Essentiality and Injectivity
Appl Categor Struct DOI 10.1007/s10485-008-9165-0
Essentiality and Injectivity H. Barzegar · M. M. Ebrahimi · M. Mahmoudi
Received: 9 January 2008 / Accepted: 9 September 2008 © Springer Science + Business Media B.V. 2008
Abstract Essentiality is an important notion closely related to injectivity. Depending on a class M of morphisms of a category A, three different types of essentiality are considered in literature. Each has its own benefits in regards with the behaviour of M-injectivity. In this paper we intend to study these different notions of essentiality and to investigate their relations to injectivity and among themselves. We will see, among other things, that although these essential extensions are not necessarily equivalent, they behave almost equivalently with regard to injectivity. Keywords Essential extension · Injectivity · Injective hull Mathematics Subject Classifications (2000) 08B30 · 18G05
1 Introduction The general theory of algebras and categories borrow techniques, ideas, and inspiration from older, more specialized branches of mathematics such as groups, rings, and modules. The notion of injectivity is a useful notion in any category and essentiality is a notion tightly related to it; see, for example, [3, 6, 9], and Ebrahimi et al. (unpublished manuscripts).
H. Barzegar · M. M. Ebrahimi (B) · M. Mahmoudi Department of Mathematics and Center of Excellence in Algebraic and Logical Structures in Discrete Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran e-mail: [email protected] H. Barzegar e-mail: [email protected] M. Mahmoudi e-mail: [email protected]
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Let M be a class of morphisms of a category A (for simplicity we let M to be a subclass of the class Mono of monomorphisms) which is isomorphism closed (contains all isomorphisms and is closed under composition with isomorphisms). If m : A → B ∈ M, we say that A, or m, is an M-subobject of B and B, or m, is an M-extension of A. Recall the usual pre-order relation on the class of all M-extensions of an object A: For M-extensions m : A A1 and n : A A2 of A, m ≤ n ⇔ ∃ f : A1 → A2 such that f m = n One can easily see that the relation m∼n⇔m≤n & n≤m is an equivalence relation and if M is right regular (for every f ∈ M and a morphism k, k is an isomorphism whenever kf = f ) then m ∼ n if and only if A1 A2 . Also the above pre-order gives a partial order on the class of all M-extensions of A (modulo isomorphism). So, from now on, M-morphisms refer to these partially ordered classes. Recall the following types of essential extensions usually used in literature: m
Definition 1.1 An M-morphism M X will be called: m
f
m
f
m
f
(1) Me1 -essential if:
M X → Y ∈ M ⇒ f ∈ M.
(2) Me2 -essential if:
M X → Y ∈ Mono ⇒ f ∈ Mono.
(3) Me3 -essential if:
M X → Y ∈ M ⇒ f ∈ Mono.
Notice that, Me1 -essentiality is usually used for an arbitrary class M of morphisms in an arbitrary category (see [3, 6, 9]); Me2 is the one usually used in Universal Algebra; and Me3 have been used, for example, when M is the class of pure monomorphisms (see [2, 4, 7, 8], Barzegar and Ebrahimi, unpublished manuscript). Note that we will use the same notation Mei (i = 1,2,3) for the classes of the above types of essential monomorphisms as well as for the prefix of any notion related to them. Thus, for example, Mei -injective hull refers to M-injective hull defined with respect to Mei -essentiality. Remark 1.2 In his recent paper [10], Walter Tholen takes right regularity to be another type of essentiality. His Proposition 2.1 shows that Me1 implies right regularity and under some mild condition they are equivalent. Let us also recall that Bernhard Banaschewski in [3] says that injectivity “Well Behaves” if the following propositions hold: Proposition 1.3 (I) Let M be a class of morphisms in a category A. Then, for every A ∈ A, the following conditions are equivalent: (I1) A is M-injective. (I2) A is M-absolute retract. (I3) A has no proper Me1 -essential extension.
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Proposition 1.4 (E) Every A ∈ A has an Me1 -injective hull. Proposition 1.5 (H) For an M-extension B of A, the following conditions are equivalent: (H1) B is an Me1 -injective hull of A. (H2) B is a maximal Me1 -essential extension of A. (H3) B is a minimal M-injective M-extension of A. He then gives some set of sufficient, but not necessary, conditions for the Well Behaviour of injectivity. Mehdi Ebrahimi in [6] and Walter Tholen in [9] further study these propositions, but again considering only Me1 -essentiality. Here, we first see that the above classes of essential extensions are not generally the same, and give some sufficient conditions under which they are the same. Then the three above mentioned Well Behaviour Theorems of Injectivity with respect to all the above types of essentialities are studied. Finally, we compare the behaviour of injectivity with respect to these notions of essentiality, and show that they behave almost equivalently with respect to injectivity.
2 Essentiality In this section, giving sufficient conditions, and examples, under which the classes of essential extensions mentioned in Definition 1.1 are the same, we mention examples showing that the notion of Me3 -essential is strictly weaker than the other two and these three classes are actually different. Definition 2.1 Let E and M be two classes of morphisms of a category A. We say that (i) A has (unique) (E , M)-factorization if every morphism f can be decomposed (uniquely) as me where e ∈ E and m ∈ M. (ii) A fulfills Mei -Banaschewski’s condition if for every f ∈ M there is a morphism g ∈ A with g f ∈ Mei . (iii) A has M-transferability if any pair f : A → B, g : A → C, with f ∈ M, can be completed to the following commutative diagram with u ∈ M f
A → B g↓ ↓v u C → D Notice that if A has pushouts, then the above M-transferability is equivalent to pushouts transferring M-morphisms. Remark 2.2 If A has (E , Mono)-factorization, f : A → B ∈ M is an Me2 -essential extension if and only if it is so on E -morphisms; in the sense that an E -morphism g : B → C is a monomorphism whenever g f ∈ Mono. The same is true for Me3 essentiality whenever M is left cancellable, in the sense that g f ∈ M implies f ∈ M. For, let f be Me3 -essential extension on E -morphisms and g f ∈ M. Factorizing g as
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me and using the left cancellability of M, we get ef ∈ M from mef ∈ M. Now, Me3 essentiality on E -morphisms implies that e, and hence g, is a monomorphism. The same result holds for Me1 when A has (E , M)-factorization and M is composition closed and left cancellable. One can easily see that each Mei (i = 1, 2) is a subclass of Me3 and Me3 = Me1 if (E31 ): f, g f ∈ M, with g ∈ Mono yields g ∈ M. Also we have Lemma 2.3 If any of the following conditions is satisfied, then Me3 = Me2 : m
f
(E32 ) A has an (E , Mono)-factorization and if M X → Y ∈ Mono, with m ∈ M and f ∈ E , then f m ∈ M. (E32 ) For every monomorphism f : A → B there exists a morphism g such that g f ∈ M. (E32 ) Me2 -Banaschewski’s condition holds and A has Mono-transferability. hold, f be an Me3 -essential morphism, and g f be a monomorphism. Proof Let E32 By Me2 -Banaschewski’s condition, there exists a morphism h such that hf is in Me2 . The morphism f being in Me3 implies that h is a monomorphism and hence, by Mono-transferability, there exists a triple (P, p, q) for which pg = qh and p ∈ Mono. Since hf is in Me2 and qhf = pg f is a monomorphism, so is qh = pg. Thus, g is a monomorphism and the proof is complete.
Example 2.4 (i) As we have already mentioned, if M is taken to be the whole class Mono of monomorphisms, all the three notions of Mei -essentiality are equivalent. To see another such instance, consider the class M of all dense monomorphisms with respect to a (categorical) closure operator on A (see [5]). Let A have an (E , Mono)-factorization. Since M satisfies the conditions (E31 ) and (E32 ) given above (see the easy Exercise 2.F(b) in [5]) we get that all the three essential notions are equivalent. (ii) In the following example of M, the three classes of Mei -essential extensions are actually different (see also Theorem 4.2 of Barzegar and Ebrahimi, unpublished manuscript). Recall that, for a semigroup S, a set A is an S-act if there is an action μ: A×S→ A (a, s) → as := μ(a, s), satisfying a(st) = (as)t. An S-system of equations, or an S-sequence, = {xs = as | s ∈ S, as ∈ A} on an S-act A is said to be solvable in an extension B of A if there is some b ∈ B such that for all s ∈ S, b s = as . Now, we say that A is sequentially pure or (s-pure) in an extension B of A if every is solvable in A whenever it is solvable in B. A monomorphism f : A → B is s-pure if f (A) is s-pure in B. Now, let M be the class of all s-pure monomorphisms of S-acts. If S is a left zero semigroup (st = s) with at least two elements s0 , s1 , and B = {a0 , a1 , 0, b } is the right S-act in which a0 , a1 , 0 are zero (fixed) elements and
Essentiality and injectivity
bs0 = a0 , bs1 = a1 and bs = 0 for all other elements of S, then A = {a0 , a1 } is Me3 essential in B which is not an Me2 -essential. If S is a decomposable semigroup as I ∪ J = S, for non-empty ideals I and J of S with I ∩ J = ∅, and A is an S-act which does not have a zero, then A is Me2 -essential (and hence, Me3 -essential) in A0 = A ∪ {0} which is not Me1 -essential. We close this section by giving some easily proved properties of the classes Mei needed in the sequel. Lemma 2.5 Let M be closed under composition. Then: (i) Each class Mei is isomorphism closed. (ii) Both classes Mei (i=1,2) are closed under composition. (iii) For f, g ∈ M, if g f ∈ Mei , then so is g, and also so is f if A satisfies Mtransferability for i = 3, M-transferability, and M is left cancellable for i = 1, and Mono-transferability for i = 2. (iv) Let f : A → E be an M-morphism and E be an injective object such that there is a monomorphism g : E → C with g f an Mei -essential (i = 1, 2, 3). Then f is an Mei -essential (i = 1, 2, 3), too. Proof We just prove (iii) and (iv). (iii) Essentiality of g is easy to check. We prove that f is essential. Let hf ∈ M. By M-transferability, there is a triple (P, p, q) such that pg = qh and q ∈ M. Then p ∈ Mono. Now commutativity of M-transferability diagram implies h ∈ Mono. The proof for i = 1, 2 is similar. (iv) Since E is injective, it is absolute retract. Consider h as a left inverse of g. So hg f = f , and hence h ∈ Mono, by essentiality of g f . Thus h, and so g, is an isomorphism. Now that f = g−1 (g f ), (i) proves the result.
3 Injectivity and Essentiality In this section we discuss the three Well-Behaviour Theorems of Injectivity with respect to all the above three types of essentiality. The results regarding the first definition of essentiality, Me1 , can be compared to those in [3, 9], and Ebrahimi et al. (unpublished manuscripts). Since Banaschewski’s condition is crucial in the study of the well-behaviour of injectivity, let us first recall the following notion which is, in some situations, a stronger condition (see [9]). Definition 3.1 We say that A fulfills the M-chain condition if for every direct system ((Aα )α∈I , ( fαβ )α≤β∈I ) whose index set I is a well-ordered chain with a least element 0, and f0α ∈ M for all α, there is a (so called “upper bound”) family (hα : Aα → A)α∈I with h0 ∈ M and hβ fαβ = hα . Theorem 3.2 Let E be a class of morphisms in A which is isomorphism closed, composition closed and right regular. If A is E -cowell-powered and has unique
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(E , M)-factorization ((E , Mono)-factorization and M is left cancellable) and M fulfills M-chain condition, then A satisfies Me1 (Me2 )-Banaschewski’s condition. Proof Let f : A → B be an M-morphism and consider = {g : B → C | g ∈ E , g f ∈ M}. One can show, by Zorn’s Lemma, that has a maximal element g : B → D, and g f is Me1 -essential. The proof for Me2 -essential is similar.
Corollary 3.3 Let A be E pi-cowell-powered and have unique (E pi, M)-factorization ((E pi, Mono)-factorization). Then, the M-chain condition implies Me1 (Me2 )Banaschewski’s condition. Now we discuss the First Theorem of Injectivity (with respect to all the three types of essentialities). First recall the following notions. Definition 3.4 (i) An object A is M-injective if it is injective with respect to M-morphisms, that is, for every M-morphism f : B → C and morphism g : B → A there exists a morphism h : C → A such that hf = g. (ii) An object A is said to be an M-retract (or simply a retract if M = Mono) of its M-extension f : A → B if it has a left inverse g : B → A, called a retraction. An object A is said to be M-absolutely retract if it is an M-retract of each of its M-extensions. The following lemma, the proof of which is standard using Mei -Banaschewski’s condition, gives all the ingredients of the First Theorem of Injectivity. Lemma 3.5 (i) Every M-injective object is M-absolutely retract. The converse is true if Mtransferability is satisfied (or if A has enough M-injectives). (ii) Every M-absolutely retract object has no proper Mei -essential extension (i=1,2,3). The converse is true if Mei -Banaschewski’s condition is satisfied. Theorem 3.6 (The First Theorem of Injectivity) If A fulfills Mei -Banaschewski’s condition (i = 1, 2, 3) and has M-transferability, then the following are equivalent for each object A: (i) A is M-injective. (ii) A is M-absolutely retract. (iii) A has no proper Mei -essential extension (i = 1, 2, 3). To discuss the Second Theorem of Injectivity, recall the following definitions. Definition 3.7 (i) An Mei -injective hull of an object A is an Mei -essential extension f : A → E in which E is M-injective. (ii) An Mei -essential morphism f is said to be maximal if g f ∈ Mei implies that g is an isomorphism.
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(iii) An M-injective M-extension f : A → B of A is said to be minimal if any Mmorphism g : C → B from an M-injective M-extension h : A → C to B (that is, with gh = f ) is an isomorphism. We are now ready to give the Second Theorem of Injectivity as follows. Although some parts of the theorem can be combined but we prefer not to do so, showing exactly which condition is used for what statement. Theorem 3.8 (The Second Theorem of Injectivity) (i) If E is an M-absolutely retract Me1 -essential extension of A, then E is a maximal Me1 -essential extension of A. (ii) Let A satisfy Mei -Banaschewski’s condition and M be composition closed. Then any maximal Mei -essential extension of A is M-absolutely retract. (iii) An Mei -injective hull of A is a maximal Mei -essential extension of A, where for i = 2, 3, Mei (or M) is assumed to be right regular. (iv) Let A satisfy Mei -Banaschewski’s condition and M-transferability, and M be composition closed. Then every maximal Mei -essential extension of A is an Mei -injective hull of A. (v) If M is composition closed, then every Mei -injective hull of A is a minimal M-injective extension of A. (vi) If A has an Me1 -injective hull, then every minimal M-injective M-extension of A is an Me1 -injective hull of A. (vii) Let Mei (or M) be right regular. If the category A has Mei -injective hulls and M is left cancellable, then every minimal M-injective M-extension is an Mei injective hull. Proof (i) Let f : A → E ∈ M with E an M-absolutely retract and g f be in Me1 . Then g ∈ M and hence it has a left inverse h. Since g f is an Me1 -essential monomorphism, h, and thus g, is an isomorphism. (ii) Let f : A → E be a maximal Mei -essential extension of A and g : E → B be an M-morphism. By Mei -Banaschewski’s condition, there is a morphism h : B → C such that hg f is an Mei -essential monomorphism. By maximality of f , hg is an isomorphism and hence g has a left inverse. (iii) For Me1 apply Lemma 3.5 (i) and (i) above. For Me3 , let f : A → E be an Me3 -injective hull and g : E → C be a morphism with g f ∈ Me3 . Then, by M-injectivity of E, there is a morphism k : C → E such that kg f = f . Thus, by right regularity, kg is an isomorphism, and so g has a left inverse h. Now, the rest of the proof is like (i). The case of Me2 is proved similarly. (iv) Apply (ii) above and the First Theorem of Injectivity. (v) Let f : A → E be an Mei -injective hull of A and f = gh for h : A → C, g : C → E ∈ M where C is M-injective. Since C is an M-injective object, g has a left inverse k, by Lemma 3.5. By Lemma 2.5(iii), g is an Mei -essential extension, so k and hence g is an isomorphism. (vi) Let f : A → E be a minimal M-injective M-extension of A and g : A → E1 be an Me1 -injective hull of A. So there exists a morphism k : E1 → E such that kg = f , which is in M. By minimality, k is an isomorphism.
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(vii) Let f : A → E be a minimal M-injective M-extension of A and g : A → E1 be an Mei -injective hull of A. Then, there are morphisms h : E1 → E and k : E → E1 such that hg = f and kf = g. Thus, (kh)g = g belongs to Mei , and so by (iii), kh is an isomorphism. Now h ∈ M, by left cancellability, and minimality of f implies that h is an isomorphism.
Now we are going to discuss the Third Theorem of Injectivity. First recall the following notion from [9]. Definition 3.9 (i) We say that A has M-bounds if any small (non-empty) family (hα : A → Bα )α∈I of M-morphisms has an upper bound; that is, there is an M-morphism h : A → B which factors over all hα ,s. (ii) The category A is Mei -cowell-powered if every object A admits only a set of Mei -essential extensions. Lemma 3.10 (i) Let Mei (or M) be right regular and A be Mei -cowell-powered and fulfill Mei -Banaschewski’s condition. If A has M-bounds, then every A ∈ A has a maximal Mei -essential extension. (ii) Let Mei (or M) be right regular and M be composition closed. If A satisfies Mei -Banaschewski’s condition and M-transferability, then any two maximal Mei -essential extensions of A are isomorphic. (iii) Let A be Me1 -cowell-powered and fulfill Me1 -Banaschewski’s condition and Me1 (or M) be right regular. If A has M-bounds and satisfies M-transferability then any two minimal M-injective extensions of A are isomorphic. Proof (i) Given A, put = { fα : A → Aα | α ∈ I, fα ∈ Mei }. By cowell-poweredness, is a non-empty poset under the usual order given in Introduction. Since A has M bounds, any chain { fαβ } in has an upper bound h : A → Y in M. Now, by Mei -Banaschewski’s condition, there is a morphism g : Y → B such that gh is in Mei , and so gh is an upper bound of the given chain in . By Zorn’s Lemma, has a maximal element, say f , which can be shown to be a maximal Mei -essential extension of A. In fact, g f ∈ Mei gives f ≤ g f and hence, by maximality of f , f = g f and so, by right regularity, g is an isomorphism. (ii) Let f : A → E1 and g : A → E2 be two maximal Mei -essential extensions of A. By M-transferability, there are morphisms p, q such that pg = q f and q ∈ M. By composition closedness of M and Mei - Banaschewski’s condition, there is a morphism h such that hq f is an Mei -essential extension and hence hq is an isomorphism. Since hpg = hq f , again by maximality of g, hp is an isomorphism. Therefore E1 is isomorphic to E2 . (iii) Let f : A → E and f : A → E be two minimal M-injective M-extensions of A. By (i) above, there exists g : A → E1 which is a maximal Me1 -essential extension of A. It can be shown that E1 has no proper Me1 -essential extension, and hence, by the First Theorem of Injectivity 3.6, E1 is M-injective. On the other hand, there exists k : E1 → E such that kg = f . Since g is Me1 -essential,
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k is an M-morphism. Now, by minimality of f , k is an isomorphism. So E and E1 and, similarly, E and E are isomorphic.
Now, as a consequence of the above results, one can easily get the Third Theorem of Injectivity stated as follows. Theorem 3.11 (The Third Theorem of Injectivity) Let M be composition closed and Mei (or M) be right regular. Then (i) A has enough Mei -injective hulls if and only if A has M-bounds, fulfills Mei Banaschewski’s condition, has M-transferability, and is Mei -cowell-powered. (ii) Any two Mei -injective hulls of an object A are isomorphic. Proof (i) The only part is clear and for the converse apply Lemma 3.10 and Theorem 3.8(iv). (ii) Suppose f and g are two Mei -injective hulls of A. By Theorem 3.8(iii), f is a maximal Mei -essential extension. Now, since there exists a morphism k such that kf = g, we get that k is an isomorphism.
4 Conclusion In this final section we compare the behaviour of injectivity with respect to the three notions of essentiality and conclude that M-injectivity actually behaves almost equivalently with respect to them. Theorem 4.1 Let Mei (i = 1, 2, 3) (or M) be right regular and M be left cancellable. Then M-injectivity is well behaved for Mei -essential if and only if A has enough Mei injective hulls. Proof We just prove that M is composition closed; the rest follows from what we have proved so far. Let f : A → B and g : B → C be two M-morphisms and h : A → E be an Mei -injective hull of A. There exists h1 : B → E, and so h2 : C → E, such that h1 f = h and h2 g = h1 . Then h2 g f = h ∈ M. Now, Since M is left
cancellable, g f ∈ M. We trivially get the following lemma from Lemma 2.3. Lemma 4.2 If A has Mono-transferability, then A fulfills Me2 -Banaschewski’s condition if and only if (i) A fulfills Me3 -Banaschewski’s condition. (ii) Me3 = Me2 . Lemma 4.3 (i) Let A fulfill Me1 -Banaschewski’s condition. Then, E is a maximal Me3 -essential extension of A if and only if E is a maximal Me1 -essential extension of A.
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(ii) Let A fulfill Me2 -Banaschewski’s condition. Then, E is a maximal Me3 -essential extension of A if and only if E is a maximal Me2 -essential extension of A. Proof (i) (⇒) Let f : A → E be a maximal Me3 -essential extension of A. By Me1 Banaschewski’s condition, there exists g : E → C such that g f is in Me1 . By the maximality of f , g is an isomorphism and so f is in Me1 . Now, since f is a maximal Me3 -essential extension, it is clear that f is a maximal Me1 -essential extension. (⇐) Let f : A → E be a maximal Me1 -essential extension. Then, it is in Me3 . Suppose g f is in Me3 . By Me1 -Banaschewski’s condition, there exists h such that hg f is in Me1 . So, h is a monomorphism and hg is an isomorphism. Therefore h, and hence g, is an isomorphism. (ii) The proof is similar to the proof of (i).
Corollary 4.4 (i) If A does not have proper Me3 -essential extension, then it does not have, both, proper Me1 -essential extension and proper Me2 -essential extension. (ii) If A does not have proper Mei -essential extension (i = 1, 2) and satisfies Mei Banaschewski’s condition (i = 1, 2), then it does not have proper Me3 -essential extension. Proof Part (i) is easy. To prove part (ii), since A does not have proper Mei -essential extensions (for i=1,2), we get that every g for which g = gid A is an Mei -essential, is an isomorphism. So id A is a maximal Mei -essential. Now by Lemma 4.3, id A is a maximal Me3 -essential. Let f : A → B be an Me3 -essential. Then f id A is so and thus f is an isomorphism.
Lemma 4.5 (i) Every Mei -injective hull (i = 1, 2) of A is an Me3 -injective hull of A. (ii) If A satisfies Mei -Banaschewski’s condition (i = 1, 2), then every Me3 -injective hull of A is an Mei -injective hull (i = 1, 2) of A. Proof (i) is clear. To get (ii), let f be an Me3 -injective hull. Then it is a maximal Me3 essential extension, by Theorem 3.8(iii). By Mei -Banaschewski’s condition (i = 1, 2) there is a morphism g such that g f is an Mei -essential (i=1,2) and hence it is Me3 essential. Then g is an isomorphism and thus f is Mei -essential.
Now we close the paper with the final result showing that M-injectivity behaves almost equivalently with respect to the three different notions of essentiality. The proof of this theorem follows from Theorem 4.1 and Lemma 4.5. We should remark that, although the left cancellability mentioned in the statement of the theorem is not a strong condition, the result can also be proved without this, but using some other results of this paper. Theorem 4.6 Let Mei (i = 1, 2, 3) (or M) be right regular and M be left cancellable. Then (i) Injectivity well behaves with respect to Me1 if and only if it well behaves for Me3 and fulfills Me1 -Banaschewski’s condition.
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(ii) Injectivity well behaves with respect to Me2 if and only if it well behaves for Me3 and fulfills Me2 -Banaschewski’s condition. (iii) Injectivity well behaves for Me1 and fulfills Me2 -Banaschewski’s condition if and only if it well behaves for Me2 and fulfills Me1 -Banaschewski’s condition. Acknowledgements The authors acknowledge the support from Shahid Beheshti University. Also their sincere thanks goes to the referee for reading the paper at least twice and giving very helpful suggestions, and to Professor Walter Tholen for pointing us his recent paper [10] (see Remark 1.2).
References 1. Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990) 2. Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, New York (1974) 3. Banaschewski, B.: Injectivity and essential extensions in equational classes of algebras. Queen Pap. Pure Appl. Math. 25, 131–147 (1970) 4. Banaschewski, B.: Equational compactness of G-sets. Canad. Math. Bull. 17(3), 11–18 (1974) 5. Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators, with Applications to Topology, Algebra, and Discrete Mathematics. Mathematics and its Applications. Kluwer, Dordrecht (1995) 6. Ebrahimi, M.M.: Algebra in a Grothendieck topos: injectivity in quasi-equational classes. J. Pure Appl. Algebra 26(3), 269–280 (1982) 7. Ebrahimi, M.M.: Equational compactness of sheaves of algebras on a Noetherian locale. Algebra Universalis (16), 318–330 (1983) 8. Mahmoudi, M., Ebrahimi, M.M.: Purity and equational compactness of projection algebras. Appl. Categ. Structures 9(4), 381–394 (2001) 9. Tholen, W.: Injective objects and cogenerating sets. J. Algebra 73(1), 139–155 (1981) 10. Tholen, W.: Injectivity versus exponentiability. http://www.math.yorku.ca/∼tholen/Injectivity_vs _exponentiability.pdf (2008)
Essentiality and Injectivity H. Barzegar · M. M. Ebrahimi · M. Mahmoudi
Received: 9 January 2008 / Accepted: 9 September 2008 © Springer Science + Business Media B.V. 2008
Abstract Essentiality is an important notion closely related to injectivity. Depending on a class M of morphisms of a category A, three different types of essentiality are considered in literature. Each has its own benefits in regards with the behaviour of M-injectivity. In this paper we intend to study these different notions of essentiality and to investigate their relations to injectivity and among themselves. We will see, among other things, that although these essential extensions are not necessarily equivalent, they behave almost equivalently with regard to injectivity. Keywords Essential extension · Injectivity · Injective hull Mathematics Subject Classifications (2000) 08B30 · 18G05
1 Introduction The general theory of algebras and categories borrow techniques, ideas, and inspiration from older, more specialized branches of mathematics such as groups, rings, and modules. The notion of injectivity is a useful notion in any category and essentiality is a notion tightly related to it; see, for example, [3, 6, 9], and Ebrahimi et al. (unpublished manuscripts).
H. Barzegar · M. M. Ebrahimi (B) · M. Mahmoudi Department of Mathematics and Center of Excellence in Algebraic and Logical Structures in Discrete Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran e-mail: [email protected] H. Barzegar e-mail: [email protected] M. Mahmoudi e-mail: [email protected]
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Let M be a class of morphisms of a category A (for simplicity we let M to be a subclass of the class Mono of monomorphisms) which is isomorphism closed (contains all isomorphisms and is closed under composition with isomorphisms). If m : A → B ∈ M, we say that A, or m, is an M-subobject of B and B, or m, is an M-extension of A. Recall the usual pre-order relation on the class of all M-extensions of an object A: For M-extensions m : A A1 and n : A A2 of A, m ≤ n ⇔ ∃ f : A1 → A2 such that f m = n One can easily see that the relation m∼n⇔m≤n & n≤m is an equivalence relation and if M is right regular (for every f ∈ M and a morphism k, k is an isomorphism whenever kf = f ) then m ∼ n if and only if A1 A2 . Also the above pre-order gives a partial order on the class of all M-extensions of A (modulo isomorphism). So, from now on, M-morphisms refer to these partially ordered classes. Recall the following types of essential extensions usually used in literature: m
Definition 1.1 An M-morphism M X will be called: m
f
m
f
m
f
(1) Me1 -essential if:
M X → Y ∈ M ⇒ f ∈ M.
(2) Me2 -essential if:
M X → Y ∈ Mono ⇒ f ∈ Mono.
(3) Me3 -essential if:
M X → Y ∈ M ⇒ f ∈ Mono.
Notice that, Me1 -essentiality is usually used for an arbitrary class M of morphisms in an arbitrary category (see [3, 6, 9]); Me2 is the one usually used in Universal Algebra; and Me3 have been used, for example, when M is the class of pure monomorphisms (see [2, 4, 7, 8], Barzegar and Ebrahimi, unpublished manuscript). Note that we will use the same notation Mei (i = 1,2,3) for the classes of the above types of essential monomorphisms as well as for the prefix of any notion related to them. Thus, for example, Mei -injective hull refers to M-injective hull defined with respect to Mei -essentiality. Remark 1.2 In his recent paper [10], Walter Tholen takes right regularity to be another type of essentiality. His Proposition 2.1 shows that Me1 implies right regularity and under some mild condition they are equivalent. Let us also recall that Bernhard Banaschewski in [3] says that injectivity “Well Behaves” if the following propositions hold: Proposition 1.3 (I) Let M be a class of morphisms in a category A. Then, for every A ∈ A, the following conditions are equivalent: (I1) A is M-injective. (I2) A is M-absolute retract. (I3) A has no proper Me1 -essential extension.
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Proposition 1.4 (E) Every A ∈ A has an Me1 -injective hull. Proposition 1.5 (H) For an M-extension B of A, the following conditions are equivalent: (H1) B is an Me1 -injective hull of A. (H2) B is a maximal Me1 -essential extension of A. (H3) B is a minimal M-injective M-extension of A. He then gives some set of sufficient, but not necessary, conditions for the Well Behaviour of injectivity. Mehdi Ebrahimi in [6] and Walter Tholen in [9] further study these propositions, but again considering only Me1 -essentiality. Here, we first see that the above classes of essential extensions are not generally the same, and give some sufficient conditions under which they are the same. Then the three above mentioned Well Behaviour Theorems of Injectivity with respect to all the above types of essentialities are studied. Finally, we compare the behaviour of injectivity with respect to these notions of essentiality, and show that they behave almost equivalently with respect to injectivity.
2 Essentiality In this section, giving sufficient conditions, and examples, under which the classes of essential extensions mentioned in Definition 1.1 are the same, we mention examples showing that the notion of Me3 -essential is strictly weaker than the other two and these three classes are actually different. Definition 2.1 Let E and M be two classes of morphisms of a category A. We say that (i) A has (unique) (E , M)-factorization if every morphism f can be decomposed (uniquely) as me where e ∈ E and m ∈ M. (ii) A fulfills Mei -Banaschewski’s condition if for every f ∈ M there is a morphism g ∈ A with g f ∈ Mei . (iii) A has M-transferability if any pair f : A → B, g : A → C, with f ∈ M, can be completed to the following commutative diagram with u ∈ M f
A → B g↓ ↓v u C → D Notice that if A has pushouts, then the above M-transferability is equivalent to pushouts transferring M-morphisms. Remark 2.2 If A has (E , Mono)-factorization, f : A → B ∈ M is an Me2 -essential extension if and only if it is so on E -morphisms; in the sense that an E -morphism g : B → C is a monomorphism whenever g f ∈ Mono. The same is true for Me3 essentiality whenever M is left cancellable, in the sense that g f ∈ M implies f ∈ M. For, let f be Me3 -essential extension on E -morphisms and g f ∈ M. Factorizing g as
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me and using the left cancellability of M, we get ef ∈ M from mef ∈ M. Now, Me3 essentiality on E -morphisms implies that e, and hence g, is a monomorphism. The same result holds for Me1 when A has (E , M)-factorization and M is composition closed and left cancellable. One can easily see that each Mei (i = 1, 2) is a subclass of Me3 and Me3 = Me1 if (E31 ): f, g f ∈ M, with g ∈ Mono yields g ∈ M. Also we have Lemma 2.3 If any of the following conditions is satisfied, then Me3 = Me2 : m
f
(E32 ) A has an (E , Mono)-factorization and if M X → Y ∈ Mono, with m ∈ M and f ∈ E , then f m ∈ M. (E32 ) For every monomorphism f : A → B there exists a morphism g such that g f ∈ M. (E32 ) Me2 -Banaschewski’s condition holds and A has Mono-transferability. hold, f be an Me3 -essential morphism, and g f be a monomorphism. Proof Let E32 By Me2 -Banaschewski’s condition, there exists a morphism h such that hf is in Me2 . The morphism f being in Me3 implies that h is a monomorphism and hence, by Mono-transferability, there exists a triple (P, p, q) for which pg = qh and p ∈ Mono. Since hf is in Me2 and qhf = pg f is a monomorphism, so is qh = pg. Thus, g is a monomorphism and the proof is complete.
Example 2.4 (i) As we have already mentioned, if M is taken to be the whole class Mono of monomorphisms, all the three notions of Mei -essentiality are equivalent. To see another such instance, consider the class M of all dense monomorphisms with respect to a (categorical) closure operator on A (see [5]). Let A have an (E , Mono)-factorization. Since M satisfies the conditions (E31 ) and (E32 ) given above (see the easy Exercise 2.F(b) in [5]) we get that all the three essential notions are equivalent. (ii) In the following example of M, the three classes of Mei -essential extensions are actually different (see also Theorem 4.2 of Barzegar and Ebrahimi, unpublished manuscript). Recall that, for a semigroup S, a set A is an S-act if there is an action μ: A×S→ A (a, s) → as := μ(a, s), satisfying a(st) = (as)t. An S-system of equations, or an S-sequence, = {xs = as | s ∈ S, as ∈ A} on an S-act A is said to be solvable in an extension B of A if there is some b ∈ B such that for all s ∈ S, b s = as . Now, we say that A is sequentially pure or (s-pure) in an extension B of A if every is solvable in A whenever it is solvable in B. A monomorphism f : A → B is s-pure if f (A) is s-pure in B. Now, let M be the class of all s-pure monomorphisms of S-acts. If S is a left zero semigroup (st = s) with at least two elements s0 , s1 , and B = {a0 , a1 , 0, b } is the right S-act in which a0 , a1 , 0 are zero (fixed) elements and
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bs0 = a0 , bs1 = a1 and bs = 0 for all other elements of S, then A = {a0 , a1 } is Me3 essential in B which is not an Me2 -essential. If S is a decomposable semigroup as I ∪ J = S, for non-empty ideals I and J of S with I ∩ J = ∅, and A is an S-act which does not have a zero, then A is Me2 -essential (and hence, Me3 -essential) in A0 = A ∪ {0} which is not Me1 -essential. We close this section by giving some easily proved properties of the classes Mei needed in the sequel. Lemma 2.5 Let M be closed under composition. Then: (i) Each class Mei is isomorphism closed. (ii) Both classes Mei (i=1,2) are closed under composition. (iii) For f, g ∈ M, if g f ∈ Mei , then so is g, and also so is f if A satisfies Mtransferability for i = 3, M-transferability, and M is left cancellable for i = 1, and Mono-transferability for i = 2. (iv) Let f : A → E be an M-morphism and E be an injective object such that there is a monomorphism g : E → C with g f an Mei -essential (i = 1, 2, 3). Then f is an Mei -essential (i = 1, 2, 3), too. Proof We just prove (iii) and (iv). (iii) Essentiality of g is easy to check. We prove that f is essential. Let hf ∈ M. By M-transferability, there is a triple (P, p, q) such that pg = qh and q ∈ M. Then p ∈ Mono. Now commutativity of M-transferability diagram implies h ∈ Mono. The proof for i = 1, 2 is similar. (iv) Since E is injective, it is absolute retract. Consider h as a left inverse of g. So hg f = f , and hence h ∈ Mono, by essentiality of g f . Thus h, and so g, is an isomorphism. Now that f = g−1 (g f ), (i) proves the result.
3 Injectivity and Essentiality In this section we discuss the three Well-Behaviour Theorems of Injectivity with respect to all the above three types of essentiality. The results regarding the first definition of essentiality, Me1 , can be compared to those in [3, 9], and Ebrahimi et al. (unpublished manuscripts). Since Banaschewski’s condition is crucial in the study of the well-behaviour of injectivity, let us first recall the following notion which is, in some situations, a stronger condition (see [9]). Definition 3.1 We say that A fulfills the M-chain condition if for every direct system ((Aα )α∈I , ( fαβ )α≤β∈I ) whose index set I is a well-ordered chain with a least element 0, and f0α ∈ M for all α, there is a (so called “upper bound”) family (hα : Aα → A)α∈I with h0 ∈ M and hβ fαβ = hα . Theorem 3.2 Let E be a class of morphisms in A which is isomorphism closed, composition closed and right regular. If A is E -cowell-powered and has unique
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(E , M)-factorization ((E , Mono)-factorization and M is left cancellable) and M fulfills M-chain condition, then A satisfies Me1 (Me2 )-Banaschewski’s condition. Proof Let f : A → B be an M-morphism and consider = {g : B → C | g ∈ E , g f ∈ M}. One can show, by Zorn’s Lemma, that has a maximal element g : B → D, and g f is Me1 -essential. The proof for Me2 -essential is similar.
Corollary 3.3 Let A be E pi-cowell-powered and have unique (E pi, M)-factorization ((E pi, Mono)-factorization). Then, the M-chain condition implies Me1 (Me2 )Banaschewski’s condition. Now we discuss the First Theorem of Injectivity (with respect to all the three types of essentialities). First recall the following notions. Definition 3.4 (i) An object A is M-injective if it is injective with respect to M-morphisms, that is, for every M-morphism f : B → C and morphism g : B → A there exists a morphism h : C → A such that hf = g. (ii) An object A is said to be an M-retract (or simply a retract if M = Mono) of its M-extension f : A → B if it has a left inverse g : B → A, called a retraction. An object A is said to be M-absolutely retract if it is an M-retract of each of its M-extensions. The following lemma, the proof of which is standard using Mei -Banaschewski’s condition, gives all the ingredients of the First Theorem of Injectivity. Lemma 3.5 (i) Every M-injective object is M-absolutely retract. The converse is true if Mtransferability is satisfied (or if A has enough M-injectives). (ii) Every M-absolutely retract object has no proper Mei -essential extension (i=1,2,3). The converse is true if Mei -Banaschewski’s condition is satisfied. Theorem 3.6 (The First Theorem of Injectivity) If A fulfills Mei -Banaschewski’s condition (i = 1, 2, 3) and has M-transferability, then the following are equivalent for each object A: (i) A is M-injective. (ii) A is M-absolutely retract. (iii) A has no proper Mei -essential extension (i = 1, 2, 3). To discuss the Second Theorem of Injectivity, recall the following definitions. Definition 3.7 (i) An Mei -injective hull of an object A is an Mei -essential extension f : A → E in which E is M-injective. (ii) An Mei -essential morphism f is said to be maximal if g f ∈ Mei implies that g is an isomorphism.
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(iii) An M-injective M-extension f : A → B of A is said to be minimal if any Mmorphism g : C → B from an M-injective M-extension h : A → C to B (that is, with gh = f ) is an isomorphism. We are now ready to give the Second Theorem of Injectivity as follows. Although some parts of the theorem can be combined but we prefer not to do so, showing exactly which condition is used for what statement. Theorem 3.8 (The Second Theorem of Injectivity) (i) If E is an M-absolutely retract Me1 -essential extension of A, then E is a maximal Me1 -essential extension of A. (ii) Let A satisfy Mei -Banaschewski’s condition and M be composition closed. Then any maximal Mei -essential extension of A is M-absolutely retract. (iii) An Mei -injective hull of A is a maximal Mei -essential extension of A, where for i = 2, 3, Mei (or M) is assumed to be right regular. (iv) Let A satisfy Mei -Banaschewski’s condition and M-transferability, and M be composition closed. Then every maximal Mei -essential extension of A is an Mei -injective hull of A. (v) If M is composition closed, then every Mei -injective hull of A is a minimal M-injective extension of A. (vi) If A has an Me1 -injective hull, then every minimal M-injective M-extension of A is an Me1 -injective hull of A. (vii) Let Mei (or M) be right regular. If the category A has Mei -injective hulls and M is left cancellable, then every minimal M-injective M-extension is an Mei injective hull. Proof (i) Let f : A → E ∈ M with E an M-absolutely retract and g f be in Me1 . Then g ∈ M and hence it has a left inverse h. Since g f is an Me1 -essential monomorphism, h, and thus g, is an isomorphism. (ii) Let f : A → E be a maximal Mei -essential extension of A and g : E → B be an M-morphism. By Mei -Banaschewski’s condition, there is a morphism h : B → C such that hg f is an Mei -essential monomorphism. By maximality of f , hg is an isomorphism and hence g has a left inverse. (iii) For Me1 apply Lemma 3.5 (i) and (i) above. For Me3 , let f : A → E be an Me3 -injective hull and g : E → C be a morphism with g f ∈ Me3 . Then, by M-injectivity of E, there is a morphism k : C → E such that kg f = f . Thus, by right regularity, kg is an isomorphism, and so g has a left inverse h. Now, the rest of the proof is like (i). The case of Me2 is proved similarly. (iv) Apply (ii) above and the First Theorem of Injectivity. (v) Let f : A → E be an Mei -injective hull of A and f = gh for h : A → C, g : C → E ∈ M where C is M-injective. Since C is an M-injective object, g has a left inverse k, by Lemma 3.5. By Lemma 2.5(iii), g is an Mei -essential extension, so k and hence g is an isomorphism. (vi) Let f : A → E be a minimal M-injective M-extension of A and g : A → E1 be an Me1 -injective hull of A. So there exists a morphism k : E1 → E such that kg = f , which is in M. By minimality, k is an isomorphism.
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(vii) Let f : A → E be a minimal M-injective M-extension of A and g : A → E1 be an Mei -injective hull of A. Then, there are morphisms h : E1 → E and k : E → E1 such that hg = f and kf = g. Thus, (kh)g = g belongs to Mei , and so by (iii), kh is an isomorphism. Now h ∈ M, by left cancellability, and minimality of f implies that h is an isomorphism.
Now we are going to discuss the Third Theorem of Injectivity. First recall the following notion from [9]. Definition 3.9 (i) We say that A has M-bounds if any small (non-empty) family (hα : A → Bα )α∈I of M-morphisms has an upper bound; that is, there is an M-morphism h : A → B which factors over all hα ,s. (ii) The category A is Mei -cowell-powered if every object A admits only a set of Mei -essential extensions. Lemma 3.10 (i) Let Mei (or M) be right regular and A be Mei -cowell-powered and fulfill Mei -Banaschewski’s condition. If A has M-bounds, then every A ∈ A has a maximal Mei -essential extension. (ii) Let Mei (or M) be right regular and M be composition closed. If A satisfies Mei -Banaschewski’s condition and M-transferability, then any two maximal Mei -essential extensions of A are isomorphic. (iii) Let A be Me1 -cowell-powered and fulfill Me1 -Banaschewski’s condition and Me1 (or M) be right regular. If A has M-bounds and satisfies M-transferability then any two minimal M-injective extensions of A are isomorphic. Proof (i) Given A, put = { fα : A → Aα | α ∈ I, fα ∈ Mei }. By cowell-poweredness, is a non-empty poset under the usual order given in Introduction. Since A has M bounds, any chain { fαβ } in has an upper bound h : A → Y in M. Now, by Mei -Banaschewski’s condition, there is a morphism g : Y → B such that gh is in Mei , and so gh is an upper bound of the given chain in . By Zorn’s Lemma, has a maximal element, say f , which can be shown to be a maximal Mei -essential extension of A. In fact, g f ∈ Mei gives f ≤ g f and hence, by maximality of f , f = g f and so, by right regularity, g is an isomorphism. (ii) Let f : A → E1 and g : A → E2 be two maximal Mei -essential extensions of A. By M-transferability, there are morphisms p, q such that pg = q f and q ∈ M. By composition closedness of M and Mei - Banaschewski’s condition, there is a morphism h such that hq f is an Mei -essential extension and hence hq is an isomorphism. Since hpg = hq f , again by maximality of g, hp is an isomorphism. Therefore E1 is isomorphic to E2 . (iii) Let f : A → E and f : A → E be two minimal M-injective M-extensions of A. By (i) above, there exists g : A → E1 which is a maximal Me1 -essential extension of A. It can be shown that E1 has no proper Me1 -essential extension, and hence, by the First Theorem of Injectivity 3.6, E1 is M-injective. On the other hand, there exists k : E1 → E such that kg = f . Since g is Me1 -essential,
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k is an M-morphism. Now, by minimality of f , k is an isomorphism. So E and E1 and, similarly, E and E are isomorphic.
Now, as a consequence of the above results, one can easily get the Third Theorem of Injectivity stated as follows. Theorem 3.11 (The Third Theorem of Injectivity) Let M be composition closed and Mei (or M) be right regular. Then (i) A has enough Mei -injective hulls if and only if A has M-bounds, fulfills Mei Banaschewski’s condition, has M-transferability, and is Mei -cowell-powered. (ii) Any two Mei -injective hulls of an object A are isomorphic. Proof (i) The only part is clear and for the converse apply Lemma 3.10 and Theorem 3.8(iv). (ii) Suppose f and g are two Mei -injective hulls of A. By Theorem 3.8(iii), f is a maximal Mei -essential extension. Now, since there exists a morphism k such that kf = g, we get that k is an isomorphism.
4 Conclusion In this final section we compare the behaviour of injectivity with respect to the three notions of essentiality and conclude that M-injectivity actually behaves almost equivalently with respect to them. Theorem 4.1 Let Mei (i = 1, 2, 3) (or M) be right regular and M be left cancellable. Then M-injectivity is well behaved for Mei -essential if and only if A has enough Mei injective hulls. Proof We just prove that M is composition closed; the rest follows from what we have proved so far. Let f : A → B and g : B → C be two M-morphisms and h : A → E be an Mei -injective hull of A. There exists h1 : B → E, and so h2 : C → E, such that h1 f = h and h2 g = h1 . Then h2 g f = h ∈ M. Now, Since M is left
cancellable, g f ∈ M. We trivially get the following lemma from Lemma 2.3. Lemma 4.2 If A has Mono-transferability, then A fulfills Me2 -Banaschewski’s condition if and only if (i) A fulfills Me3 -Banaschewski’s condition. (ii) Me3 = Me2 . Lemma 4.3 (i) Let A fulfill Me1 -Banaschewski’s condition. Then, E is a maximal Me3 -essential extension of A if and only if E is a maximal Me1 -essential extension of A.
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(ii) Let A fulfill Me2 -Banaschewski’s condition. Then, E is a maximal Me3 -essential extension of A if and only if E is a maximal Me2 -essential extension of A. Proof (i) (⇒) Let f : A → E be a maximal Me3 -essential extension of A. By Me1 Banaschewski’s condition, there exists g : E → C such that g f is in Me1 . By the maximality of f , g is an isomorphism and so f is in Me1 . Now, since f is a maximal Me3 -essential extension, it is clear that f is a maximal Me1 -essential extension. (⇐) Let f : A → E be a maximal Me1 -essential extension. Then, it is in Me3 . Suppose g f is in Me3 . By Me1 -Banaschewski’s condition, there exists h such that hg f is in Me1 . So, h is a monomorphism and hg is an isomorphism. Therefore h, and hence g, is an isomorphism. (ii) The proof is similar to the proof of (i).
Corollary 4.4 (i) If A does not have proper Me3 -essential extension, then it does not have, both, proper Me1 -essential extension and proper Me2 -essential extension. (ii) If A does not have proper Mei -essential extension (i = 1, 2) and satisfies Mei Banaschewski’s condition (i = 1, 2), then it does not have proper Me3 -essential extension. Proof Part (i) is easy. To prove part (ii), since A does not have proper Mei -essential extensions (for i=1,2), we get that every g for which g = gid A is an Mei -essential, is an isomorphism. So id A is a maximal Mei -essential. Now by Lemma 4.3, id A is a maximal Me3 -essential. Let f : A → B be an Me3 -essential. Then f id A is so and thus f is an isomorphism.
Lemma 4.5 (i) Every Mei -injective hull (i = 1, 2) of A is an Me3 -injective hull of A. (ii) If A satisfies Mei -Banaschewski’s condition (i = 1, 2), then every Me3 -injective hull of A is an Mei -injective hull (i = 1, 2) of A. Proof (i) is clear. To get (ii), let f be an Me3 -injective hull. Then it is a maximal Me3 essential extension, by Theorem 3.8(iii). By Mei -Banaschewski’s condition (i = 1, 2) there is a morphism g such that g f is an Mei -essential (i=1,2) and hence it is Me3 essential. Then g is an isomorphism and thus f is Mei -essential.
Now we close the paper with the final result showing that M-injectivity behaves almost equivalently with respect to the three different notions of essentiality. The proof of this theorem follows from Theorem 4.1 and Lemma 4.5. We should remark that, although the left cancellability mentioned in the statement of the theorem is not a strong condition, the result can also be proved without this, but using some other results of this paper. Theorem 4.6 Let Mei (i = 1, 2, 3) (or M) be right regular and M be left cancellable. Then (i) Injectivity well behaves with respect to Me1 if and only if it well behaves for Me3 and fulfills Me1 -Banaschewski’s condition.
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(ii) Injectivity well behaves with respect to Me2 if and only if it well behaves for Me3 and fulfills Me2 -Banaschewski’s condition. (iii) Injectivity well behaves for Me1 and fulfills Me2 -Banaschewski’s condition if and only if it well behaves for Me2 and fulfills Me1 -Banaschewski’s condition. Acknowledgements The authors acknowledge the support from Shahid Beheshti University. Also their sincere thanks goes to the referee for reading the paper at least twice and giving very helpful suggestions, and to Professor Walter Tholen for pointing us his recent paper [10] (see Remark 1.2).
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