@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 11, No. 2, 2010

pp. 67-88

Convergence semigroup categories

H. Boustique, P. Mikusiński and G. Richardson

Abstract. Properties of the category consisting of all objects of the
form (X, S, λ) are investigated, where X is a convergence space, S is a
commutative semigroup, and λ : X × S → X is a continuous action.
A ”generalized quotient” of each object is defined without making the
usual assumption that for each fixed g ∈ S, λ(., g) : X → X is an
injection.

2000 AMS Classification: 54A20, 54B15, 54B30

Keywords: convergence space, convergence semigroup, continuous action,
categorical properties.

1. Introduction and Preliminaries

. The notion of a topological group acting continuously on a topological space
has been the subject of numerous research articles. Park [13, 14] and Rath
[16] studied these concepts in the larger category of convergence spaces. This
is a more natural category to work in since the homeomorphism group on a
space can be equipped with a coarsest convergence structure making the group
operations continuous. Moreover, unlike in the topological context, quotient
maps are productive in the category of all convergence spaces with continuous
maps as morphisms. This property played a key role in the proof of several
results contained in [3]; for example, see Theorem 4.5 [3].

Given a topological semigroup acting on a topological space, Burzyk et al.
[5] introduced a ”generalized quotient space.” Elements of this space are equiv-
alence classes determined by an abstraction of the method used to construct
the rationals from the integers. Generalized quotient spaces are used in the
study of generalized functions [10, 11, 12]. Moreover, generalized quotients
in the category of convergence spaces are defined in [3] for the case whenever
λ(., g) : X → X is injective, where λ is a continuous action of a convergence
semigroup S on a convergence space X. Generalized quotients are defined and
studied here without the requirement that λ(., g) is injective. Furthermore,



68 H. Boustique, P. Mikusiński and G. Richardson

the category consisting of objects of the form (X, S, λ) is investigated. The
terminology used here involving categories can be found in Adamek et al. [1].

Basic definitions and concepts needed in the area of convergence spaces are
given in this section. Let X be a set, 2X the power set of X, and let F(X)
denote the set of all filters on X. Recall that B ⊆ 2X is a base for a filter on
X provided B 6= ∅, ∅ /∈ B, and B1, B2 ∈ B implies that there exists B3 ∈ B
such that B3 ⊆ B1 ∩ B2. Moreover, [B] denotes the filter on X whose base is
B; that is, [B] = {A ⊆ X : B ⊆ A for some B ∈ B}. Fix x ∈ X, define ẋ to
be the filter whose base is B = {{x}}. If f : X → Y and F ∈ F(X), then f→F
denotes the image filter on Y whose base is {f(F) : F ∈ F}.

A convergence structure on X is a function q : F(X) → 2X obeying:

(CS1) x ∈ q(ẋ) for each x ∈ X
(CS2) x ∈ q(F) implies that x ∈ q(G) whenever F ⊆ G.

The pair (X, q) is called a convergence space. The more intuitive notation

F
q
−→ x is used for x ∈ q(F). A map f : (X, q) → (Y, p) between two conver-

gence spaces is called continuous whenever F
q
−→ x implies that f→F

p
−→ f(x).

Let CONV denote the category whose objects consist of all the convergence
spaces, and whose morphisms are all the continuous maps between objects. The
collection of all objects in CONV is denoted by |CONV|. If p and q are two

convergence structures on X, then p ≤ q means that F
p
−→ x whenever F

q
−→ x.

In this case, p(q) is said to be coarser(finer) than q(p), respectively. Also, for
F, G ∈ F(X), F ≤ G means that F ⊆ G, and F(G) is called coarser(finer)
than G(F), respectively.

It is well-known that CONV possesses initial and final convergence struc-
tures. In particular, if (Xj, qj) ∈ |CONV| for each j ∈ J, then the prod-

uct convergence structure r on X = ×
j∈J

Xj is given by H
r
−→ x = (xj) iff

π→j H
qj
−→ xj for each j ∈ J, where πj denotes the j

th projection map. Also, if

f : (X, q) → Y is a surjection, then the quotient convergence structure σ

on Y is defined by: H
σ
−→ y iff there exists x ∈ f−1(y) and F

q
−→ x such that

f→F = H. In this case, σ is the finest convergence structure on Y making
f : (X, q) → (Y, σ) continuous. Convergence quotient maps are defined and
studied by Kent [7]. Moreover, Beattie and Butzmann [2] and Preuss [15] are
good references for convergence space results.

Unlike the category of all topological spaces, CONV is cartesian closed and
thus has suitable function spaces. In particular, let (X, q), (Y, p) ∈ |CONV|
and let C(X, Y ) denote the set of all continuous functions from X to Y . Define
ω : (X, q)×C(X, Y ) → (Y, p) to be the evaluation map given by ω(x, f) = f(x).
There exists a coarsest convergence structure c on C(X, Y ) such that w is

jointly continuous. More precisely, c is defined by : Φ
c
−→ f iff w→(F × Φ)

p
−→

f(x) whenever F
q
−→ x. This compatibility between (X, q) and (C(X, Y ), c) is

an example of a continuous action in CONV.
Let SG denote the category whose objects consist of all the semigroups

(with an identity element), and whose morphisms are all the homomorphisms



Convergence semigroup categories 69

between objects. Further, (S, ., p) is said to be a convergence semigroup
provided : (S, .) ∈ |SG|, (S, p) ∈ |CONV|, and γ : (S, p) × (S, p) → (S, p)
is continuous, where γ(x, y) = x.y. Let CSG be the category whose objects
consist of all the convergence semigroups, and whose morphisms are all the
continuous homomorphisms between objects.

An action of a semigroup on a topological space is used to define ”generalized
quotients” in [5]. Below is Rath’s [16] definition of an action in the convergence
space context. Let (X, q) ∈ |CONV|, (S, ., p) ∈ |CSG|, λ : X × S → X, and
consider the following conditions:

(A1) λ(x, e) = x for each x ∈ X (e is the identity element)
(A2) λ(λ(x, g), h) = λ(x, g.h) for each x ∈ X, g, h ∈ S
(A3) λ : (X, q) × (S, ., p) → (X, q) is continuous.

Then (S, .)((S, ., p)) is said to act(act continuously) on (X, q) whenever A1-
A2 (A1-A3) are satisfied and, in this case, λ is called the action (continuous
action), respectively. For sake of brevity, (X, S) ∈ A(AC) denotes the fact
that (S, .) is commutative and (S, ., p) ∈ |CSG| acts (acts continuously) on
(X, q) ∈ |CONV|, respectively. Moreover, (X, S, λ) ∈ A indicates that the
action is λ.

Remark 1.1. Fix a set X; then the set of all convergence structures on X
with the ordering p ≤ q defined above is a complete lattice. Indeed, if (X, qj) ∈

|CONV|, j ∈ J, then sup
j∈J

qj = q
1 is given by F

q1
−→ x iff F

qj
−→ x, for each

j ∈ J. Dually, inf
j∈J

qj = q
0 is defined by F

q0
−→ x iff F

qj
−→ x, for some j ∈ J.

It is easily verified that if ((X, qj), (S, ., p), λ) ∈ AC for each j ∈ J, then both
((X, q1), (S, ., p), λ) and ((X, q0), (S, ., p), λ) belong to AC.

The notion of ”generalized quotients” determined by a commutative semi-
group acting on a topological space is investigated in [5]. Elements of the
semigroup in [5] are assumed to be injections on the given topological space.

Lemma 1.2 ([5]). Suppose that (S, X, λ) ∈ A, and λ(., g) : X → X is an
injection, for each g ∈ S. Define (x, g) ≈ (y, h) on X × S iff λ(x, h) = λ(y, g).
Then ≈ is an equivalence relation on X × S.

In the context of Lemma 1.2, let 〈(x, g)〉 be the equivalence class containing
(x, g), B(X, S) denote the quotient set (X×S)/ ≈, and define ϕ : (X×S, r) →
B(X, S) to be the canonical map, where r = q × p is the product convergence
structure. Equip B(X, S) with the convergence quotient structure σ. Then

K
σ
−→ 〈(y, h)〉 iff there exist (x, g) ≈ (y, h) and H

r
−→ (x, g) such that ϕ→H = K.

Properties of (B(X, S), σ) are investigated in [3].
Whenever the hypothesis that λ(., g) is an injection for each g ∈ S fails, one

can still define a generalized quotient by extending ≈ to an equivalent relation
as defined in (R2) below.



70 H. Boustique, P. Mikusiński and G. Richardson

Assume that ((X, q), (S, ., p), λ) ∈ A, and consider the following relations:

(R1) (x, f) ≈ (y, g) in X × S iff λ(x, g) = λ(y, f)
(R2) (x, f) ∼ (y, g) in X × S iff there exists (zi, hi) ∈ X × S, satisfying

(x, f) ≈ (z1, h1) ≈ (z2, h2) ≈ · · · ≈ (zn, hn) ≈ (y, g), for some n ≥ 1
(R3) x ≃ y in X iff there exists g ∈ S such that λ(x, g) = λ(y, g).

According to Lemma 1.1 [5], (R1) is an equivalence relation provided that
λ(., g) is an injection for each g ∈ S. However, (R2) is an equivalence relation
without assuming that λ(., g) is an injection for each g ∈ S. Moreover, it easily
follows that (R3) is an equivalence relation.

Given ((X, q), (S, ., p), λ) ∈ A; denote X∗ = X/ ≃, ξ : X → X∗ the canonical
map ξ(x) = [x], and let q∗ be the quotient structure on X∗ in CONV deter-
mined by ξ : (X, q) → X∗. Define ϕ : X × S → B(X, S) = (X × S)/ ∼ to be
the canonical map ϕ(x, g) = 〈(x, g)〉, and let σ denote the quotient structure on
B(X, S) in CONV determined by ϕ : (X × S, r) → B(X, S), where r = q × p.
Likewise, define ϕ∗ : X∗ × S → B(X∗, S) = (X∗ × S)/ ≈ by ϕ∗([x], g) =
〈([x], g)〉. Let r∗ denote the product structure on X∗ × S, and let σ∗ be the
quotient structure on B(X∗, S) determined by ϕ∗ : X∗ ×S → B(X∗, S). Define
λ∗ : X∗ ×S → X∗ by λ∗([x], g) = [λ(x, g)], and denote η : B(X, S) → B(X∗, S)
by η(〈(x, g)〉) = 〈([x], g)〉. It is shown below that these definitions are well-
defined, and their properties are investigated.

Lemma 1.3. Assume that ((X, q), (S, ., p), λ) ∈ A. Then

(a) (x, f) ∼ (y, g) iff there exists h ∈ S such that (x, hf) ≈ (y, hg)
(b) x ≃ y iff for each g ∈ S, λ(x, g) ≃ λ(y, g)
(c) λ∗(., g) : X∗ → X∗ is an injection, for each fixed g ∈ S
(d) η : B(X, S, λ) → B(X∗, S, λ∗) is a bijection.

Proof. (a): Suppose that (x, f) ∼ (y, g). Then there exists (zi, hi) ∈ X × S,
1 ≤ i ≤ n, such that (x, f) ≈ (z1, h1), (z1, h1) ≈ (z2, h2),...,(zn, hn) ≈ (y, g).
Verification is illustrated whenever n = 2; that is, (x, f) ≈ (z1, h1), (z1, h1) ≈
(z2, h2), and (z2, h2) ≈ (y, g). Then λ(x, h1) = λ(z1, f), λ(z1, h2) = λ(z2, h1),
and λ(z2, g) = λ(y, h2). It is shown that (x, h2h1f) ≈ (y, h2h1g). Indeed,
since (S, .) is commutative, λ(x, h2h1g) = λ(λ(x, h1), h2g) = λ(λ(z1, f), h2g) =
λ(λ(z1, h2), fg) = λ(λ(z2, h1), fg) = λ(λ(z2, g), h1f) = λ(λ(y, h2), h1f) =
λ(y, h2h1f). Hence (x, h2h1f) ≈ (y, h2h1g) and, in general h = h1h2 · · · hn.

Conversely, assume that (x, hf) ≈ (y, hg) for some h ∈ S. It is shown
that (x, f) ≈ (λ(y, h), hg) and (λ(y, h), hg) ≈ (y, g). Indeed, by hypothe-
sis, λ(x, hg) = λ(y, hf) = λ(λ(y, h), f) and thus (x, f) ≈ (λ(y, h), hg). Also,
λ(λ(y, h), g) = λ(y, hg) and hence (λ(y, h), hg) ≈ (y, g). It follows that (x, f) ∼
(y, g).

(b): Suppose that x ≃ y; then there exists h ∈ S such that λ(x, h) = λ(y, h).
Assume that g ∈ S. Then (λ(x, g), h) ≈ (λ(y, g), h)). Indeed, since (S, .) is
commutative, λ(λ(x, g), h) = λ(λ(x, h), g) = λ(λ(y, h), g) = λ(λ(y, g), h), and
thus λ(x, g) ≃ λ(y, g). Conversely, x = λ(x, e) ≃ λ(y, e) = y.



Convergence semigroup categories 71

(c): Recall that X∗ = X/ ≃= {[x] : x ∈ X} and λ∗([x], g) = [λ(x, g)].
First, note that λ∗ is well-defined. Indeed, if x ≃ y and g ∈ S, then by (b),
λ(x, g) ≃ λ(y, g) implies that λ∗([x], g) = λ∗([y], g), and thus λ∗ is well-defined.
Next, it is shown that for g ∈ S fixed, λ∗(., g) is an injection. Suppose that
λ∗([x], g) = λ∗([y], g); then λ(x, g) ≃ λ(y, g) and it follows by (R3) that there
exists h ∈ S such that λ(λ(x, g), h) = λ(λ(y, g), h). Equivalently, λ(x, gh) =
λ(y, gh), and again by (R3), x ≃ y. Hence λ∗(., g) is an injection.

(d): Recall that η(〈(x, g)〉) = 〈([x], g)〉, x ∈ X, g ∈ S. First, observe that η is
well-defined. Indeed, assume that (x, g) ∼ (y, h); then by (a), there exists k ∈ S
such that (x, kg) ≈ (y, kh). Hence λ(x, kh) = λ(y, kg), and thus λ∗([x], kh) =
[λ(x, kh)] = [λ(y, kg)] = λ∗([y], kg). Therefore ([x], kg) ≈ ([y], kh) on X∗ × S,
and thus by (a) and (c), ([x], g) ≈ ([y], h). It follows that 〈([x], g)〉 = 〈([y], h)〉,
and hence η is well-defined.

Next, it is shown that η is an injection. Suppose that 〈([x], g)〉 = η(〈(x, g)〉) =
η(〈(y, h)〉) = 〈([y], h)〉; then ([x], g) ≈ ([y], h) in X∗ × S. Hence [λ(x, h)] =
λ∗([x], h) = λ∗([y], g) = [λ(y, g)], and thus λ(x, h) ≃ λ(y, g). According to
(R3), there exists k ∈ S such that λ(x, hk) = λ(λ(x, h), k) = λ(λ(y, g), k) =
λ(y, gk). Hence (x, gk) ≈ (y, hk), and by part (a), (x, g) ∼ (y, h). Therefore
〈(x, g)〉 = 〈(y, h)〉, and thus η is an injection. Clearly η is a surjection, and
consequently η is a bijection. �

2. Action Categories

. Consider the triple (X, (S, .), λ), where X ∈ |SET|, (S, .) ∈ |SG|, (S, .) is
commutative, and λ : X × S → X is an action. Define X to be the category
whose objects consist of all triples (X, (S, .), λ), and whose morphisms are all
pairs (f, k) : (X, (S, .), λ) → (Y, (T, .), µ) obeying:

(B1) f : X → Y is a map, k : (S, .) → (T, .) is a homomorphism
(B2) f ◦ λ = µ ◦ (f × k).

Furthermore, let C denote the category consisting of all objects ((X, q),
(S, ., p), λ) ∈ AC, and whose morphisms (f, k) : ((X, q), (S, .), λ) → ((Y, qY ),
(T, ., pT ), µ) satisfy:

(C1) f : (X, q) → (Y, qY ) is continuous, k : (S, ., p) → (T, ., pT ) is a continu-
ous homomorphism

(C2) f ◦ λ = µ ◦ (f × k).

Clearly idX ×idS : ((X, q), (S, ., p), λ) → ((X, q), (S, ., p), λ) is the identity mor-
phism in C. Also, observe that the composition of two C-morphisms is again a C-
morphism. Indeed, suppose that (f, k) : ((X, q), (S, ., p), λ) → ((Y, qY ), (T, ., pT ),
µ) and (h, l) : ((Y, qY ), (T, ., pT ), µ) → ((Z, qZ), (R, ., pR), δ) are two C-morphisms.
Clearly (C1) is satisfied. It remains to verify (C2). Since (f, k) and (h, l) each
obeys (C2), f ◦ λ = µ ◦ (f × k) and h ◦ µ = δ ◦ (h × l). If (x, g) ∈ X × S,
then δ ◦ (h × l) ◦ (f × k)(x, g) = δ ◦ (h × l)(f(x), k(g)) = (h ◦ µ)(f(x), k(g)) =
h(µ ◦ (f × k))(x, g) = h(f ◦ λ)(x, g) = ((h ◦ f) ◦ λ)(x, g), and it follows that
(h ◦ f) ◦ λ = δ ◦ (h × l) ◦ (f × k). Hence (C2) is valid and C is a category;
likewise, X , is also a category.



72 H. Boustique, P. Mikusiński and G. Richardson

If U : C → X denotes the faithful functor defined by U((X, q), (S, ., p), λ) =
(X, (S, .), λ), then (C, U) is a concrete category over the base category X . Let
D denote the full subcategory of C consisting of all objects ((X, q), (S, ., p), λ) ∈
|C| such that λ(., g) : X → X is an injection, for each fixed g ∈ S. Then
(D, U ◦ E) is also a concrete category over X , where E : D → C denotes the
inclusion functor.

Theorem 2.1. The category D is reflective in C.

Proof. Given ((X, q), (S, .p), λ) ∈ |C|, consider ((X∗, q∗), (S, ., p), λ∗), where
(X∗, q∗) ∈ |CONV| and λ∗ are defined in section 1. Then λ∗ : X∗ × S → X∗
is an action, and by Lemma 1.2(c), λ∗(., g) : X∗ → X∗ is an injection. Recall
that q∗ is the quotient structure in CONV determined by the canonical map
ξ : (X, q) → (X∗, q∗), ξ(x) = [x].

Since quotient maps are productive in CONV, ξ × idS is a quotient map.
Moreover, observe that λ∗ ◦(ξ ×idS) = ξ ◦λ, ξ ◦λ is continuous, and hence λ∗ :
(X∗, q∗)×(S, ., p) → (X∗, q∗) is a continuous map. If follows that ((X∗, q∗), (S, .p),
λ∗) ∈ |D|. Moreover, (ξ, idS) : ((X, q), (S, ., p), λ) → ((X∗, q∗), (S, ., p), λ∗) is
a C morphism. Clearly (C1) is satisfied, and as mentioned above, ξ ◦ λ =
λ∗ ◦ (ξ × idS); hence (ξ, idS) is a C-morphism.

Assume that (f, k) : ((X, q), (S, ., p), λ) → ((Y, qY ), (T, ., pT ), µ) is a C-
morphism, where ((Y, qY ), (T, ., pT ), µ) ∈ |D|. Then f ◦ λ = µ ◦ (f × k). Define
f∗ : X∗ → Y by f∗([x]) = f(x). Observe that f∗ is well-defined. Indeed, if
x1 ∈ [x], then by (R3), λ(x, g) = λ(x1, g) for some g ∈ S. Hence µ(f(x), k(g)) =
(µ ◦ (f × k))(x, g) = (f ◦ λ)(x, g) = (f ◦ λ)(x1, g) = (µ ◦ (f × k))(x1, g) =
µ(f(x1), k(g)), and thus by (R3), f(x) ≃ f(x1). Since Y∗ = Y , f∗ : X∗ → Y
is well-defined. Moreover, f∗ ◦ ξ = f is continuous, ξ : (X, q) → (X∗, q∗) is
a quotient map in CONV, and thus it follows that f∗ is continuous. Finally,
(µ ◦ (f∗ × k))([x], g) = µ(f(x), k(g)) = (µ ◦ (f × k))(x, g) = (f ◦ λ)(x, g) =
(f∗ ◦ λ∗)([x], g), for each ([x], g) ∈ X∗ × S, and thus f∗ ◦ λ∗ = µ ◦ (f∗ × k).
Hence (f∗, k) : ((X∗, q∗), (S, ., p), λ∗) → ((Y, q

Y ), (T, ., pT ), µ) is a D-morphism.
Therefore D is a reflective subcategory of C. �

Theorem 2.2. The concrete category (C, U) over X is topological.

Proof. Assume that (fj, kj) : ((X, (S, .), λ) → U((Xj, qj), (Sj, ., pj), λj), j ∈ J,

is a source in X . Define F
q
−→ x (G

p
−→ g) iff for each j ∈ J, f→j F

qj
−→

fj(x)(k
→

j G
pj
−→ kj(g)), respectively. Then q(p) is the initial structure in CONV

(CSG) determined from fj : X → (Xj, qj) (kj : (S, .) → (Sj, ., pj)), j ∈ J,
respectively. Next, it is shown that λ : (X, q) × (S, p) → (X, q) is continuous.

Assume that F
q
−→ x and G

p
−→ g; then f→j F

qj
−→ fj(x) and k

→

j G
pj
−→ kj(g),

for each j ∈ J. Employing the hypothesis, fj ◦ λ = λj ◦ (fj × kj), it fol-

lows that f→j (λ
→(F × G)) = λ→j (fj × kj)

→(F × G) = λ→j (f
→

j F × k
→

j G)
qj
−→

λj(fj(x), kj(g)) = (λj ◦ (fj × kj))(x, g) = fj(λ(x, g)), for each j ∈ J. Hence

λ→(F×G)
q
−→ λ(x, g), and thus λ is a continuous action. Then ((X, q), (S, ., p), λ)



Convergence semigroup categories 73

∈ |C|, and thus (fj, kj) : ((X, q), (S, ., p), λ) → ((Xj, qj), (Sj, ., pj), λj) is a C-
morphism.

Finally, suppose that (f, k) : U((Z, qZ), (T, ., pT ), µ) → U((X, q), (S, ., p), λ)
is a X-morphism such that (fj × kj) ◦ (f × k) : ((Z, q

Z), (T, ., pT ), µ) →
((Xj, qj), (Sj, ., pj), λj) is a C-morphism. Since q(p) is the initial structure in
CONV (CSG) determined by fj : X → (Xj, qj) (kj : (S, .) → (Sj, ., pj)), j ∈ J,
it follows that f : (Z, qZ) → (X, q) and k : (T, ., pT ) → (S, ., p) are continu-
ous, respectively. Moreover, p and q are the unique structures possessing these
properties. Since (f, k) is an X-morphism, f ◦ µ = λ ◦ (f × k), and thus (f, k)
is a C-morphism. Therefore (C, U) is topological. �

Remark 2.3. Theorem 2.2 shows that products exist in C. In particular, sup-
pose that ((Xj, qj), (Sj, ., pj), λj) ∈ |C|, for each j belonging to set J. Denote
the product set by X = ×

j∈J
Xj, the product semigroup by (S, .) = ×

j∈J
(Sj, .), the

jth projection map by πj1 × πj2 : X × S → Xj × Sj, and define λ : X × S → X
by λ((xj)j∈J , (gj)j∈J ) = (λj(xj, gj))j∈J . It follows that λ is an action, and
(X, (S, .), λ) ∈ |X|. Observe that for each j ∈ J, πj1 ◦ λ = λj ◦ (πj1 × πj2), and
thus (πj1, πj2) : (X, (S, .), λ) → U((Xj, qj), (Sj, ., pj), λj) is an X-morphism.
Then by Theorem 2.2, ((X, q), (S, ., p), λ) is the unique U-initial lift, where q(p)
are product structures in CONV (CSG), respectively.

It was shown in Theorem 2.2 that every U-structured source has a unique
U-initial lift. This also implies that every U-structured sink has a unique U-final
lift; for example, see Theorem 21.9 [1]. Quotient morphisms in C are considered
in the next theorem. Recall that if ((X, q), (S, ., p), λ) ∈ |C|, then B(X, S) =
(X×S)/ ∼ denotes the generalized quotient, where ∼ is the equivalence relation
defined in (R2). Let r = q×p; then ϕ : (X×S, r) → (B(X, S), σ) is the quotient
map ϕ(x, g) = 〈(x, g)〉, and σ is the corresponding quotient structure in CONV.

Theorem 2.4. Assume that ((X, q), (S, ., p), λ) ∈ |C|. Then

(a) a surjective X-morphism (f, k) : U((X, q), (S, .p), λ) → (Y, (T, .), µ) has
a unique U-final lift to a quotient map (f, k) : ((X, q), (S, ., p), λ) →
((Y, qY ), (T, ., pT ), µ) in C.

(b) (ϕ, idS) : ((X ×S, r), (S, ., p), Λ) → ((B(X, S), σ), (S, ., p), λB) is a quo-
tient map in C, where Λ((x, g), h) := (λ(x, h), g) and λB(〈(x, g)〉, h) =
〈(λ(x, h), g)〉.

(c) (η, idS) : ((B(X, S, λ), σ), (S, ., p), λB) → ((B(X∗, S, λ∗), (S, ., p), λ
∗

B)
is a C-isomorphism, where λ∗B(〈([x], g)〉, h) = 〈(λ∗([x], h), g)〉 =
〈([λ(x, h)], g)〉.

Proof. (a): Fix (y, t) ∈ Y × T , and define H
q
Y

−−→ y (K
p
T

−−→ t) iff there exists

F
q
−→ x ∈ f−1(y) (G

p
−→ g ∈ k−1(t)) such that f→F = H (k→G = K), respec-

tively. Then qY (pT ) is the quotient structure in CONV (CSG). It is shown that
the action µ : (Y, qY )×(T, ., pT ) → (Y, qY ) is continuous. Indeed, suppose that

H
q
Y

−−→ y and K
p
T

−−→ t; then there exist F
q
−→ x ∈ f−1(y) and G

p
−→ g ∈ k−1(t)



74 H. Boustique, P. Mikusiński and G. Richardson

such that f→F = H and k→G = K. Since (f, k) is an X-morphism, µ◦(f ×k) =
f ◦ λ, and thus µ→(H × K) = µ→(f→F × k→G) = (µ ◦ (f × k))→(F × G) =

(f ◦ λ)→(F × G)
q
Y

−−→ (f ◦ λ)(x, g) = (µ ◦ (f × k))(x, g) = µ(y, t). Therefore µ
is continuous, and (f, k) is a C-morphism.

Next, suppose that (F, K) : U((Y, qY ), (T, ., pT ), µ) → U((Z, qZ), (R, ., pR), δ)
is an X-morphism such that (F, K) ◦ (f, k) : ((X, q), (S, ., p), λ) → (Z, qZ),
(R, ., pR), δ) is a C-morphism. Since qY and pT are quotient structures in CONV
and CSG, F and K are continuous maps, and thus (F, K) is a C-morphism.
Moreover qY and pT are unique and hence (f, k) is a quotient map in C.

(b): Observe that Λ is an action. Indeed, Λ((x, g), e) = (λ(x, e), g) =
(x, g). Moreover, Λ(Λ((x, g), h), k) = Λ((λ(x, h), g), k) = (λ(λ(x, h), k), g) =
(λ(x, hk), g) = Λ((x, g), hk), and thus Λ is an action. Note that Λ is the
composition of the continuous maps :((x, g), h) 7→ ((x, h), g) 7→ (λ(x, h), g).
Therefore Λ is a continuous action, and thus ((X × S, r), (S, ., p), Λ) ∈ |C|.

First, it is shown that λB is well-defined. It must be shown that if (x, g) ∼
(x1, g1), then λB(〈(x, g)〉, h) = λB(〈(x1, g1)〉, h). According to Lemma 1.3(a),
there exists k ∈ S such that (x, kg) ≈ (x1, kg1), or λ(x, kg1) = λ(x1, kg). Note
that λ(λ(x, h), kg1) = λ(λ(x, kg1), h) = λ(λ(x1, kg), h) = λ(λ(x1, h), kg), and
thus (λ(x, h), kg) ≈ (λ(x1, h), kg1). Again, by Lemma 1.3(a), (λ(x, h), g) ∼
(λ(x1, h), g1), and thus λB is well-defined.

Next, λB is an action. Indeed, λB(〈(x, g)〉, e) = 〈(λ(x, e), g)〉 = 〈(x, g)〉, and
λB(λB(〈(x, g)〉, h), k) = λB(〈(λ(x, h), g)〉, k) = 〈(λ(λ(x, h), k), g)〉 = 〈(λ(x, hk),
g)〉 = λB(〈(x, g)〉, hk). Hence λB is an action.

Since ϕ (idS) are quotient maps in CONV (CSG), respectively, it remains
to show that ϕ ◦ Λ = λB ◦ (ϕ × idS). Let ((x, g), h) ∈ (X × S) × S; then
(λB ◦ (ϕ × idS))((x, g), h) = λB(〈(x, g)〉, h) = 〈(λ(x, h), g)〉 = 〈Λ((x, g), h)〉 =
(ϕ ◦ Λ)((x, g), h). Therefore ϕ ◦ Λ = λB ◦ (ϕ × idS), and thus by part (a), λB is
continuous, and (ϕ, idS) : ((X × S, r), (S, ., p), Λ) → ((B(X, S), σ), (S, ., p), λB)
is a quotient map in C.

(c): Recall from Lemma 1.3(d) that η is a bijection, where η : (B(X, S, λ), σ) →
((B(X∗, S, λ∗), σ∗) is defined by η(〈(x, g)〉 = 〈([x], g)〉. It is shown that η is a
homeomorphism in CONV. Observe that the diagram below commutes:

(X × S, r)
ϕ
- (B(X, S), σ)

(X∗ × S, r∗)

ξ × idS

? ϕ∗
- (B(X∗, S), σ∗)

η

?

Since ϕ is a quotient map in CONV, η is continuous iff η ◦ ϕ is continuous.
However, η ◦ ϕ = ϕ∗ ◦ (ξ × idS) is continuous by construction, and thus η is
continuous. Also, ϕ∗ is a quotient map in CONV, and hence η

−1 is continuous
iff η−1 ◦ ϕ∗ is continuous. Since ξ × idS is a quotient map, η

−1 ◦ ϕ∗ is continu-
ous iff (η−1 ◦ ϕ∗) ◦ (ξ × idS) is continuous. However, the latter map is simply



Convergence semigroup categories 75

ϕ, and hence η−1 ◦ ϕ∗ is continuous. Therefore η
−1 is continuous, and thus

η : (B(X, S), σ) → (B(X∗, S), σ∗) is a homeomorphism in CONV.
Since η is a homeomorphism, ((B(X, S), σ), (S, ., p), λB) ∈ |C|, ((B(X∗, S), σ∗),

(S, ., p), λ∗B) ∈ |C|, and it remains to show that η ◦ λB = λ
∗

B ◦ (η × idS) and
η−1 ◦ λ∗B = λB ◦ (η

−1 × idS). Let (〈(x, g)〉, h) ∈ B(X, S) × S; then λ
∗

B ◦
(η × idS)(〈(x, g)〉, h) = λ

∗

B(〈([x], g)〉, h) = 〈(λ∗([x], h), g)〉 = 〈([λ(x, h)], g)〉 =
η(〈(λ(x, h), g)〉) = (η◦λB)(〈(x, g)〉, h). Hence η◦λB = λ

∗

B ◦(η×idS). Moreover,
since η is a bijection, λB = η

−1 ◦λ∗B ◦(η ×idS) and λB ◦(η
−1 ×idS) = η

−1 ◦λ∗B.
Therefore (η, idS) is a C-isomorphism. �

Remark 2.5. (i): Assume that ((X, q), (S, ., p), λ) ∈ |C|. An object in CONV
is called Hausdorff whenever each filter converges to at most one element. It is
shown in Theorem 4.1 [3] that (B(X, S, λ), σ) is Hausdorff iff (X, q) is Hausdorff,
provided that λ(., g) is an injection for each g ∈ S. Since λ∗(., g) is an injection
for each g ∈ S, and (B(X, S, λ), σ) and (B(X∗, S, λ∗), σ∗) are homeomorphic,
it follows that (B(X, S, λ), σ) is Hausdorff iff (X∗, q∗) is Hausdorff.

(ii): Suppose that (f, k) : ((X, q), (S, ., p), λ) → ((Y, qY ), (T, ., pT ), µ) is an
isomorphism in C. In particular, f−1 (k−1) is an isomorphism in CONV (CSG),
respectively. Moreover, since f ◦ λ = µ ◦ (f × k), then µ = f ◦ λ ◦ (f−1 × k−1)
and λ = f−1 ◦ µ◦ (f × k). Hence each action can be found from the other. It is
not enough just to assume that f and k are isomorphisms in CONV and CSG,
respectively.

3. Extensions

Consideration is given to embedding objects in C into objects which have
nicer properties such as compactness. Recall that a convergence space (X, q) is
said to be Hausdorff whenever each filter converges to at most one element. A

(Hausdorff) convergence space (X, q) is regular (T3) provided clqF
q
−→ x when-

ever F
q
−→ x, respectively. Moreover, (X, q) is compact if each ultrafilter on

X q-converges. Given (X, q) ∈ |CONV|; (X, πq) denotes the pretopological

modification of (X, q), where F
πq
−→ x iff F ≥ ∩{G : G

q
−→ x} (neighborhood

filter at x). It is shown that in Theorem 1 [18] that (X, q) ∈ |CONV| has a T3
compactification iff πq is a Hausdorff completely regular topology, and q and
πq agree on ultrafilter convergence. In this case, there exists a T3 compactifi-
cation (X∗, q∗, δ) such that δ : (X, q) → (X∗, q∗) is a dense embedding, and if
f : (X, q) → (Y, ρ) is a continuous function, where (Y, ρ) is compact T3, then
there exists a continuous map f∗ : (X∗, q∗) → (Y, ρ) such that f∗ ◦ δ = f

Lemma 3.1. Suppose that (f1, k1) and (f, k) are C-morphisms, F(K) is a
morphism in CONV (CSG), respectively, f1(X) and k(S) are dense, and the
diagram below commutes. Then (F, K) is also a C-morphism.



76 H. Boustique, P. Mikusiński and G. Richardson

((X, q), (S, ., p), λ)
(f1, k1)

- ((X1, q1), (S1, ., p1), λ1)

((Y, qY ), (T, ., pT ), µ)

(f, k)

?
�

(F,
K)

Proof. It remains to show that F ◦ λ1 = µ ◦ (F × K). Fix (z, g) ∈ X1 ×
S1. Since f1(X)(k1(S)) is dense, there exists F(G) ∈ F(X)(F(S)) such that

f→1 F
q1
−→ z(k→1 G

p1
−→ g), respectively. Employing the assumptions, it follows

that (µ ◦ (F × K))(z, g) = µ(F(z), K(g)) = µ(F(lim f→1 F), K(lim k
→

1 G) =
µ(lim(F ◦ f1)

→F, lim(K ◦ k1)
→G) = lim µ→(f→F × k→G) = lim(µ ◦ (f ×

k))→(F × G) = lim(f ◦ λ)→(F × G) = lim((F ◦ f1) ◦ λ)
→(F × G) = lim F →(f1 ◦

λ)→(F ×G) = lim F →(λ1 ◦(f1 ×k1))
→(F ×G) = lim(F ◦λ1)

→(f→1 F ×k
→

1 G) =
(F ◦ λ1)(lim(f

→

1 F × k
→

1 G)) = (F ◦ λ1)(z, g). Hence F ◦ λ1 = µ ◦ (F × K), and
thus (F, K) is a C-morphism. �

Theorem 3.2. Assume that (X, q) has a T3-compactification, and (f, k) :
((X, q), (S, ., p), λ) → ((Y, qY ), (T, ., pT ), µ) is a C-morphism, where (Y, qY ) is
compact T3, and p is the discrete structure. Then, using the notations above,
(δ, idS) is a dense C-embedding and, moreover, (f

∗, k) is a C-morphism such
that the diagram below commutes, for some λ∗:

((X, q), (S, ., p), λ)
(δ, idS)

- ((X∗, q∗), (S, ., p), λ∗)

((Y, qY ), (T, ., pT ), µ)

(f, k)

?
�

(f
∗ , k)

Proof. Fix g ∈ S. Then the subspace X∗ × {g} of X∗ × S is a T3 compact-
ification of X × {g} which possesses the continuous extension property. Let
λg : X × {g} → X be the function λg(x, g) = λ(x, g), x ∈ X. Since δ ◦ λg is
continuous, there exists a continuous extension λ∗g such that the diagram below
commutes:

X × {g}
δ ◦ λg

- X∗

X∗ × {g}

δ × idg

?

λ
∗
g

-



Convergence semigroup categories 77

Define λ∗ : X∗ × S → X∗ by λ∗(z, g) = λ∗g(z, g), for each z ∈ X
∗, g ∈ S.

Since p is the discrete structure on S, it follows that λ∗ is continuous and,
moreover, since the diagram above commutes, (λ∗ ◦(δ ×idS))(x, g) = (λ

∗

g ◦(δ ×
idg))(x, g) = (δ ◦ λg)(x, g) = (δ ◦ λ)(x, g), for each (x, g) ∈ X × S. Hence λ

∗ ◦
(δ ×idS) = δ ◦λ, and thus (δ, idS) : ((X, q), (S, ., p), λ) → ((X

∗, q∗), (S, ., p), λ∗)
is a C-morphism.

Next, it is shown that λ∗ : X∗ × S → X∗ is a action. As shown above,
λ∗◦(δ×idS) = δ◦λ, and thus if x ∈ X, then λ

∗(δ(x), e)) = (λ∗◦(δ×idS))(x, e) =
(δ ◦ λ)(x, e) = δ(λ(x, e)) = δ(x). Moreover, if z ∈ X∗ − δ(X), then since δ(X)

is dense in X, there exists a filter F on X such that δ→F
q
∗

−→ z . However, λ∗

is continuous whenever p is the discrete structure, and thus λ∗→(δ→F × ė)
q
∗

−→
λ∗(z, e). Moreover, λ∗→(δ→F × ė) = (λ∗→ ◦(δ ×idS))

→(F × ė) = (δ ◦λ)→(F ×

ė) = δ→(λ→(F × ė)) = δ→F
q
∗

−→ z. Since (X∗, q∗) is Hausdorff, λ∗(z, e) = z.
Let z ∈ X and g, h ∈ S; it is shown that λ∗(λ∗(z, g), h) = λ∗(z, gh). First,

suppose that z = δ(x). Then λ∗(λ∗(δ(x), g), h) = λ∗(λ∗ ◦ (δ × idS)(x, g), h) =
λ∗((δ ◦ λ)(x, g), h) = λ∗(δ(λ(x, g)), h) = (λ∗ ◦ (δ × idS))(λ(x, g), h) = (δ ◦
λ)(λ(x, g), h) = δ(λ(λ(x, g), h)) = δ(λ(x, gh)) = (δ ◦ λ)(x, gh) = (λ∗ ◦ (δ ×
idS))(x, gh) = λ

∗(δ(x), gh). Hence λ∗(λ∗(δ(x), g), h) = λ∗(δ(x), gh). Further,
assume that z ∈ X∗ − δ(X); it is shown that λ∗(λ∗(z, g), h) = λ∗(z, gh).

There exists a filter F on X such that δ→F
q
∗

−→ z. Since λ∗ is continuous

whenever p is the discrete structure, λ∗→(δ→F × ġ)
q
∗

−→ λ∗(z, g). Employing
λ∗◦(δ×idS) = δ◦λ, λ

∗→(δ→F ×ġ) = [λ∗◦(δ×idS)]
→(F ×ġ) = (δ◦λ)→(F ×ġ),

and thus λ∗→[(δ◦λ)→(F×ġ)×ḣ]
q
∗

−→ λ∗(λ∗(z, g), h). However, λ∗→[(δ◦λ)→(F×
ġ) × ḣ] = [λ∗ ◦ (δ × idS)]

→(λ→(F × ġ) × ḣ) = (δ ◦ λ)→(λ→(F × ġ) × ḣ) =

δ→[λ→(λ→(F × ġ) × ḣ)] = δ→(λ→(F × ˙gh)) = [λ∗ ◦ (δ × idS)]
→(F × ˙gh) =

λ∗→(δ→F × ġh)
q
∗

−→ λ∗(z, gh). Since (X∗, q∗) is Hausdorff, it follows that
λ∗(λ∗(z, g), h) = λ∗(z, gh), and thus λ∗ is a continuous action.

It remains to show that (δ−1
δ(X)

, idS) : ((δ(X), q
∗|δ(X)), (S, ., p), λ

∗|δ(X)×S) →

((X, q), (S, ., p), λ) is a C-morphism. Since δ is a embedding, only (δ−1 ◦
λ∗)|δ(X)×S = λ ◦ (δ

−1
δ(X)

× idS) must be verified. Using the fact that λ
∗ ◦

(δ × idS) = δ ◦ λ and δ is an embedding, λ
∗|δ(X)×S = δ ◦ λ ◦ (δ

−1
δ(X)

× idS) and

δ−1
δ(X)

◦ λ∗|δ(X)×S = λ ◦ (δ
−1
δ(X)

× idS). Hence (δ
−1
δ(X)

, idS) is also a C-morphism,

and thus (δ, idS) : ((X, q), (S, ., p), λ) → ((X
∗, q∗), (S, ., p), λ∗) is an embedding

in C.
Finally, since (δ, idS), (f, k) are C-morphisms and δ(X) is dense in X

∗, it
follows from Lemma 3.1 that (f∗, k) is a C-morphism. �

Suppose that (X, q) has the T3 compactification (X∗, q∗, δ) mentioned above,

and assume that ((X, q), (S, ., p), λ) ∈ |C|. Define G
p
∗

−→ g iff for each H
q
∗

−→ z,

λ∗→(H × G)
q
∗

−→ λ∗(z, g).



78 H. Boustique, P. Mikusiński and G. Richardson

Corollary 3.3. Assume that ((X, q), (S, ., p), λ) ∈ |C|, and (X∗, q∗, δ) is the T3
compactification of (X, q) described above. Then p∗ is the coarsest structure on
S such that ((X∗, q∗), (S, ., p∗), λ∗) ∈ |C|.

Proof. First, it is shown that (S, p∗) ∈ |CONV|. Fix g ∈ S, Suppose that

H
q
∗

−→ z; then according to Theorem 3.1, λ∗ : (X∗, q∗) × (S, p) → (X∗, q∗)

is continuous whenever p has the discrete structure. Hence λ∗→(H × ġ)
q
∗

−→

λ∗(z, g), and thus ġ
p
∗

−→ g. Moreover, it easily follows that if K
p
∗

−→ g and

L ≥ K, then L
p
∗

−→ g. Hence (S, p∗) ∈ |CONV|. Next, assume that Gi
p
∗

−→ gi,

i = 1, 2, and H
q
∗

−→ z; then λ∗→(H × G1G2) = λ
∗→(λ∗→(H × G1) × G2)

q
∗

−→

λ∗(λ∗(z, g1), g2) = λ
∗(z, g1g2). Hence G1G2

p
∗

−→ g1g2, and thus (S, ., p
∗) ∈

|CSG|. It follows from the definition that p∗ is the coarsest structure such that
((X∗, q∗), (S, ., p∗), λ∗) ∈ |C|. �

Let (X, q) ∈ |CONV|. It is shown in [17] that if (X, q) is Hausdorff, then

there exists a Hausdorff compactification (X̂, q̂), where j : (X, q) → (X̂, q̂)

denotes a dense embedding, and X̂ = j(X) ∪ {α : α = G is an ultrafilter on X

which fails to q-converge}. Let F ∈ F(X); then F̂ denotes the filter on X̂

whose base is {F̂ : F ∈ F}, and F̂ = j(F) ∪ {α ∈ X̂ : F ∈ α}. Define

H
q̂
−→ j(x) iff H ≥ F̂ for some F

q
−→ x, and H

q̂
−→ α iff H ≥ Ĝ for some α = G.

According to [17], if f : (X, q) → (Y, qY ) is continuous, (Y, qY ) is compact T3,

then f̂ : (X̂, q̂) → (Y, qY ) is a continuous extension of f, where f̂(j(x) = f(x)

and f̂(α) = lim f→F in (Y, qY ). Moreover, it is easily verified that the above
results are valid whenever (X, q) fails to be Hausdorff.

Given any (X, q) ∈ |CONV|, it is shown in Proposition 2.1 [8] that there
exists a finest regular convergence rq which is coarser than q. Define x ∼

y iff ẋ
rq
−→ y; let sX be the set of all equivalence classes, and denote the

corresponding quotient map by Φ : X → sX. Let sq be the quotient structure
in CONV determined by: Φ : (X, rq) → (sX, sq). According to Proposition 1.3
[9], (sX, sq) is T3. Moreover, it is shown in [8] and [9] that if f is continuous,
then the maps below are continuous and the diagram commutes:

(X, q)
idX
- (X, rq)

ΦX
- (sX, sq)

(Y, p)

f

? idY
- (Y, rp)

f

? ΦY
- (sY, sp)

fs

?

, where fs([x]) = f(x).

In particular, if f : (X, q) → (Y, qY ) is continuous and (Y, qY ) is compact T3,
then the maps below are continuous and the diagram commutes:



Convergence semigroup categories 79

(X, q)
j
- (X̂, q̂)

id
X̂- (X̂, rq̂)

Φ
- (sX̂, sq̂)

(Y, qY )

f

?
idY
- (Y, qY )

f̂

?
idY
- (Y, qY )

f̂

?
idY
- (Y, qY )

f̂s

?

Theorem 3.4. Assume that (f, k) : ((X, q), (S, ., p), λ) → ((Y, qY ), (T, ., pT ), µ)
is a C-morphism, where (X, q) is Hausdorff, p is the discrete structure on S,

and (Y, qY ) is compact T3. Using the notations defined above, (f̂, k) and (f̂s, k)
are C-morphisms such that the diagram below commutes:

((X, q), (S, ., p), λ)
(j, idS)

- ((X̂, rq̂), (S, ., p), λ̂)
(Φ, idS)

- ((sX̂, sq̂), (S, ., p), λ̂s)

((Y, qY ), (T, ., pT ), µ)

(f̂, k))

? �

(f̂s
, k)

(f, k)
-

Proof. It follows from (3.1) that f̂, f̂s are continuous, and the diagram com-

mutes. Using the notations given above, define λ̂ : X̂ × S → X̂ by λ̂(j(x), g) =

(j ◦ λ)(x, g), and if α = F, λ̂(α, g) = lim(j ◦ λ)→(F × ġ) in (X̂, q̂). Observe

that λ̂(Â × B) ⊆ clrq̂(j ◦ λ)(A × B) whenever A ⊆ X and B ⊆ S, and thus,

since p is discrete, λ̂ : (X̂, q̂) × (S, p) → (X̂, rq̂) is continuous. According to

Theorem 6.3 [8], r(q̂ × p) = rq̂ × p, and thus λ̂ : (X̂, rq̂) × (S, p) → (X̂, rq̂)

is continuous. Moreover, define λ̂s : sX̂ × S → sX̂ by λ̂s([z], g) = [λ̂(z, g)].

Note that λ̂s is well-defined. Indeed, if z1 ∼ z2, then ż1
rq̂
−→ z2, and since

λ̂ : (X̂, rq̂) × (S, p) → (X̂, rq̂) is continuous, λ̂ ˙(z1, g) = λ̂
→(ż1 × ġ)

rq̂
−→ λ̂(z2, g).

Hence λ̂(z1, g) ∼ λ̂(z2, g), and thus λ̂s is well-defined. Observe that the diagram
below commutes and Φ × idS is a quotient map:

(X̂, rq̂) × (S, p)
λ̂
- (X̂, rq̂)

(sX̂, sq̂) × (S, p)

Φ × idS

?
λ̂s
- (sX̂, sq̂)

Φ

?

Since Φ × idS is a quotient map and Φ × λ̂ is continuous, it follows that λ̂s
is continuous. Then ((X̂, rq̂), (S, ., p), λ̂), ((sX̂, sq̂), (S, ., p), λ̂s) ∈ |C|, and it is



80 H. Boustique, P. Mikusiński and G. Richardson

straightforward to verify that (j, idS) and (Φ, idS) are C-morphisms. Moreover,

employing Lemma 3.1, (f̂, k) and (f̂s, k) are also C-morphisms. �

A commutative semigroup can be embedded in a group iff it is cancellative.
One way of constructing the group is by means of equivalence classes of or-
dered pairs just as one forms the rationals from the integers. Let CG denote
the category whose objects consist of all the commutative convergence groups,
and having all the continuous group homomorphisms as its morphisms. Let
(S, .) ∈ |SG| be commutative and cancellative; then a special case of Theorem
1.24 [6] shows that (S, .) is embedded in the group (S̄, .), where elements in S̄
can be expressed in the form gh−1, for some g, h ∈ S. The natural injection is
denoted by j : (S, .) → (S̄, .), where j(g) = g for all g ∈ S. This notation is
used below.

Lemma 3.5. Assume that (S, ., p) ∈ |CSG| is commutative and cancellative.
Then there is a finest structure p̄ on S̄ such that (S̄, ., p̄) ∈ |CG| and j : (S, p) →
(S̄, p̄) is continuous.

Proof. Let H ∈ F(S̄); then H−1 denotes the filter on S̄ whose base is {H−1 :

H ∈ H}. Define p̄ on S̄ as follows: K
p̄
−→ gh−1 iff there exist G

p
−→ g1 and H

p
−→ h1

such that K ≥ (j→G)(j→H)−1 and g1h
−1
1 = gh

−1. Clearly ˙gh−1
p̄
−→ gh−1 since

ġ
p
−→ g and ḣ

p
−→ h. Also, if L ≥ K and K

p̄
−→ gh−1, then L

p̄
−→ gh−1. Hence

(S̄, p̄) ∈ |CONV|.
Next, it is shown that multiplication in (S̄, ., p̄) is a continuous operation.

Suppose that Ki
p̄
−→ gih

−1
i , Gi

p
−→ gi, Hi

p
−→ hi and Ki ≥ (j

→Gi)(j
→Hi)

−1,
i = 1, 2. Since (S, .) is commutative and j is a homomorphism, K1.K2 ≥
(j→G1.j

→G2) (j
→H1)

−1(j→H2)
−1 = j→(G1.G2)(j

→(H1.H2))
−1. However,

G1.G2
p
−→ g1g2, H1.H2

p
−→ h1h2, and thus K1.K2

p̄
−→ (g1g2)(h1h2)

−1 = (g1h
−1
1 )

(g2h
−1
2 ). Hence multiplication in (S̄, ., p̄) is continuous.

Finally, inversion is a continuous operation. Indeed, suppose that K
p̄
−→ gh−1,

G
p
−→ g, H

p
−→ h, and K ≥ (j→G)(j→H)−1. Then K−1 ≥ (j→H)(j→G)−1

p̄
−→

hg−1 = (gh−1)−1, and thus (S̄, .p̄) ∈ |CG|
Assume that (S̄, ., r) ∈ |CG| such that j : (S, p) → (S̄, r) is continuous and

K
p̄
−→ gh−1. Then there exist G

p
−→ g1, H

p
−→ h1 such that K ≥ (j

→G)(j→H)−1

and g1h
−1
1 = gh

−1. It follows that j→H
r
−→ g1, j

→H
r
−→ h1, and since

(S̄, ., r) ∈ |CG|, (j→G).(j→H)−1
r
−→ g1h

−1
1 . Therefore K

r
−→ gh−1, and thus

p̄ ≥ r. �

Suppose that ((X, q), (S, ., p), λ) ∈ |C|, and recall that (B(X, S), σ) denotes
the generalized quotient space determined by ϕ : (X × S, r) → (X × S)/ ∼=
B(X, S). It is shown in Theorem 2.4(b) that ((B(X, S), σ), (S, ., p), λB) ∈ |C|,
where λB(〈(x, g)〉, h) = 〈(λ(x, h), g)〉. The next result shows that (S̄, ., p̄) also
acts continuously on (B(X, S), σ).



Convergence semigroup categories 81

Lemma 3.6. Assume that ((X, q), (S, ., p), λ) ∈ |C|, where (S, .) is cancellative,
and j : (S, ., p) → (S̄, p̄) denote the natural injection. Then there exists a
continuous action λ̄B such that ((B(X, S), σ), (S̄, ., p̄), λ̄B) ∈ |C| and (idB, j) :
((B(X, S), σ), (S, ., p), λB) → ((B(X, S), σ), (S̄, ., p̄), λ̄B) is a C-morphism.

Proof. Define λ̄B : B(X, S)×S̄ → B(X, S) by λ̄B(〈(x, g)〉, kl
−1) = 〈(λ(x, k), gl)〉),

where x ∈ X and g, k, l ∈ S. First, it is shown that λ̄B is well-defined. Assume
that (x, g) ∼ (x1, g1) and kl

−1 = k1l
−1
1 . Employing Lemma 1.3(a), there exists

h ∈ S such that (x, hg) ≈ (x1, hg1), and thus λ(x, hg1) = λ(x1, hg). Then
(λ(x, k), hgl) ≈ (λ(x1, k1), hg1l1); indeed, λ(λ(x, k), hg1l1) = λ(x, khg1l1) =
λ(λ(x, hg1), kl1) = λ(λ(x1, hg), kl1) = λ(λ(x1, hg), k1l) = λ(x1, hgk1l) =
λ(λ(x1, k1), hgl). Hence (λ(x, k), hgl) ≈ (λ(x1, k1), hg1l1), and again by Lemma
1.3(a), (λ(x, k), gl) ∼ (λ(x1, k1), g1l1). Therefore, λ̄B is well-defined.

Observe that λ̄B is an action. Indeed, λ̄B(〈(x, g)〉, e) = 〈(λ(x, e), g)〉 =
〈(x, g)〉, and λ̄B(λ̄B(〈(x, g)〉, k1l

−1
1 ), k2l

−1
2 ) = λ̄B(〈(λ(x, k1), gl1)〉, k2l

−1
2 ) =

〈(λ(λ(x, k1), k2), gl1l2)〉 =
〈(λ(x, k1k2), gl1l2)〉 = λ̄B(〈(x, g)〉, k1l

−1
1 k2l

−1
2 ). Hence λ̄B is an action. Fur-

thermore, λ̄B : (B(X, S), σ) × (S̄, p̄) → (B(X, S), σ) is continuous. Sup-

pose that H
σ
−→ 〈(x, g)〉 and M

p̄
−→ kl−1; then there exist (x1, g1) ∼ (x, g),

k1l
−1
1 = kl

−1, F
q
−→ x1, G

p
−→ g1, K

p
−→ k1, and L

p
−→ l1 such that H ≥

ϕ→(F×G) and M ≥ K.L−1. It follows that λ̄B
→

(H×M) ≥ λ̄B
→

(ϕ→(F×G)×

K.L−1) = ϕ→(λ→(F × K) × G.L)
σ
−→ ϕ(λ(x1, k1), g1l1) = 〈(λ(x1, k1), g1l1)〉 =

λ̄B(〈(x1, g1)〉, k1l
−1
1 ) = λ̄B(〈(x, g)〉, kl

−1) since λ̄B is well-defined. Therefore
λ̄B is continuous, and thus ((B(X, S), σ), (S̄, ., p̄), λ̄B) ∈ |C|

Finally, (idB, j) : ((B(X, S), σ), (S, ., p), λB) → ((B(X, S), σ), (S̄, ., p̄), λ̄B) is
a C-morphism since idB ◦ λB = λ̄B ◦ (idB × j). �

Lemma 3.7. Suppose that (f, k) : ((X, q), (S, ., p), λ) → ((Y, qY ), (T, ., pT ), µ)
is C-morphism, and assume that (T, ., pT ) ∈ |CG|. If (x, g), (y, h) ∈ X × S and
(x, g) ∼ (y, h), then µ(f(x), (k(g))−1)) = µ(f(y), (k(h))−1).

Proof. Since (x, g) ∼ (y, h), it follows from Lemma 1.3(a) that there exists
l ∈ S such that (x, lg) ≈ (y, lh); hence λ(x, lh) = λ(y, lg) and thus f(λ(x, lh)) =
f(λ(y, lg)). Observe that µ(f(x), (k(g))−1) = µ(f(x), (k(g))−1k(lh)(k(lh))−1) =
µ(µ(f(x), k(lh)), (k(g))−1(k(lh))−1) = µ((µ◦(f×k))(x, lh), (k(g))−1(k(lh))−1).
Since (f, k) is a C-morphism, f ◦ λ = µ ◦ (f × k), and thus (µ ◦ (f × k))(x, lh) =
f(λ(x, lh)) = f(λ(y, lg)) = ((µ ◦ (f × k))(y, lg). Therefore, µ(f(x), (k(g))−1) =
µ((µ ◦ (f × k))(y, lg), (k(g))−1(k(lh))−1) =
µ(µ(f(y), k(l)k(g)), (k(g))−1(k(h))−1(k(l))−1) = µ(f(y), (k(h))−1). �

Define β : (X, q) → (B(X, S), σ) by β(x) = 〈(x, e)〉, and observe that β is

continuous. Indeed, if F
q
−→ x, then β→F = ϕ→(F × ė)

σ
−→ ϕ(x, e) = 〈(x, e)〉.

Hence β is continuous. Therefore (β, j) : ((X, q), (S, ., p), λ) → ((B(X, S), σ),
(S̄, ., p̄), λ̄B) is a C-morphism since (λ̄B ◦ (β × j))(x, g) = λ̄B(〈(x, e)〉, g) =
〈(λ(x, g), e)〉 = β(λ(x, g)) = (β ◦ λ)(x, g).



82 H. Boustique, P. Mikusiński and G. Richardson

The following result was proved in the topological context under the as-
sumptions that λ(., g) is injective for each fixed g ∈ S, and S is equipped with
the discrete topology [4]. Here j : (S, .) → (S̄, .) is an embedding, and conse-
quently (S, .) must be cancellative. A canonical map is used in [4] which is not
necessarily an embedding but does not require that (S, .) be cancellative.

Theorem 3.8. Assume ((X, q), (S, ., p), λ) ∈ |C|, (S, .) is cancellative, and let
j : (S, ., p) → (S̄, ., p̄) denote the injection j(g) = g given in Lemma 3.5. If
(f, k) is a C-morphism and (T, ., pT ) ∈ |CG|, then there exists a C-morphism
(F, K) such that the diagram below is commutative:

((X, q), (S, ., p), λ)
(β, j)

- ((B(X, S), σ), (S̄, ., p̄), λ̄B)

((Y, qY ), (T, ., pT ), µ)

(f, k)

?
�

(F,
K)

Proof. Define F : B(X, S) → Y by F(〈(x, g)〉) = µ(f(x), (k(g))−1), and K :
S̄ → T as K(mn−1) = k(m)(k(n))−1. If follows from Lemma 3.7 that F is
well-defined, and it is easily shown that K is a well-defined group homomor-
phism.

Observe that F is continuous. Indeed, assume that H
σ
−→ 〈((x, g))〉; then

there exist (x1, g1) ∼ (x, g), F
q
−→ x1, G

p
−→ g1 such that H ≥ ϕ

→(F ×G). Hence

F →H ≥ F →(ϕ→(F × G)) = µ→(f→F × (k→G)−1)
q
Y

−−→ µ(f(x1), (k(g1))
−1) =

F(〈(x1, g1)〉) = F(〈(x, g)〉), and thus F is continuous. It easily follows that K
is continuous and the diagram commutes. It remains to show that F ◦ λ̄B =
µ◦(F ×K). Note that (µ◦(F ×K))(〈(x, g)〉, mn−1) = µ(µ(f(x), (k(g))−1), k(m)
(k(n))−1) = µ(f(x), k(m)(k(gn))−1). Since (f, k) is a C-morphism, f ◦ λ =
µ ◦ (f × k), and therefore f(λ(x, m)) = µ(f(x), k(m)).
Hence (F◦λ̄B)(〈(x, g)〉, mn

−1) = F(〈(λ(x, m), gn)〉) = µ(f(λ(x, m)), (k(gn))−1) =
µ(µ(f(x), k(m)), (k(gn))−1) = µ(f(x), k(m)(k(gn))−1), and thus F ◦ λ̄B =
µ ◦ (F × K). Therefore, (F, K) is a C-morphism. �

Corollary 3.9. Suppose that the hypotheses of Theorem 3.8 are satisfied except
that (T, .) is cancellative, commutative, and (T, ., pT ) ∈ |CSG|. Then there
exists a C-morphism (F, K) such that the diagram below is commutative:



Convergence semigroup categories 83

((X, q), (S, ., p), λ)
(βX, jS)

- ((B(X, S), σX), (S̄, ., p̄), λ̄B)

((Y, qY ), (T, ., pT ), µ)

(f, k)

?
(βY , jT )

- ((B(Y, T ), σY ), (T̄ , ., p̄T ), µ̄B)

(F, K)

?

4. Examples and Special Cases

First, some examples of generalized quotient spaces are presented, and then
the work is concluded with some results pertaining to the case whenever the
semigroup is generated by a single element. In the following examples, N de-
notes a natural number, and let N designate the set of all natural numbers.

Example 4.1. Let X = C∞(RN) be the space of all continuous complex-
valued functions defined on RN, and equipped with the topology of uniform
convergence on compact sets. Denote

S = {f ∈ C∞(RN) : f has compact support, f ≥ 0, and

∫

f = 1}.

Then S is a semigroup with respect to the convolution

(f ∗ g)(u) =

∫

RN

f(v)g(u − v)dv.

Define the action λ of S on the space X by convolution, that is,

λ(x, f) = x ∗ f.

Note that λ is not injective. Then, for x, y ∈ X,

x ≃ y iff x ∗ f = y ∗ f for some f ∈ S.

Under this equivalence relation some functions will be identified with 0 and,
moreover, B(X∗, S) is a space of generalized functions. Clearly, not all contin-
uous functions can be identified with elements of B(X∗, S), and thus B(X∗, S)
is a proper extension of X∗. Every element of B(X∗, S) has derivatives of all
orders defined by

Dα〈([x], g)〉 = 〈([Dαx], g)〉.

It is not difficult to check that this operation is well-defined and continuous.

Consider the situation when S is generated by a single map. Let X be a
nonempty set, and let g : X → X be a non-injective map. Then S = {gn :
n = 0, 1, 2, ...} is a commutative semigroup with respect to composition, and

x ≃ y iff there exists an n such that λ(x, gn) = λ(y, gn).

If g is a surjection, then λ∗(., g
n) : X∗ → X∗ is a surjection for every n ∈ N.

Consequently, B(X∗, S) = X∗ (identifiable); hence, in order to produce a



84 H. Boustique, P. Mikusiński and G. Richardson

proper extension of X∗, it is necessary to start with a g that is not surjec-
tive.

Example 4.2. Let X be a normed space, T : X → X a norm preserving
linear operator, and assume that S is the semigroup generated by T . Since T
is injective, X∗ = X. For sake of brevity, denote 〈(x, T

n)〉 by x
T n

∈ B(X, S),
and define

∥

∥

∥

x

T n

∥

∥

∥
= ‖x‖ .

If x
T n

= y
T m

, then T mx = T ny, and hence
∥

∥

∥

x

T n

∥

∥

∥
= ‖x‖ = ‖T mx‖ = ‖T ny‖ = ‖y‖ =

∥

∥

∥

y

T m

∥

∥

∥
.

Therefore, ‖·‖ is well-defined in B(X, S). One can also verify that ‖·‖ is a norm
in B(X, S); in particular,

∥

∥

∥

x

T n
+

y

T m

∥

∥

∥
=

∥

∥

∥

∥

T mx + T ny

T m+n

∥

∥

∥

∥

= ‖T mx + T ny‖

≤ ‖T mx‖ + ‖T ny‖ = ‖x‖ + ‖y‖

=
∥

∥

∥

x

T n

∥

∥

∥
+
∥

∥

∥

y

T m

∥

∥

∥
.

Consequently, this extension of a normed space (X, ‖·‖) produces a normed
space (B(X, S), ‖·‖). Moreover, the map defined by

T
x

T n
=

T x

T n

is a norm preserving bijection on B(X, S).
In particular, given a vector space X, let {e1, e2, ...} be a Hamel basis, and

define a norm by
∥

∥

∥

∥

∥

∥

k
∑

j=1

αjepj

∥

∥

∥

∥

∥

∥

=
k
∑

j=1

|αj|.

Then

T (

k
∑

j=1

αjepj ) =

k
∑

j=1

αjepj+1

and

U(

k
∑

j=1

αjepj ) =

k
∑

j=1

αje2pj

are norm preserving operators on X. In both cases, B(X, S) is a proper exten-
sion of X.



Convergence semigroup categories 85

Example 4.3. (i) Consider an arbitrary nonempty set Z and denote X = ZN.
Define g : X → X by g((xn)) = (xn+1), and let S = {g

n : n = 0, 1, 2, ...}.
Then

(xn) ≃ (yn) iff there exists an n0 such that xn = yn for all n ≥ n0.

That is, two sequences are equivalent whenever they are eventually equal. Note
that g is a surjection, and thus B(X∗, S) = X∗ .

(ii) Let E be a vector space, X = EN, and denote

S = {(αn) ∈ {0, 1}
N : Card{n ∈ N : αn 6= 0} < N0}.

Then S is a commutative semigroup with respect to termwise multiplication,
and define

λ((xn), (αn)) = (αnxn).

Observe that λ(., (αn)) is neither injective nor surjective, and B(X∗, S) = X∗.

Recall that a continuous surjection f : (X, q) → (Y, p) in CONV is called

a quotient map if whenever F
p
−→ y, then there exists G

q
−→ x ∈ f−1(y) such

that f→G = F. Moreover, a continuous surjection f is said to be a proper

map provided that for each ultrafilter F on X, f→F
p
−→ y implies that F

q
−→ x,

for some x ∈ f−1(y). Given that S = {gn : n = 0, 1, 2, ...}, two special cases
are considered below by requiring that g be either a quotient or proper map.

Theorem 4.4. Assume that (X, q) ∈ |CONV|, g : (X, q) → (X, q) is a quotient
map in CONV, S = {gn : n = 0, 1, 2, ...}, and define the action by λ(x, gn) =
gn(x), n = 0, 1, 2, .... Then

(a) λ∗(., g
n) : (X∗, q∗) → (X∗, q∗) is a homeomorphism, for each fixed

n = 0, 1, 2, ...
(b) β∗ : (X∗, q∗) → (B(X∗, S), σ∗) is a homeomorphism whenever S is

equipped with the discrete topology, where β∗([x]) = 〈([x], e)〉.

Proof. (a): Denote λ∗([x], g) = [λ(x, g)] by g∗([x]) = [g(x)]. Consider the
commutative diagram below:

(X, q)
g
- (X, q)

(X∗, q∗)

ξ

? g∗
- (X∗, q∗)

ξ

?

Since ξ is a quotient map in CONV, g∗ is continuous iff g∗ ◦ ξ is continuous.
However, g∗ ◦ ξ = ξ ◦ g is continuous, and thus g∗ is continuous. Moreover,
g∗ is injective. Indeed, if [g(x)] = g∗([x]) = g∗([z]) = [g(z)], then g(x) ≃ g(z)
and thus gn+1(x) = gn(g(x)) = gn(g(z)) = gn+1(z) for some n ≥ 0, and thus



86 H. Boustique, P. Mikusiński and G. Richardson

[x] = [z]. Hence g∗ is injective. Since g is onto, it follows that g∗ is a continuous
bijection. Next, it is shown that g−1

∗
is continuous.

Again, since ξ is a quotient map, g−1
∗

is continuous iff g−1
∗

◦ ξ is continuous.
Since the diagram commutes, g∗ ◦ ξ = ξ ◦ g, and thus ξ = g

−1
∗

◦ ξ ◦ g. Assume

that F
q
−→ x; it must be shown that (g−1

∗
◦ ξ)→F

q∗
−→ g−1

∗
([x]). Since g is a

quotient map in CONV, there exists H
q
−→ z such that g→H = F and g(z) = x.

Then ξ→H
q∗
−→ [z], and thus (g−1

∗
◦ ξ)→F = (g−1

∗
◦ ξ ◦ g)→H = ξ→H

q∗
−→ [z].

Since g(z) = x, g∗([z]) = [x], and thus (g
−1
∗

◦ ξ)(x) = g−1
∗

([x]) = [z]. Hence
(g∗ ◦ ξ)

→F → (g−1
∗

◦ ξ)(x)), and thus g−1
∗

◦ ξ is continous. Therefore g−1
∗

is
continuous, and hence g∗ is a homeomorphism. Further, since the finite com-
position of quotient maps is again a quotient in CONV, gn is also a quotient
map, and it follows that (gn)∗ is also a homeomorphism, n = 0, 1, 2, ....
(b): Note that if 〈([x], e)〉 = β∗([x]) = β∗([z]) = 〈([z], e)〉, then [x] = λ∗([x], e) =
λ∗([z], e) = [z], and thus β∗ is injective. Given 〈([z], g

n)〉 ∈ B(X∗, S), since g
n

is onto, gn(x) = z for some x ∈ X. Then β∗([x]) = 〈([x], e)〉 = 〈([z], g
n)〉, and

thus β∗ is a bijection. Observe that β∗ = ϕ∗ ◦ σ, where σ([x]) = ([x], e), and
hence β∗ is a continuous bijection.

It remains to show that β−1
∗

is continuous. Assume that F ∈ F(X∗) such that

β→
∗
F

σ∗
−→ β∗([x]); it must be shown that F

q∗
−→ [x]. Since ϕ∗ : (X∗ × S, r∗) →

(B(X∗, S), σ∗) is a quotient map in CONV and β
→

∗
F

σ∗
−→ β([x]) = 〈([x], e)〉,

there exists K
r∗
−→ ([z], gn) such that ϕ→

∗
K = β→

∗
F, where r∗ = q∗ × p and

〈([z], gn)〉 = 〈([x], e)〉. In particular, λ∗([z], e) = λ∗([x], g
n), or [z] = [gn(x)].

Since S has the discrete topology, there exists H
q∗
−→ [z] such that H × ġn ≤ K,

and thus ϕ→
∗
(H × ġn) ≤ ϕ→

∗
K = β→

∗
F. Fix H ∈ H; then there exists F ∈ F

such that β∗(F) ⊆ ϕ∗(H×{g
n}). If [s] ∈ F , then β∗([s]) = 〈([s], e)〉 = 〈([t], g

n)〉
for some [t] ∈ H. In particular, [gn(s)] = λ∗([s], g

n) = λ∗([t], e) = [t], and thus

(gn)∗(F) ⊆ H. Therefore (g
n)→

∗
F ≥ H

q∗
−→ [z], and since (gn)∗ is a homeo-

morphism, F
q∗
−→ (gn)−1

∗
([z]). However, (gn)∗([x]) = [g

n(x)] = [z], and thus

(gn)−1
∗

([z]) = [x]. Hence F
q∗
−→ [x], and it follows that β∗ is a homeomor-

phism. �

The final result is the analogue of Theorem 4.4 whenever g is a proper map.

Object (X, q) ∈ |CONV| is said to be a Choquet space provided that F
q
−→ x

whenever each ultrafilter G ≥ F, G
q
−→ x; that is, q−convergence is determined

by q−convergence of the ultrafilters on X. Given any (X, q) ∈ |CONV|, define

q̂ by : F
q̂
−→ x iff for each ultrafilter G ≥ F, G

q
−→ x. Note that q̂ and q agree

on ultrafilter convergence. Moreover, it follows that q̂ is the finest Choquet
structure on X which is coarser than q. Suppose that f : (X, q) → (Y, p) is a

map between two convergence spaces such that f→F
p
−→ f(x) whenever F is

an ultrafilter on X for which F
q
−→ x. It is easily shown that f : (X, q̂) → (Y, p̂)

is continuous. Verification of the next result makes use of the preceding fact
along with an argument similar to that given in Theorem 4.4. The proof is
omitted.



Convergence semigroup categories 87

Theorem 4.5. Suppose that (X, q) ∈ |CONV|, g : (X, q) → (X, q) is a proper
map, S = {gn : n = 0, 1, 2, ...}, and action λ(x, gn) = gn(x), n ≥ 0. Then

(a) λ∗(., g
n) : (X∗, q̂∗) → (X∗, q̂∗) is a homeomorphism, for each fixed

n ≥ 0
(b) β∗ : (X∗, q̂∗) → (B(X∗, S), γ∗) is a homeomorphism whenever S has

the discrete topology p, and γ∗ denotes the quotient structure in CONV
determined from ϕ∗ : (X∗, q̂∗) × (S, p) → (X∗ × S)/ ≈.

Employing Theorem 4.5, along with the fact that a continuous surjection
g : (X, q) → (X, q) is a proper map whenever (X, q) ∈ |CONV| is compact and
Hausdorff, gives the concluding result.

Corollary 4.6. Assume that (X, q) ∈ |CONV| is compact, Hausdorff , and
g : (X, q) → (X, q) is a continous surjection. Suppose that S = {gn : n =
0, 1, 2, ...} has the discrete topology. Then β∗ : (X∗, q̂∗) → (B(X∗, S), γ∗) is a
homeomorphism.

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Received November 2009

Accepted September 2010

Gary Richardson (garyr@mail.ucf.edu)
Department of Mathematics, University of Central Florida,Orlando, FL 32816,
USA, fax: (407) 823-6253, tel: (407) 823-2753


	Convergence semigroup categories. By H. Boustique, P. Mikusinski and G. Richardson