CATEGORICAL PROPERTIES OF SEQUENTIALLY DENSE MONOMORPHISMS OF SEMIGROUP ACTS
ABSTRACT Let M be a class of (mono)morphisms in a category A. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair (A,M). In this paper we take A to be the category ActS of Sacts, for a semigroup S, and M d to be the class of sequentially dense monomorphisms (of interests to computer scientists, too) and study the categorical properties, such as limits and colimits, of the pair (A,M). Injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, and the authors and got information about injectivity relative to monomorphisms.

Article: CDENSE INJECTIVITY IN ActS
[Show abstract] [Hide abstract]
ABSTRACT: To study mathematical notions, such as injectivity with respect to the class M of (mono)morphisms in a category A, one needs to have some categorical and algebraic information about the pair (A, M). In this paper, we take A to be the category ActS of acts over a semigroup S, C to be an arbitrary closure operator in the category ActS, and M d to be the class of Cdense monomorphisms resulting from a closure operator C and first study some categorical properties of the pair (ActS, M d). Then injectivity with respect to the class of Cdense monomorphisms is studied. The class of sequentially dense monomorphisms resulting from a special closure operator (sequential closure operator) and injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, Mahmoudi and the author. Some of these results generalize some of the results about the class of sequentially dense monomorphisms.AsianEuropean Journal of Mathematics 09/2013; 3(5).  SourceAvailable from: Leila Shahbaz[Show abstract] [Hide abstract]
ABSTRACT: An important notion related to injectivity with respect to monomorphisms or any other class M of morphisms in a category A is essentialness. In this paper, taking A to be the category of right acts over a semigroup S, C to be an arbitrary closure operator in the category ActS, and M d to be the class of Cdense monomorphisms resulting from a closure operator C, we study the properties of M d essential monomorphisms and we show the existence of a maximal M d essential extension for any given act. Finally, the behavior of M d injectivity in the sense that the three so called Wellbehavedness propositions hold is studied. We show that the idempotency and weak hereditariness of a closure operator C are sufficient, but not necessary, conditions for the wellbehavedness of M d injectivity. The class of sequentially dense monomorphisms resulting from a special closure operator (sequential closure operator) and injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, Mahmoudi, Moghaddasi, and the author. Some of these results generalize some of the results about the class of sequentially dense monomorphisms.Italian journal of pure and applied mathematics. 10/2013;  SourceAvailable from: Leila Shahbaz[Show abstract] [Hide abstract]
ABSTRACT: sdense monomorphisms and injectivity with respect to these monomorphisms were rst introduced and studied by Giuli for acts over the monoid (N ∞ , min). Ebrahimi, Mahmoudi, Moghaddasi, and Shahbaz generalized these notions to acts over a general semigroup. In this paper, we study atness with respect to the class of sdense monomorphisms. The theory of atness properties of acts over monoids has been of major interest over the past some decades, but so far there are not any papers published on this subject that relate specically to the class of sdense monomorphisms. We give some sucient conditions for sdense atness of semigroup acts. Also, we characterize a large number of semigroups over which sdense atness coincides with atness. This gives a useful criterion for atness of acts over such semigroups. In fact it is shown that the study of sdense atness is also useful in the study of ordinary atness of acts.Quasigroups and Related Systems. 10/2013; 21:201206.
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TAIWANESE JOURNAL OF MATHEMATICS
Vol. 15, No. 2, pp. 543557, April 2011
This paper is available online at http://www.tjm.nsysu.edu.tw/
CATEGORICAL PROPERTIES OF SEQUENTIALLY DENSE
MONOMORPHISMS OF SEMIGROUP ACTS
Mojgan Mahmoudi and Leila Shahbaz
Abstract. Let M be a class of (mono)morphisms in a category A. To study
mathematical notions, such as injectivity, tensor products, flatness, one needs
to have some categorical and algebraic information about the pair (A,M).
In this paper we take A to be the category ActS of acts over a semigroup
S, and Mdto be the class of sequentially dense monomorphisms (of interest
to computer scientists, too) and study the categorical properties, such as limits
and colimits, of the pair (A,M). Injectivity with respect to this class of
monomorphisms have been studied by Giuli, Ebrahimi, and the authors who
used it to obtain information about injectivity relative to monomorphisms.
1. INTRODUCTION AND PRELIMINARIES
Let M be a class of (mono)morphisms of a category A. To study mathematical
notions, such as injectivity and flatness, one needs to have some categorical and
algebraic information about the pair (A,M) (see [1, 3, 14]).
In this paper we take A to be the category ActS of (right) acts over a semigroup
S and Mdto be the class of sequentially dense monomorphisms, to be defined in
Section 2, and study the categorical properties of this pair which are usually related
to the behaviour of Mdinjectivity (see [13]).
In the following we first recall some facts about the category ActS needed in
this paper.
Let S be a semigroup and A be a set. If we have a mapping (called the action
of S on A)
µ :
A × S → A
(a,s) ?→ as := µ(a,s)
Received October 15, 2007, accepted September 7, 2009.
Communicated by WenFong Ke.
2000 Mathematics Subject Classification: 08B25, 18A20, 18A30, 20M30, 20M50.
Key words and phrases: Sequential closure, Sequential dense.
543
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Mojgan Mahmoudi and Leila Shahbaz
such that a(st) = (as)t for a ∈ A,s,t ∈ S, we call A a (right) Sact or a (right)
act over S.
If S is a monoid with identity 1, we usually also require that a1 = a for a ∈ A.
A subset A?of an Sact A is called a subact of A, written as A?≤ A, if a?s ∈ A?
for all s ∈ S and a?∈ A?.
The semigroup S itself becomes an Sact by taking its operation as its action.
A subact of the Sact S is a right ideal of the semigroup S. A subset K ⊆ S
is called a left ideal of S if SK ⊆ K, and an ideal or a twosided ideal of S if
SK ⊆ K and KS ⊆ K. Also note that if S does not have an identity, one can
attach an identity 1 to it to get a monoid, or an Sact, S1= S ∪ {1}.
Also, recall that an element a of an Sact A is said to be a fixed or a zero
element if as = a, for all s ∈ S.
A homomorphism (or an equivariant map, or an Smap) from an Sact A to an
Sact B is a function from A to B such that for each a ∈ A,s ∈ S, f(as) = f(a)s.
Since the identity maps and the composition of two equivariant maps are equiv
ariant, we have the category ActS of all right Sacts and Smaps between them.
An Sact B containing (an isomorphic copy of) an Sact A as a subact is called
an extension of A.
As a very interesting example of acts, used in computer science as a convenient
means of algebraic specification of process algebras (see [7], [8]), consider the
monoid (N∞,·,∞), where N is the set of natural numbers and N∞= N ∪ {∞}
with n < ∞,∀n ∈ N and m · n = min{m,n} for m,n ∈ N∞. Then an N∞act
is called a projection algebra (see [7, 10, 12]).
Let A be an Sact. An equivalence relation ρ on A is called an Sact congru
ence, or simply a congruence on A, if aρa?implies asρa?s for a,a?∈ A, s ∈ S.
If ρ is a congruence on A, then the factor set A/ρ = {[a]ρ: a ∈ A} is clearly an
Sact, called the factor act of A by ρ, with the action given by [a]ρs = [as]ρ, for
s ∈ S, a ∈ A.
If H ⊆ A × A then we denote the congruence generated by H by ρ(H); it is
the smallest congruence on A containing H. One can see that xρ(H)y if and only
if either x = y or there exist s1,s2,...,sn∈ S1, a1,...,an, b1,...,bn∈ A such
that (ai,bi) ∈ H or (bi,ai) ∈ H, and
x = a1s1
b2s2= a3s3
b1s1= a2s2
···
bnsn= y
···
b3s3= a4s4
Now we give some categorical ingredients of ActS needed in the sequel (see
also [5, 11]).
The class of Sacts is an equational class, and so the category ActS is complete
(has all products and equalizers). In fact, limits in this category are computed as in
the category Set of setsand equippedwitha natural action. In particular, theterminal
object of ActS is the singleton {0}, with the obvious Saction. Also, for Sacts
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Categorical Properties of Sequentially Dense Monomorphisms
545
A,B, their cartesian product A×B with the Saction defined by (a,b)s = (as,bs)
is the product of A and B in ActS.
The pullback of a given diagram
A
f
??
C
g
??B
in ActS is the subact P = {(c,a) : c ∈ C,a ∈ A,g(c) = f(a)} of C × A, and
pullback maps pC: P → C, pA: P → A are restrictions of the projection maps.
Notice that for the case where g is an inclusion, P can be taken as f−1(C).
All colimits in ActS exist and are calculated as in Set with the natural action
of S on them. In particular, ∅ with the empty action of S on it, is the initial
object of ActS. Also, the coproduct of Sacts A,B is their disjoint union A?B =
(A×{1})∪(B×{2}) with the obvious action, and coproduct injections are defined
naturally.
The pushout of a given diagram
A
g
??
f
??
C
B
in ActS is the factor act Q = (B ? C)/θ, where θ is the congruence relation
on B ? C generated by all pairs (uBf(a),uCg(a)), a ∈ A, where uB : B →
B ? C,uC : C → B ? C are the coproduct injections. Also, the pushout maps
are given as q1= γuC: C → (B ? C)/θ, q2= γuB: B → (B ? C)/θ, where
γ : B ?C → (B ?C)/θ is the canonical epimorphism. Multiple pushouts in ActS
are constructed analogously.
Recall that for a family {Ai : i ∈ I} of Sacts, each with a unique fixed
element 0, the direct sum ⊕i∈IAiis defined to be the subact of the product?
indices.
Free objects in ActS exist. In fact X × S1, where S1is S with an identity
adjoined, with the action (x,t)s = (x,ts) is the free Sact on the set X.
Cofree objects exist in ActS. In fact XS1= {f  f : S1→ X is a function}
with the action given by (fs)(t) = f(st) is the cofree Sact on the set X.
A morphism in ActS is a monomorphism if and only if it is oneone (so
sometimes we consider monomorphisms as inclusion maps), and epimorphisms in
ActS are exactly onto Smaps. These follow from the existence of free and cofree
Sacts, respectively.
i∈IAi
consisting of all (ai)i∈I such that ai= 0 for all i ∈ I except a finite number of
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Mojgan Mahmoudi and Leila Shahbaz
We also need to mention the construction of general limits and colimits which
will be needed in the sequel.
Let A : I → ActS be a diagram in ActS (I is a small category and A is
a functor) determining the acts A(α) = Aα, for α ∈ I = Obj(I), and Smaps
gαβ: Aα→ Aβ, for λ : α → β in Mor(I). Recall that the limit of this diagram is
lim
← −αAα:=?
λ∈Mor(I)Eλ, where for λ : α → β in Mor(I),
Eλ= {a = (aα)α∈I∈
?
α
Aα: gαβpα(a) = pβ(a)}
and pα,pβare the α,βth projection maps of the product. Also, the limit Smaps
are qα=: pαlim
Also, the colimit of the above diagram is obtained as lim
where θ is the congruence generated by
← −αAα: lim
← −αAα→ Aα.
− →αAα=:?
α∈IAα/θ,
H = {(uα(aα),uβgαβ(aα)) : aα∈ Aα,α → β ∈ Mor(I)}.
The colimit Smaps are gα:= γθuα: Aα→ lim
injection maps and γθis the canonical epimorphism of the quotient.
Recall that a directed system of Sacts and Smaps is a family (Bα)α∈Iof S
acts indexed by an updirected set I endowed by a family (gαβ: Bα→ Bβ)α≤β∈Iof
Smaps such that given α ≤ β ≤ γ ∈ I we have gβγgαβ= gαγ, and also gαα= id.
Note that the directed colimit (which is usually called the direct limit in literature)of
a directedsystem ((Bα)α∈I,(gαβ)α≤β∈I) inActS is given as lim
where the congruence ρ is given by (bα,bβ) ∈ ρ if and only if there exists δ ≥ α,β
such that uδgαδ(bα) = uδgβδ(bβ), where uα’s are injection maps of the coproduct.
Notice that the family gα= γρuα: Bα→ lim
gαfor α ≤ β, where γρ:?
2. SEQUENTIAL CLOSURE OPERATOR
− →αAαwhere uα’s are the coproduct
− →αBα=?
αBα/ρ,
− →αBαof Smaps satisfies gβgαβ=
− →αBαis the canonical epimorphism.
αBα→ lim
In this section we introduce and briefly study a closure operator; the dense
monomorphisms resulting from it are the subject of study in this paper. First note
that, denoting the lattice of all subacts of an Sact B by SubB, following [2] for the
general definition of closure operators on a category (which is not a priori assumed
to be idempotent), we get:
Definition 2.1. A family C = (CB)B∈Act−S, with CB : SubB → SubB,
taking the subbact A ≤ B to CB(A), is called a closure operator on ActS if it
satisfies the following laws:
(c1) (Extension) A ≤ CB(A),
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Categorical Properties of Sequentially Dense Monomorphisms
547
(c2) (Monotonicity) A1≤ A2implies CB(A1) ≤ CB(A2),
(c3) (Continuity) f(CB(A)) ≤ CC(f(A)), for all morphisms f : B → C.
Now, one has the usual two classes of monomorphisms related to the notion of
a closure operator as follows:
Definition 2.2. Let A ≤ B be in ActS. We say that A is Cclosed in B if
CB(A) = A, and it is Cdense in B if CB(A) = B. Also, an Smap f : A → B
is said to be Cdense (Cclosed) if f(A) is a Cdense (Cclosed) subact of B.
We take Mcto be the set of all Cclosed, and Mdto be the set of all Cdense
monomorphisms.
Definition 2.3. A closure operator C is said to be:
(a) Weakly hereditary if for every Sact B and every A ≤ B, A is Cdense in
CB(A).
(b) Hereditary if for every Sact B and A1≤ A2≤ B,
CA2(A1) = CB(A1) ∩ A2.
(c) Grounded if for every Sact B, CB(∅) = ∅.
(d) Additive if for every Sact B, CB(A1∪ A2) = CB(A1) ∪ CB(A2).
(e) Productive if for every family of subacts Aiof Bi, taking A =?
(f) Idempotent if CB(CB(A)) = CB(A) for all Sacts B and A ≤ B.
(g) Discrete if CB(A) = A for every A ≤ B.
(h) Trivial if CB(A) = B for every A ≤ B.
Now, we introduce the sequential closure operator on the category of Sacts
and investigate some of its properties (see also [9] and [4]).
iAiand
B =?
iBi, CB(A) =?
iCBi(Ai).
Definition 2.4. The sequential closure operator Cd= (Cd
is defined as
Cd
B)B∈Act−Son ActS
B(A) = {b ∈ B : bS ⊆ A}
for any subact A of an Sact B.
Notice that for the case where S is a monoid, every subact A of B is Cdclosed,
and A is Cddense in B if and only if A = B. Note that, by Definition 2.2, a
subact A of an Sact B is Cddense, which will also be called sequentially dense
or sdense, in B if bS ⊆ A for each b ∈ B. An Smap f : A → B is said to be
sdense if f(A) is an sdense subact of B.
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Mojgan Mahmoudi and Leila Shahbaz
Remark 2.5. For each subact A of an Sact B, Cd
of B with the property TS ⊆ A.
We now prove some of the properties of this closure operator.
B(A) is the largest subact T
Theorem 2.6. The closure operator Cdis hereditary, weakly hereditary, pro
ductive, grounded if S ?= ∅, discrete if and only if S is a monoid, and also trivial
if and only if S is empty.
Proof.
We just prove some parts of this result; the remainder are also straight
forward. For hereditariness, let A1 ≤ A2 ≤ B and a ∈ Cd
A1,a ∈ A2. Thus aS ⊆ A1,a ∈ B. Hence a ∈ Cd
a ∈ Cd
For the last part, we see that if S = ∅ then Cd
If S ?= ∅, let s ∈ S. Then, taking sets A ⊂ B as Sacts with the identity action,
and b ∈ B − A, we have bs = b ?∈ A. Thus Cd
Corollary 2.7. If A ≤ B ≤ C then Cd
As the following result shows, Cdis not idempotent in general.
A2(A1). Then aS ⊆
B(A1) ∩ A2. Conversely, let
A2(A1).
B(A) = {b ∈ B : bS ⊆ A} = B.
B(A1) ∩ A2. Then a ∈ A2,aS ⊆ A1. Thus a ∈ Cd
B(A) ?= B.
B(A) ⊆ Cd
C(A).
Theorem 2.8. The closure operator Cdis idempotent if and only if S2= S.
Proof.
Let Cdbe idempotent. Since
S1= Cd
S1(S) = Cd
S1(Cd
S(S2)) ⊆ Cd
S1(Cd
S1(S2)) = Cd
S1(S2)
and S1S ⊆ S2which means that S ⊆ S2. The converse is obvious.
Lemma 2.9. A (right) ideal I of S is sdense, that is Cd
if S2⊆ I.
S(I) = S, if and only
Theorem 2.10. The closure operator Cdis additive if and only if for every
element b in an Sact B, bS is join prime in the lattice Sub(B).
Proof.
Let A and D be subacts of an Sact B and b ∈ Cd
Then, bS ⊆ A ∪ D and hence, bS being ∨prime, bS ⊆ A or bS ⊆ D. Thus,
b ∈ Cd
hence Cd, is additive.
Conversely, let Cd, and hence each Cd
where A and D are subacts of B. Then, by monotonicity and additivity,
B(A ∪ D).
B(A) ∪ Cd
B(D). This, using monotonicity of Cd, shows that each Cd
B, and
B, be additive. Let b ∈ B and bS ⊆ A∪D,
Cd
B(bS) ⊆ Cd
B(bS), b ∈ Cd
B(A ∪ D) = Cd
B(A) or b ∈ Cd
B(A) ∪ Cd
B(D). Thus, bS ⊆ A or bS ⊆ D,
B(D).
Now, since b ∈ Cd
proving that bS is join prime in Sub(B).
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Categorical Properties of Sequentially Dense Monomorphisms
549
Theorem 2.11. If S has a left identity element e, then Cdis additive.
Proof.
Note that, in this case, for any subact A of an Sact B, b ∈ Cd
and only if be ∈ A. This is because, bs = b(es) = (be)s, for each s ∈ S.
Now, if A and D are subacts of B and bS ⊆ A ∪ D, for b ∈ B, then one can
easily see that bS ⊆ A or bS ⊆ D, depending on be ∈ A or be ∈ D, respectively.
B(A) if
3. CATEGORICAL PROPERTIES OF sDENSE MONOMORPHISMS
In this final section we study some categorical and algebraic properties of the
category ActS with respect to sequentially dense monomorphisms. We study the
composition, limit, and colimit properties in the following three subsections.
3.1. Composition properties of sdense monomorphisms
In this subsection we investigate some properties of the class Md, mostly to
do with the composition of dense monomorphisms. These properties and the ones
given in the next two subsections are normally used to study injectivity, and of
course other mathematical notions.
The class Mdis clearly isomorphism closed; that is, contains all isomorphisms
and is closed under composition with isomorphisms. But, unfortunately Mdis not
always closed under composition:
Lemma 3.1. The class Md is closed under composition if and only if the
Cdclosure operator is idempotent.
Proof.
If the composition of sdense monomorphisms is an sdense monomor
phism then, since the inclusion maps S2?→ S and S ?→ S1are clearly sdense, we
get that S2is sdense in S1. Hence, S = 1S ⊆ S2and so, by Theorem 2.8, Cdis
idempotent. For the converse, let A ≤ B and B ≤ D be sdense subacts. Then,
D = Cd
D(B) = Cd
As the above result shows, the composition of sdense monomorphisms need not
be sdense. For example take a semigroupS with S2?= S and consider the inclusion
maps S2?→ S and S ?→ S1given in the above proof (see Theorem 2.8). But the
following useful result shows that the composition of an sdense monomorphism
with a surjective morphism is sdense.
D(Cd
B(A)) ⊆ Cd
D(Cd
D(A)) = Cd
D(A). Thus D = Cd
D(A).
Proposition 3.2. The composition of an sdense morphism with a surjective
morphism is an sdense morphism.
Proof.
Let f : A → B be an sdense monomorphism and g : B → C be a
surjective Smap. We want to show that for each c ∈ C and s ∈ S, cs ∈ Im(gf).
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Mojgan Mahmoudi and Leila Shahbaz
Let c ∈ C. Since g is surjective, there exists b ∈ B such that c = g(b). Now,
since f is sdense, for each s ∈ S,bs = f(a) for some a ∈ A. Thus cs = g(b)s =
g(bs) = g(f(a)) = (gf)(a) ∈ Im(gf). Hence gf is sdense.
Now, let g : A → B be a surjective Smap and f : B → C be an sdense
monomorphism. Take c ∈ C,s ∈ S. Since f is sdense, there exists b ∈ B such
that f(b) = cs. Since g is surjective, there exists a ∈ A such that g(a) = b. Now,
cs = f(b) = f(g(a)) = (fg)(a) ∈ Im(fg). Hence fg is sdense.
The following result shows that Mdis right (left) cancellable, in the sense that
for monomorphisms f and g if gf ∈ Mdthen g ∈ Md(f ∈ Md).
Proposition 3.3. The class Mdis right and left cancellable.
Proof.
For the right cancellability, let gf be in Md for monomorphisms
f : A → B, g : B → C. Take s ∈ S,c ∈ C. Since gf is sdense, there exists
a ∈ A such that (gf)(a) = cs. Now, g(f(a)) = cs, cs ∈ Img. Thus g ∈ Md. For
the left cancellability, let b ∈ B, s ∈ S, and so g(bs) ∈ C. Since gf is sdense,
there exists a ∈ A such that gf(a) = g(bs). Now, since g is a monomorphism,
f(a) = bs and hence bs ∈ Imf.
Proposition 3.4. Let f : A → B ∈ActS. Then there are unique (always up to
isomorphism) morphisms e,m ∈ActS such that:
(1) (right Mdfactorization) f =me, where m : C→B∈Md, e : A → C, and
(2) (diagonalization property) for every commutative diagram
A
e
??
u
??D
g
??
C
w
?
???
?
?
?
?
?
m
??
B
v
??E
in ActS with g : D → E ∈ Md, there is a uniquely determined morphism
w : C → D with gw = vm and we = u.
Take f : A → B, and let C = f(A) ∪ BS. Define e : A → C
by e(a) = f(a) for a ∈ A, and take m : C → B to be the inclusion map. Then
f = me. To see (2), define w : C → D by w(f(a)) = u(a), w(bs) = v(bs) =
v(b)s. Then w is welldefined, for, if bs = b?s?then v, being welldefined, we get
w(bs) = v(bs) = v(b?s?) = w(b?s?); and if f(a) = f(a?) then gu(a) = vf(a) =
vf(a?) = gu(a?) and so u(a) = u(a?) since g is a monomorphism; and if f(a) = bs
then gwf(a) = gu(a) = vf(a) = v(bs) = gw(bs) which gives wf(a) = w(bs),
Proof.
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Categorical Properties of Sequentially Dense Monomorphisms
551
since g is a monomorphism. It is clear, by the definition of w, that gw = vm
and we = u. Also, w is unique by this property, since having w?: C → D with
gw = vm = gw?, we get that w = w?because g is a monomorphism.
To show the uniquenessof m and e, let there also exist morphisms m?: C?→ B
and e?: A → C?satisfying conditions (1) and (2) above. Then, there are w : C →
C?, w?: C?→ C such that we = e?, m?w = idBm and w?e?= e, mw?=
idBm?. Hence mw?w = m, which makes w?w = idCsince m is a monomorphism.
Similarly, ww?= idC. This means that e and e?, also m and m?, are isomorphic.
3.2. Limits of sdense monomorphisms
In this subsection we will investigate the behaviour of dense monomorphisms
with respect to limits.
Proposition 3.5. The class Mdis closed under products.
Proof.
Let (fi: Ai→Bi)i∈Ibe a family of sdense monomorphisms. Consider
the commutative diagram
?
i∈IAi
pi
??
f
??
?
i∈IBi
p?
??
i
Ai
fi
??Bi
We show that f = (fi)i∈I:?
bs = (bis)i∈I∈ Imf. Hence f is sdense. It is obvious that f is a monomorphism.
So f ∈ Md.
Proposition 3.6. The class Mdis closed under Mdpullbacks.
i∈IAi→?
i∈IBiis an sdense monomorphism. Let
b = (bi)i∈I ∈?
i∈IBiand s ∈ S. Since each fiis sdense, bis ∈ Imfi. Now
Proof.
Consider the pullback diagram
f−1(C)??i
??
f=ff−1(C)
??
A
f
??
C
g
??B
with g,f ∈ Md. For simplicitywe consider g to be inclusion. We have to show that
Im(gf) is sdense. Let b ∈ B, s ∈ S. Since g and f are sdense, there exist c ∈ C
and a ∈ A such that c = bs = f(a). Thus bs = c = g(c) = g(f(a)) ∈ Im(gf).
Proposition 3.7. The class Mdis stable under Mdpullbacks; in the sense that
pullback of any sdense monomorphism along any morphism is again sdense.
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Mojgan Mahmoudi and Leila Shahbaz
Proof.
Consider the pullback diagram given in the proof of the above proposi
tion with g ∈ Mdas inclusion, and let f be an arbitrary Smap. We have to show
that i is sdense. Let a ∈ A, s ∈ S. Since g is sdense, f(as) = f(a)s ∈ C. Thus
as ∈ f−1(C).
Proposition 3.8. The class Mdis closed under limits.
Proof.
Let A,B : I → ActS be diagrams in ActS determining the acts Aα,
Bα, for α ∈ I = Obj(I), and Smaps gαβ: Aα→ Aβ, g?
in Mor(I). Consider limits of these diagrams with limit maps qα: lim
q?α: lim
← −Bα → Bα. Let {fα : Aα → Bα : α ∈ I} be a family of sdense
monomorphisms such that g?
which exists by the universal property of limits. We show that f belongs to Md.
Consider the diagram
αβ: Bα→ Bβ, for α → β
← −Aα→ Aα,
← −fα: lim
αβfα= fβgαβ. Let f denote lim
← −Aα→ lim
← −Bα
lim
← −Aα
e
??
qα
??Aα
gαβ??
fα
??
Aβ
fβ
??
M
wα
?
?
?
???
?
?
?
?
?
m
??
lim
← −Bα
q?α
??Bα
g?
αβ??Bβ
where f = me is the right Mdfactorization of f, which exists by Proposition 3.4.
Since each fα∈ Md, the diagonalization property of the factorization for each α
implies that there exists wα: M → Aαsuch that fαwα= q?
the uniqueness of wα’s gives that gαβwα = wβ for each α → β. Now, by the
universal property of limits, there exists j : M → lim
α. We show that j is in fact an isomorphism. We have qαje = wαe = qαfor each
α, and so, by the universal property of limits, je = idlim
following diagram
αm,wαe = qα. Then,
← −Aαwith qαj = wαfor each
← −Aα. Also, considering the
lim
← −Aα
e
e
??
??
M
m
??
M
j
??
idM
?
?
?
?
???
?
?
?
?
?
?
m
??
lim
← −Bα
id??lim
← −Bα
we get that q?
mej = m which, by the uniquenessof the diagonalizationproperty, gives ej = idM.
Therefore, j, and hence e, is an isomorphism. But f = me and Mdis closed under
composition with isomorphisms, so f belongs to Md.
αmej = fαwαej = fαqαj = fαwα= q?
αm for each α, and hence
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Categorical Properties of Sequentially Dense Monomorphisms
553
3.3. Colimits of sdense monomorphisms
This subsection is devoted to the study of the behaviour of sdense monomor
phisms with respect to colimits.
Proposition 3.9. The class Mdis closed under coproducts.
Proof.
Consider the diagram
Ai
ui
??
fi
??Bi
u?
i
??
?
i∈IAi
f
???
i∈IBi
in which {fi : Ai → Bi : i ∈ I} is a family of sdense monomorphisms. Let
f :?
We have to show that f is an sdense monomorphism. Let b ∈?
ai∈ Aisuch that fi(ai) = bis, and hence u?
bs = u?
because u?
i∈IAi→?
i∈IBibe the Smap satisfying f(ui(ai)) = u?
which exists by the universal property of coproducts; in fact, f(ai,i) = (fi(ai),i).
ifi(ai), for ai∈ Ai,
i∈IBi, s ∈ S.
Then thereexistsi ∈ I, bi∈ Bisuch that b = u?
i(bi). Since fiis sdense, thereexists
ifi(ai) = u?
i(bis) = u?
i(bi)s = bs. Now,
ifi(ai) = fui(ai) ∈ Imf. Thus f is sdense. Also, f is a monomorphism,
iand fi, i ∈ I are monomorphisms.
Proposition 3.10. Let {fi: Bi→ A : i ∈ I} be a family of sdense Smaps.
Then f :?
Proof.
Consider the diagram
i∈IBi→ A is an sdense Smap.
Bi
ui
??
fi
??A
?
i∈IBi
f
???
?
?
?
?
?
?
?
?
where f :?
of Md(Proposition 3.3), we get f ∈ Md.
Proposition 3.11. The class Mdis closed under direct sums.
Proof.
Let {fi: Ai→ Bi: i ∈ I} be a family of sdense monomorphisms.
Then, usingProposition 3.5, we get that f = ⊕i∈Ifi=?
by Proposition 3.5, there exists (ai)i∈I∈?
i∈IBi→ A is the Smap obtained by the universal property of coprod
ucts. Then, since fui= fibelongs to Md, applying the proof of right cancellability
i∈Ifi: ⊕i∈IAi→ ⊕i∈IBi
is an sdense monomorphism. More precisely, if (bi)i∈I∈ ⊕i∈IBiand s ∈ S then,
i∈IAiwith f((ai)i∈I) = (bi)i∈Is. But,
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554
Mojgan Mahmoudi and Leila Shahbaz
(ai)i∈I∈ ⊕i∈IAi, because for all i with bi= 0 we have 0 = bis = f(ai) = fi(ai)
and so ai= 0, since fiis a monomorphism.
We recall the following lemma from [6]. First recall that it is said that pushouts
transfer monomorphisms in a category if for a pushout diagram
A
f
??
g
??C
h
??
B
k
??D
if g is a monomorphism then so is k.
Lemma 3.12. Pushouts transfer monomorphisms in ActS.
Proposition 3.13. In ActS, pushouts transfer sdense monomorphisms.
Proof.
Consider the pushout diagram
A
f
??
g
??C
h
??
B
k??(B ? C)/θ
where g ∈ Md, h = γuC : C → (B ? C)/θ, k = γuB : B → (B ? C)/θ,
γ : B ? C → (B ? C)/θ is the natural epimorphism, and uB: B → B ? C,uC:
C → B ? C are coproduct injections, and θ is the congruence relation on B ? C
generated by all pairs H = {(uBf(a),uCg(a)) : a ∈ A}. We show that k belongs
to Md. Notice that by the above lemma, k is a monomorphism, so it is enough
to show that k is sdense. Let [x]θ∈ (B ? C)/θ and s ∈ S be arbitrary. Then,
x = uB(b) for some b ∈ B, or x = uC(c) for some c ∈ C. In the former case, we
have [x]θs = k(b)s = k(bs) ∈ Im(k). In the latter case, using that g is sdense,
we get a ∈ A with g(a) = cs and hence [x]θs = [uC(c)]θs = h(c)s = h(cs) =
hg(a) = kf(a) ∈ Im(k).
Proposition 3.14. The pushout of sdense monomorphisms belongs to Md.
Proof.
Applying the notations of the above proposition, with a similar argu
ment to its proof, one gets that when f and g in the pushout diagram are sdense
monomorphisms, then so is kf = hg.
Note that if the composition of sdense monomorphismswere sdense, the above
result would have been just a direct corollary of the last proposition.
Page 13
Categorical Properties of Sequentially Dense Monomorphisms
555
Proposition 3.15. The multiple pushout of sdense monomorphisms is an s
dense monomorphism. Also, multiple pushouts transfer sdense monomorphisms.
Proof.
Let {di: A → Bi: i ∈ I} be a family of sdense monomorphisms.
Recall that the multiple pushout of this family is?
where for each i ∈ I, ui: Bi→?
?
Bi, d?
exist elements a1,a2,...,an∈ A, k1...kn+2∈ I such that ui(bi) = uk1dk1(a1),
uk2dk2(a1) = uk3dk3(a2), ···, ukn+2dkn+2(an) = ui(b?
k2= k3, k4= k5, ..., kn+2= i, and hence, since each dkis a monomorphism,
bi= di(a1), a1= a2= a3= a4= a5= ... = an−1 = an, di(an) = b?
bi= di(a1) = di(a2) = di(a3) = ... = di(an) = b?
To show that each d?
s ∈ S. Then, there exist j ∈ I and bj ∈ Bj such that b = [uj(bj)]θ. Since
dj is sdense, there exists an element a ∈ A such that dj(a) = bjs. Now, bs =
[uj(bj)]θs = [uj(bjs)]θ= d?
i∈IBi/θ, where θ is the congru
ence on?
the multiple pushout maps are d?
i∈IBi/θ is the natural epimorphism.
First, we see that for each i ∈ I, d?
i(bi) = d?
i∈IBigenerated by all pairs H = {(uidi(a),ujdj(a)) : i,j ∈ I,a ∈ A},
i∈IBiis the ith coproduct injection map. Also,
i= γui: Bi→?
iis a monomorphism. Let for bi,b?
i). Then (ui(bi),ui(b?
i∈IBi/θ where γ :?
i∈IBi→
i∈
i(b?
i)) ∈ θ and thus either bi = b?
ior there
i).Therefore, k1 = i,
i. Thus
i.
idi(and hence each d?
i) is dense, let b ∈?
i∈IBi/θ and
j(bjs) = d?
jdj(a) = d?
idi(a) ∈ Im(d?
idi).
Definition 3.16. We say that a category A has Mbounds if for every set
indexed family {mi: A → Ai: i ∈ I} of Mmorphisms there is an Mmorphism
m : A → B which factors over all mi’s; that is there are di : Ai → B with
dimi= m.
Proposition 3.17. The category ActS has Mdbounds.
Proof.
Let {hα : A → Bα : α ∈ I} be a set indexed family in Mdand
h : A → B =?
αBα/θ be the multiple pushout of hα’s. Then h factors over all
hα’s, and is an sdense monomorphism, by Proposition 3.15.
Definition 3.18. We say that a category A has Mamalgamationproperty, if the
morphism m in the definition of Mbounds factors over all mi’s through members
of M; that is di’s belong to M.
Proposition 3.19. The category ActS has Mdamalgamation property.
Proof.
Since, by Proposition3.15, multiplepushouttransferssdensemonomor
phisms, we are done.
Proposition 3.20. The category ActS has Mddirected colimits.
Page 14
556
Mojgan Mahmoudi and Leila Shahbaz
Proof.
Let ((Bα)α∈I,(gαβ)α≤β∈I) be a directed system of Sacts and S
maps, and gα: Bα→ lim
hα: A → Bα, α ∈ I, with gαβhα= hβfor α ≤ β ∈ I. Let h : A → lim
the directed colimit of hαs. That is, h = lim
Then since each hαis a monomorphism, h is a monomorphism. Also, h is sdense
because for b ∈ lim
such that b = [xσ]ρand since hσis sdense, there exists an element as∈ A with
hσ(as) = xσs. Then bs = [xσ]ρs = gσ(xσ)s = gσ(xσs) = gσhσ(as) = h(as) ∈
Im(h).
Definition 3.21. We say that a category A fulfills the Mchain condition if for
every directed system ((Aα)α∈I,(fαβ)α≤β∈I) whose index set I is a wellordered
chain with the least element 0, and f0α∈ M for all α, there is a (so called “upper
bound”) family (gα: Aα→ A)α∈Iwith g0∈ M and gβfαβ= gα.
Proposition 3.22. The category ActS fulfills the Mdchain condition.
Proof.
Take A = lim
− →αAαand let gα: Aα→ A be the colimit maps. Then,
applying Proposition 3.20, we get the result.
− →αBαare the colimit maps. Take sdense monomorphisms
− →αhα= gγhγ= gαhα= gβhβ= ....
− →αBα and s ∈ S, since b ∈ lim
− →αBαbe
− →αBα, there exists xσ ∈ Bσ
Theorem 3.23. The class Mdis closed under colimits.
Proof.
Let A,B : I → ActS be diagrams in ActS determining the acts
Aα, Bα for α ∈ I = Obj(I), and the Smaps gαβ : Aα → Aβ, g?
Bβ, for α → β in Mor(I). Consider the colimits of these diagrams with the
colimit maps gα= γθuα: Aα→ lim
lim
sdense monomorphisms such that g?
is an sdense monomorphism. Recall that θ is the congruence generated by H =
{(uα(aα),uβgαβ(aα)) : aα∈ Aα, α → β ∈ Mor(I)}, and θ?is the congruence
generated by H?= {(u?α(bα),u?
notice that f[uα(aα)]θ= [u?αfα(aα)]θ?. Since each fαis a monomorphism, it is
not hard to check that f is a monomorphism. To see that f is sdense, let s ∈ S,
x = [u?α(bα)]θ? ∈ lim
aα∈ Aαwith fα(aα) = bαs. Then, gα(aα) = [uα(aα)]θ∈ lim
xs = [u?α(bαs)]θ? = g?
αβ: Bα →
− →αAα =?
αβfα= fβgαβ. We show that f =: lim
α∈IAα/θ, g?α= γθ?u?α: Bα→
− →αBα=?
α∈IBα/θ?, and assume that {fα: Aα→ Bα: α ∈ I} is a family of
− →αfα
βg?
αβ(bα)) : bα ∈ Bα, α → β ∈ Mor(I)}, and
− →αBα for some α ∈ I. Since fα is sdense, there exists
α(bαs) = g?
− →αAαand we have
αfα(aα) = fgα(aα) ∈ Im(f).
ACKNOWLEDGMENTS
The authors thank the referee for his/her very careful reading and useful com
ments. We also would like to thank Professor M. Mehdi Ebrahimi for his very good
comments and helpful conversations during this research.
Page 15
Categorical Properties of Sequentially Dense Monomorphisms
557
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Mojgan Mahmoudi and Leila Shahbaz
Department of Mathematics and Center of Excellence in Algebraic
and Logical Structures in Discrete Mathematics
Shahid Beheshti University
G. C., Tehran
Iran
Email: mmahmoudi@cc.sbu.ac.ir
lshahbaz@cc.sbu.ac.ir
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