 = Iq2 and p2s2 = q2s2 = p3s3. By continuing with this way, we have a = p1s1 = q1s1 = p2s2 = q2s2 = p3s3 = . . .= qi−1si−1 = pisi = pi ∈ {b, b′}. So b or b′ belongs to A, which is a contradiction. Thus for each 1 ≤ i ≤ n, si 6= 1, which deduced that a = p1s1 = q1s1 = p2s2 = q2s2 = p3s3 = . . .= a′. So π |A is a monomorphism. By the hypothesize π is a monomorphism and hence b = b′. An S-act A is called s-dense in an extension B, if for each b ∈ B, bS ⊆ A. Also B is said to be an s-dense essential extension of A, if B is an s-dense extension as well as essential extension of A. The following lemma is an essential test lemma for s-dense essentiality. H. Barzegar / Eur. J. Pure Appl. Math, 9 (2016), 19-26 24 Lemma 5. An s-dense extension τ : A→ B is s-dense essential if and only if for each b ∈ B \ A, b′ ∈ B, if λb = λb′ , then b = b′. Proof. (⇒) Since A is s-dense in B, Ib = Ib′ = S. Now we are done by using Proposition 3. (⇐) Consider A τ → B g → C , which τ is inclusion map and gτ is a monomorphism. Let g(b) = g(b′) such that b ∈ B \ A. For each s ∈ S we have g(bs) = g(b′s) and {bs, b′s} ∈ A. Since g | A is one to one, λb = λb′ and hence b = b′. Lemma 6. Let A have a fixed element and B be a proper essential extension of A. Then for every b ∈ B and every nonempty right ideal J of S, Ib ∩ J 6= ;. Proof. For b ∈ A, Ib = S and the result is obvious. For b /∈ A, let J be a nonempty right ideal of S and Ib∩ J = ;. By Corollary 3, Ib 6= ; and F ix(B) ⊆ A. It is clear that B′ = {bs|s ∈ J} is a subact of B and B′ ∩ A= ;. By Proposition 1, |B′| = 1 and hence for every s ∈ J , bs = b0 for some b0 ∈ B \ A. Consider s0 ∈ J . Then for every t ∈ S, b0 t = (bs0)t = b(s0 t) = b0. So b0 ∈ F ix(B) ⊆ A, which is impossible. The following two theorems are the main results of this article, which is in fact a kind of essential test lemma. In these theorems we give an (internal) characterization for essential monomorphisms (in terms of elements rather than congruences). Theorem 5. (Essential Test Lemma 1) An S-act B is an essential extension of A if and only if for every x ∈ B and y ∈ B \ A whenever the following two conditions hold then x = y: (i) For each s ∈ S with s ∈ Ix ∩ I y , we have xs = ys. (ii) If I1 = Ix \ I y and I2 = I y \ Ix , then kerλy |I1 ⊆ kerλx and kerλx |I2 ⊆ kerλy . Proof. (⇐) Let g : B→ C be a homomorphism with g|A a monomorphism, and g(x) = g(y) for x , y ∈ B . Then, clearly conditions (i) and (ii) hold for x ∈ B, y ∈ B \ A. Thus x = y , and so B is an essential extension of A. (⇒) Let B be an essential extension of A, x ∈ B, and y ∈ B \ A. Let the conditions (i) and (ii) hold. At first we show that Ix = I y . On the contrary, let Ix 6= I y . In this case, there exists t ∈ S such that a = x t ∈ A and b = y t /∈ A (or x t /∈ A,y t ∈ A). So, by (i), we have (∗) For every s ∈ Ib, as = x ts = y ts = bs. Now, consider the congruence ρ = ρ(a, b) on B and (a1, a2) ∈ ρ with a1, a2 ∈ A. Then a1 = a2 or there exist p1, p2, . . . , pn and q1,q2, . . . ,qn in B with {pi ,qi}= {a, b} and s1, s2, . . . , sn ∈ S1 such that a1 = p1s1,q1s1 = p2s2, . . . ,qnsn = a2. We prove, by induction on n, that a1 = a2,. If n = 1, then a1 = p1s1, a2 = q1s1 (where s1 6= 1, since otherwise p1 = a and hence a2 = q1 = b which is a contradiction). But, one of p1 or q1 is b, so bs1 ∈ A and hence using (∗) a1 = p1s1 = q1s1 = a2. Now, let the result be true when the path connecting a1 to a2 has length less than n. Assume we have the above path of length n≥ 1. Then: If q1s1 ∈ A then s1 6= 1 (because otherwise q1 = a and p1 = b which contradicts a1 = p1s1). Also bs1 ∈ A because it is one of p1s1 or q1s1. Thus a1 = p1s1 = q1s1. This means p2s2 = a1 and so we get a path with length n−1 which connects a1 to a2. Then, by induction hypothesis, a1 = a2. REFERENCES 25 If q1s1 /∈ A then p2 = q1 = b and q2 = p1 = a. Hence, q2s2 ∈ A. So y ts1 = bs1 = q1s1 = p2s2 = bs2 = y ts2 where {x ts1, x ts2} ⊆ A and y ts2 = y ts1 /∈ A. Thus, by (ii), we get x ts1 = x ts2 and so a1 = p1s1 = as1 = as2 = q2s2. This gives p3s3 = a1 and hence we get a path with a lower length than n which yields a1 = a2, by induction hypothesis. Therefore, a1 = a2. So ρ is identity on A. Thus it is identity on B and hence a = b, which is a contradiction. So, Ix = I y . Now using Proposition 3 deduced the result. Theorem 6. (Essential Test Lemma 2) An S-act B is an essential extension of A if and only if the following hold: (i) for every b, b′ ∈ B, if ρ(b, b′)∩ A× A=∆, then λb = λb′ . (ii) for every b, b′ ∈ B \ A, if λb = λb′ , then b = b′. (iii) C p B (A) = A. Proof. (⇒) To prove (i), let b, b′ ∈ B and ρ(b, b′) ∩ A× A= ∆. So for the canonical map π : B→ B/ρ(b, b′), π | A and so π is a monomorphism. Therefore b = b′ and hence λb = λb′ . To prove (ii), let b, b′ ∈ B \ A with λb = λb′ . For the canonical map π : B → B/ρ(b, b′), π | A is a monomorphism, indeed, let aρ(b, b′)a′(a, a′ ∈ A). So there are p1, p2, . . . , pn,q1,q2, . . . ,qn ∈ {b, b′} and s1, s2, . . . , sn ∈ S1 such that a = p1s1,q1s1 = p2s2, . . . ,qnsn = a′. Since p1s1 = a ∈ A, s1 6= 1 and hence p2s2 = q1s1 = p1s1 = a which implies s2 6= 1. By continue to this process, for each 1 ≤ i ≤ n, si 6= 1, which deduced a = a′. By essentiality π is a monomorphism and thus ρ(b, b′) = ∆ and b = b′. To prove (iii), let b ∈ C p B (A). So there exists a ∈ A such that for each s ∈ S, as = bs. Similarly, to prove (ii), for the canonical map π : B → B/ρ(a, b), π | A is a monomorphism. Thus by essentiality, π is a monomorphism and hence a = b. (⇐) By Lemma 1, it is enough to show that for every monogenic congruence ρ = ρ(b, b′) on B such that for the canonical epimorphism π : B→ B/ρ, π | A is a monomorphism, we get b = b′. Since π | A is a monomorphism, by (i), λb = λb′ and if {b, b′} ⊆ A, then b = b′. In the case where b, b′ ∈ B \ A, by (ii), b = b′. At last condition (iii) shows that the case where b ∈ A, b′ /∈ A may not occur. ACKNOWLEDGEMENTS The author gratefully acknowledges to the referee for carefully reading the paper. References [1] B. Banaschewski. Injectivity and essential extensions in equational classes of algebras. Queen’s Papers in Pure and Applied Mathematics, 25:131–147, 1970. REFERENCES 26 [2] H. Barzegar and M.M. Ebrahimi. Sequential pure monomorphism of acts over semi- groups. European journal of pure and applied mathematics, 1(4):91–99, 2009. [3] H. Barzegar, M.M. Ebrahimi, and M. Mahmoudi. Essentiality and injectivity relative to sequential purity of acts. Semigroup Forum, 79:128–144, 2009. [4] P. Berthiaume. The injective envelope of s-sets. Canadian Mathematical Bulletin, 10(2):261–273, 1967. [5] M.M. Ebrahimi. Algebra in a grothendieck topos: injectivity in quasi-equational classes. Jounal of pure and applied algebra, 26(3):269–280, 1982. [6] M.M. Ebrahimi and M. Mahmoudi. The category of m-sets. Itallian journal of Pure and Applied Mathematics, 9:123–132, 2001. [7] M.M. Ebrahimi and M. Mahmoudi. Purity and equational compactness of projection algebras. Applied Categorical Structure, 9:381–394, 2001. [8] M. Kilp, U. Knauer, and A. Mikhalev. Monoids, Acts and Categories. Walter de Gruyter, Berlin, New York, 2000. [9] M. Mahmoudi and L. Shahbaz. Sequential dense essential monomorphisms of acts over semigroups. Applied Categorical Structure, 18(5):461–471, 2010. [10] W. Tholen. Injective objects and cogenerating sets. Journal of algebra, 73(1):139–155, 1981.