Free and Cofree Acts of DcpoMonoids on Directed Complete Posets
Bull. Malays. Math. Sci. Soc.
Free and Cofree Acts of DcpoMonoids on Directed Complete Posets
Mojgan Mahmoudi 0
Halimeh Moghbeli 0
Mathematics Subject Classification 0
0 Department of Mathematics, Shahid Beheshti University , G.C., 19839 Tehran , Iran
In this paper, we study the existence of the free and cofree objects in the categories DcpoS (and CpoS) of all directed complete posets (with bottom element) equipped with a compatible right action of a dcpomonoid (cpomonoid) S, with (strict) continuous actionpreserving maps between them. More precisely, we consider all forgetful functors between these categories and the categories Dcpo of dcpo's, (CPo) of cpo's, Pos of posets, and Set of sets, and study the existence of their left and right adjoints.

06F05 · 18A40 · 20M30 · 20M50
1 Introduction and Preliminaries
The category Dcpo of Directed Complete Partial Ordered sets plays an important role
in Theoretical Computer Science, and specially in Domain Theory (see [
1
]). It has
Dedicated to Professor M. Mehdi Ebrahimi on the occasion of his 65th Birthday.
Communicated by Ang Miin Huey.
been proved that this category is complete and cocomplete (see [
1,7
]). The free dcpo
over a poset has been given in [
4,9,12
].
In this paper, we consider the free and cofree objects in the category DcpoS of all
Sdcpo’s; dcpo’s equipped with a compatible right action of a dcpomonoid S, with
continuous actionpreserving maps between them. We take the forgetful functors from
this category to the categories of dcpo’s, posets, and sets, and study the existence of
their left and right adjoints. In fact, we consider the following three squares of forgetful
functors
U1
CpoS −−−−→
U4
DcpoS −−−−→ Dcpo
⏐⏐ U3
⏐⏐ U6
PosS
⏐⏐ U9
ActS
U7
−−−−→
U10
−−−−→
Cpo
⏐⏐ U2
⏐⏐ U5
Pos
⏐⏐ U8
Set
and study the existence of the left and the right adjoints for these functors (that is,
Ui free and Ui cofree objects). We recall that the bottom square has been considered
in [
3
], where it has been shown that the horizontal forgetful functors, U7 and U10, have
both left and right adjoints, while the vertical forgetful functors, U8 and U9, have just
left adjoints (here we give a correction to the definition of the U9free functor given
in [
3
]). Also, the left adjoint to the right vertical forgetful functor, U5, in the middle
square has been found in [
4,9,12
].
Here, we show that, although all the forgetful functors in these squares have left
adjoints (for the existence of U1free, U3free, and U6free, we had to add a condition
on S), none of the vertical forgetful functors in all the above three squares has a right
adjoint. In finding the left adjoints (free objects), we observe that the definition of the
left hand side vertical free functors is the same as the right hand side ones, and just we
need to define a proper Saction. Also we prove that all the horizontal functors, except
U1, have right adjoints (cofree objects). In finding these right adjoints, we observe that
the cofree horizontal functors (when existing) are in some sense the restrictions of the
bottom horizontal cofree functors. The same is true for the free horizontal functors.
In the following, we give some preliminaries needed in the sequel. For more
information about dcpo’s we refer to [
8
], about Ssets see [
6,10
], about Sposets see [3],
and for Sdcpo’s refer to [
11
].
Let Pos denote the category of all partially ordered sets (posets) with
orderpreserving (monotone) maps between them. A nonempty subset D of a partially
ordered set is called directed, denoted by D ⊆d P, if for every a, b ∈ D there
exists c ∈ D such that a, b ≤ c; and P is called directed complete, or briefly a dcpo,
if for every D ⊆d P, the directed join d D exists in P. A dcpo which has a bottom
element ⊥ is said to be a cpo.
A dcpo map or a continuous map f : P → Q between dcpo’s is a map with the
property that for every D ⊆d P, f (D) is a directed subset of Q and f ( d D) =
d f (D). A dcpo map f : P → Q between cpo’s is called strict if f (⊥) = ⊥. Thus
we have the categories Dcpo (and Cpo) of all dcpo’s (cpo’s) with (strict) continuous
maps between them.
We repeatedly apply the following lemmas in this paper.
Lemma 1.1 [
1,5
] Let { Ai : i ∈ I } be a family of dcpo’s. Then the directed join of a
directed subset D ⊆d i∈I Ai is calculated as d D = ( d Di )i∈I where
Di = {a ∈ Ai : ∃d = (dk )k∈I ∈ D, a = di }
for all i ∈ I .
Lemma 1.2 [
8
] Let P, Q, and R be dcpo’s, and f : P × Q → R be a function of two
variables. Then f is continuous if and only if f is continuous in each variable; which
means that for all a ∈ P, b ∈ Q, fa : Q → R (b → f (a, b)) and fb : P → R
(a → f (a, b)) are continuous.
Remark 1.3 Recall that for a poset P, a nonempty subset I is called an ideal if I is an
(up)directed down subset of P, and the collection of all ideals of P is usually denoted
by Id( P). It has been stated in [
4
] that for a poset P, Id( P) is the free dcpo over P.
Notice that Id( P) is a dcpo in which the supremum of every directed subset is given
by union, and the down map ↓: P → Id( P) is the universal monotone map of the free
object.
We consider the cofree dcpo on a poset in Sect. 2.
Recall that a pomonoid is a monoid with a partial order ≤ which is compatible
with the monoid operation: for s, t, s , t ∈ S, s ≤ t , s ≤ t imply ss ≤ t t . Similarly,
a dcpo (cpo)monoid is a monoid which is also a dcpo (cpo) whose binary operation
is a (strict) continuous map.
Recall that a (right) Sact or Sset for a monoid S is a set A equipped with an
acti on A × S → A, (a, s) → as, such that a1 = a and a(st ) = (as)t, for all a ∈ A
and s, t ∈ S. Let ActS denote the category of all Sacts with actionpreserving maps
(maps f : A → B with f (as) = f (a)s, for all a ∈ A, s ∈ S).
Also, for a pomonoid S, a (right) Sposet is a poset A which is also an Sact whose
action λ : A × S → A is orderpreserving, where A × S is considered as a poset with
componentwise order. The category of all Sposets with actionpreserving monotone
maps between them is denoted by PosS.
Finally, recall that for a dcpo (cpo)monoid S, a (right) Sdcpo (Scpo) is a dcpo
(cpo) A which is also an Sact whose action λ : A × S → A is a (strict) continuous
map.
Also, by an Sdcpo map (Scpo map) between Sdcpo’s (Scpo’s), we mean a
map f : A → B which is both (strict) continuous and actionpreserving. We denote
the categories of all Sdcpo’s (Scpo’s) and Sdcpo (Scpo) maps between them by
DcpoS and CpoS, respectively.
Furthermore, notice that in the definition of an Scpo, the monotonicity of the action
implies that it is also strict (this is because, for an Scpo A, ⊥A ≤ ⊥A⊥S ≤ ⊥A1S =
⊥A). Also, by Lemma 1.2, the action λ : A × S → A is continuous if and only if the
maps λa : S → A and λs : A → A, for all a ∈ A and s ∈ S, are continuous.
2 Adjoint Relations for DcpoS
In this section, we consider the middle square of the forgetful functors. We show that
both U4 and U5 have left adjoints, while only U4 has a right adjoint. Also, it is proved
that if S satisfies a condition which we call “good,” then U6 has a left adjoint, but U6
does not have a right adjoint.
Free Sdcpo over a dcpo By a free Sdcpo on a dcpo P we mean an Sdcpo F4
together with a continuous map τ : P → F4 with the universal property that given
any Sdcpo A and a continuous map f : P → A there exists a unique Sdcpo map
f : F4 → A such that f ◦ τ = f .
Theorem 2.1 For a given dcpo P, the free Sdcpo on P is F4 = P × S, with
componentwise order and the action given by (x , s)t = (x , st ), for x ∈ P, s, t ∈ S.
Proof Recall that P × S is an Sposet, and is a dcpo (see [
1
]). Now, we show that
the action defined above on P × S is a continuous map. Applying Lemma 1.2, let
D ⊆d P × S and s ∈ S. We show that
d
D s =
d
( p,t)∈D
( p, t s).
By Lemma 1.1, d D = d D1, d D2 where D1 = Dom D and D2 = I m D are
directed subsets of P and S, respectively. Now,
d
d
d
D s =
D1,
D2 s
= ⎝
⎛ d
D1,
d
s ∈D2
⎞
s s⎠ =
d
( p,t)∈D
( p, t s).
Notice that the first equality is true by the definition of the action on P × S, also the
last equality is proved straightforward. Now, let T ⊆d S and ( p, s) ∈ P × S. Then
( p, s)
d
T
=
p, s
d
T
p,
st
d
t∈T
=
d
t∈T
( p, st ),
=
d
x∈D
where the last equality follows by applying the definition of the least upper bound.
Again, recalling that P × S is the free Sposet on the poset P, with the universal
map τ : P → P × S, given by x → (x , 1) (see [
3
]), we show that τ is continuous.
Let D ⊆d P. Then
d
d
τ
D =
D, 1
=
(x , 1) =
τ (d),
d
d∈D
where the second equality is because of the definition of the upper bound.
Finally, to prove the universal property of τ : P → P × S for Sdcpo’s, take a
continuous map f : P → B to an Sdcpo B. Then the map f : P × S → B defined
by f ( p, s) = f ( p)s, which is the unique Sposet map with f ◦ τ = f (see [
3
]), is
continuous. Applying Lemma 1.2, let first D ⊆d P and s ∈ S. Then
f
d
d
D, s
= f
D s =
f (x ) s =
f (x )s
=
where the third equality is because B is an Sdcpo. Secondly, assume that T ⊆d S
and p ∈ P, then
d
d
f
p,
T
= f ( p)
T =
f ( p)t =
f ( p, t ),
where the second equality is because B is an Sdcpo.
Corollary 2.2 The forgetful functor U4 : DcpoS → Dcpo has a left adjoint.
Cofree Sdcpo over a dcpo By a cofree Sdcpo on a dcpo P we mean an Sdcpo K4
together with a continuous map σ : K4 → P with the universal property that given
any Sdcpo A and a continuous map g : A → P there exists a unique Sdcpo map
g : A → K4 such that σ ◦ g = g.
Theorem 2.3 For a given dcpo P and dcpomonoid S, the cofree Sdcpo on P is the
set K4 = P(S), of all dcpo maps from S to P, with pointwise order and the action
given by ( f s)(t ) = f (st ), for s, t ∈ S and f ∈ P(S).
Proof First we show that P(S) is an Sdcpo. Recall that P(S) is a dcpo, and the
supremum in P(S) is calculated pointwise (see [
8
]). Also, the action defined above is
a continuous map. It is welldefined, since for f ∈ P(S) and s ∈ S, f s is continuous.
This is because for T ⊆d S,
d
( f s)
T
= f s
d
T
d
t∈T
d
t∈T
d
t∈T
= f
st
=
f (st ) =
( f s)(t ),
where the second equality is because S is a dcpomonoid, and the third equality is
because f is continuous.
To prove the continuity of the action, we apply Lemma 1.2. Let first F ⊆d P(S)
and s ∈ S. Then
d
d
F s (t ) =
F (st ) =
f (st ) =
d
f ∈F
d
f ∈F
⎛
( f s)(t ) = ⎝
d
f ∈F
⎞
f s⎠ (t ),
where the second and the last equality are because supremum in P(S) is calculated
pointwise. Therefore, d F s = d (F s). Now assume that T ⊆d S and f ∈ P(S).
Then
= f
d
t s
=
f (t s)
( f t )(s) =
f t (s),
d
t∈T
d
t∈T
as required. Consequently P(S) is an Sdcpo. Now, take the cofree map σ : P(S) → P
defined by σ ( f ) = f (1). First, we show that it is continuous. Let F ⊆d P(S). Then
σ ⎝
⎛
d
f ∈F
f ⎠
⎞
= ⎝
⎛
d
f ∈F
⎞
f ⎠ (1) =
d
f ∈F
f (1) =
σ ( f ).
Further, given a continuous map α : A → P from an Sdcpo A, the map α : A → P(S),
given by α(a)(s) = α(as), is an Sdcpo map and satisfies σ ◦ α = α. First, we show
that α is continuous. Let D ⊆d A and s ∈ S, then
d
α
D (s) = α
d
D s
Secondly, α is actionpreserving, since for all s, t ∈ S and a ∈ A we have
α(as)(t ) = α((as)t ) = α(a(st )) = α(a)(st ) = (α(a)s)(t ).
To establish the uniqueness of α, suppose that h : A → P(S) is also an Sdcpo map
such that σ ◦ h = α. Then for all a ∈ A and s ∈ S,
h(a)(s) = h(a)(s1) = (h(a)s)(1) = σ (h(a)s)
= σ (h(as)) = α(as) = α(a)(s).
Corollary 2.4 The forgetful functor U4 : DcpoS → Dcpo has a right adjoint.
Now, we consider the adjoints of U5. Recall that for a poset P, the set Id( P) of ideals
of P is the free dcpo over P (see Remark 1.3).
Corollary 2.5 The forgetful functor U5 : Dcpo → Pos has a left adjoint.
d
f ∈F
d
x∈D
d
x∈D
d
In the following, we see that the right adjoint of U5 does not necessarily exist.
Lemma 2.6 If P is a nontrivial poset with nonidentity order, which is also a dcpo,
then the cofree dcpo over P does not exist.
Proof Let P be a nontrivial dcpo in which the order is not identity, and let K ( P) be
the cofree dcpo over P as a poset. Take k : K ( P) → P to be the cofree monotone
map.
First we see that k is oneone. This is because, otherwise there exist x = y ∈ K ( P)
such that k(x ) = k(y) = p0. Then, considering the monotone map f : { } → P from
the singleton dcpo { }, defined by f ( ) = p0, we see that there exist two dcpo maps
f1, f2 : { } → K ( P), given by f1( ) = x and f2( ) = y, such that k ◦ f1 = f and
k ◦ f2 = f . This contradicts the universal property of the cofree map k.
Moreover, we see that k is a retraction, since for the monotone map i dP : P → P,
by the universal property of cofree maps, there exists a dcpo map f : P → K ( P)
with k ◦ f = i dP . Therefore, k is a poset isomorphism.
Now, since the order on P is not identity, there exist x , y ∈ P with x < y. Define
the poset map f : P(N) → P by
f (M ) =
x
y
if M is finite
otherwise
Then, by the universal property of cofree maps, there exists a unique dcpo map f :
P(N) → K ( P) with k ◦ f = f . Now, f being a composition of two dcpo maps, is a
dcpo map. But this is a contradiction, because taking the directed subset D of P(N)
consisting of all finite subsets of N, we have f d D
D = f (N) = y but
= f
d f (D) =
d {x } = x .
Corollary 2.7 The forgetful functor U5 : Dcpo → Pos does not have a right adjoint.
Now, we consider U6. First, using the above corollary, we have:
Remark 2.8 The forgetful functor U6 from DcpoS to PosS does not have a right
adjoint for a general dcpomonoid S. This is implied by taking S = {1}, and applying
Corollary 2.7.
Now, we give a condition on S under which U6 has a left adjoint.
Definition 2.9 We say that a dcpo P is good if for every directed subset D of P,
d D ∈ D.
Remark 2.10 A dcpo P is good if and only if each directed subset of P has a top
element. This condition is also equivalent to the fact that each element of P is compact.
Recall that the element x of a dcpo P is called compact if for every directed subset D
of P, x ≤ d D implies x ≤ d, for some d ∈ D.
Finite posets and Noetherian posets (satisfying ACC on chains of elements) are
examples of good dcpo’s. Also, for any poset P with discrete order, the posets P ⊕
and ⊥ ⊕ P are good dcpo’s, where P ⊕ and ⊥ ⊕ P are obtained by adding a top
element and a bottom element to P, respectively.
Theorem 2.11 Let S be a good dcpomonoid. For a given Sposet A, the free
Sdcpo on A is the dcpo Id( A) with the action λ : Id( A) × S → Id( A), given by
(I, s) → I.s =: ↓(I s), where I s = {as : a ∈ I } for I ∈ Id( A) and s ∈ S.
Proof First we show that Id( A) is an Sdcpo. Notice that, by Remark 1.3, Id( A) is a
dcpo in which the supremum of a directed subset D of Id( A) is D. Also, it is clear
that the given action is welldefined. Further, for all I ∈ Id( A) and s, t ∈ S, we have
(1) I.1 = ↓(I 1) = ↓I = I ,
(2) I.(st ) = ↓(I (st )) = ↓((I s)t ) = ↓(↓(I s))t = (I.s).t ,
where equalities are true by a straightforward computation using definitions. Now, we
show that the action is also continuous. Applying Lemma 1.2, let {Iα : α ∈ } be a
directed subset of Id( A) and s ∈ S. We have
Iα .s =
Iα .s = ↓
Iα s
= ↓
(Iαs)
=
(↓(Iαs)
d
I.t,
d
t∈T
α∈
d
α∈
where the equalities are true by straightforward calculations.
Now, assume that T ⊆d S and I ∈ Id( A). Then
=
d
t∈T
d
t∈T
where the second equality follows from the hypothesis that
that ↓ I (
d T is the maximum element of the set {↓I t : t ∈ T }. Therefore,
d T ∈ T , which gives
↓ I
d
T
=
↓(I t ),
and so Id( A) is an Sdcpo. Now, we show that ↓: A → Id( A), a → ↓a is an
Sposet map. It is clear that ↓ is orderpreserving. It is also actionpreserving, since
(↓a).s = ↓((↓a)s) = ↓(as). Finally, we show that ↓: A → Id( A) is a universal map.
Let f : A → B be an Sposet map to an Sdcpo B. Then the map f : Id( A) → B
given by f (I ) = d f (I ) is the unique Sdcpo map with f ◦ ↓ = f . To see this, first
we show that f is continuous. Let {Iα : α ∈ } be a directed subset of Id( A). Then
where the third equality follows by the definition of supremum. In fact, since f (Iα) ⊆
f α∈ Iα , we get d f (Iα) ≤ d f α∈ Iα , for all α ∈ . Also, if b ∈
f
Iα
= f
Iα
=
α∈
d
f
α∈
Iα
=
d
α∈
d
f (Iα)
=
f (Iα),
d
B is an upper bound of the set
d f (Iα) : α ∈
have x ∈ f (Iα0 ), for some α0 ∈ , and so x ≤
d f α∈ Iα ≤ b.
Also, f is actionpreserving, since for I ∈ Id( A) and s ∈ S, we have
, then for x ∈ f
α∈
d f (Iα0 ) ≤ b, which gives
Iα , we
where the third equality is because an element c is an upper bound of f (↓(I s))
if and only if it is an upper bound of f (I s). The fourth equality is because f is
actionpreserving. Also, the fifth equality is because B is an Sdcpo. Furthermore,
we have f (↓a) = d f (↓a) = f (a). To show the uniqueness of f , suppose that
h : Id( A) → B is also an Sdcpo map with h ◦ ↓ = f . Then for every I ∈ Id( A),
d
f (I ) =
f (I ) =
d
a∈I
d
a∈I
Corollary 2.12 If S is a good dcpomonoid, then the forgetful functor U6 :
DcpoS → PosS has a left adjoint.
3 Adjoint Relations for CpoS
In this section, we consider the top square of the forgetful functors. We show that U2
has a left adjoint, and if we assume that S is a cpomonoid whose identity is the bottom
element, then U1 has a left adjoint; also if S is a cpomonoid whose identity is the top
element, then U3 has a left adjoint. But, none of U1, U2, and U3 has a right adjoint.
Free Scpo on a cpo P By a free Scpo on a cpo P we mean an Scpo F1 together
with a strict continuous map τ : P → F1 with the universal property that given any
Scpo A and a strict continuous map f : P → A there exists a unique Scpo map
f : F1 → A such that f ◦ τ = f .
Theorem 3.1 Let S be a cpomonoid whose identity is the bottom element. Then for a
given cpo P and cpomonoid S, the free Scpo on P is F1 = P ×S, with componentwise
order and the action given by (x , s)t = (x , st ), for x ∈ P, s, t ∈ S.
Proof First recall that P × S with the above action and order is the free Sdcpo on
the dcpo P (see Theorem 2.1). Also, we know that P × S with the componentwise
order is a cpo (see [
1
]). Now, we show that τ : P → P × S given by x → (x , 1) is a
universal strict continuous map. Since the identity element of S is the bottom element,
we have
τ (⊥P ) = (⊥P , 1) = (⊥P , ⊥S),
which means that τ is strict. The continuity of τ was proved in Theorem 2.1. To prove
the universal property, let f : P → B be any strict continuous map to an Scpo B.
Then the map f : P × S → B defined by f ( p, s) = f ( p)s is the unique Sdcpo
map with f ◦ τ = f (see Theorem 2.1). Now, we show that f is also strict. Since f
is strict and B is an Scpo,
f (⊥P , ⊥S) = f (⊥P )⊥S = ⊥B ⊥S = ⊥B .
Corollary 3.2 If S is a cpomonoid whose identity is the bottom element, then the
forgetful functor U1 : CpoS → Cpo has a left adjoint.
Remark 3.3 The forgetful functor U1 : CpoS → Cpo does not have a right adjoint.
One can see this by noting that it does not necessarily preserve the initial object. For
example, let S be the 2element chain {1, a} with 1 < a, and aa = a, 1a = a = a1.
Then S is an Scpo and, it is the initial object of CpoS (see [
11
]), whereas the initial
object in the category Cpo is the singleton cpo.
Now, we consider U2.
Theorem 3.4 The forgetful functor U2 : Cpo → Dcpo has a left adjoint.
Proof For a dcpo P, P⊥ = ⊥ ⊕ P is the free cpo on P.
Remark 3.5 The forgetful functor U2 : Cpo → Dcpo does not have a right adjoint.
This is because, U2 does not preserve the initial object. Notice that the initial object
in Cpo is the singleton poset { }, while the initial object in Dcpo is the empty poset.
Finally, we study U3.
Free Scpo over an Sdcpo. By a free Scpo on an Sdcpo A we mean an Scpo F6
together with a Sdcpo map τ : A → F6 with the universal property that given any
Scpo B and a strict continuous map f : A → B there exists a unique Scpo map
f : F6 → A such that f ◦ τ = f .
Theorem 3.6 Let S be a cpomonoid in which the identity element is the top element.
Then the free Scpo on an Sdcpo A is A⊥ = ⊥ ⊕ A with the action defined by
a.s =
as if a ∈ A
⊥ if a = ⊥
for all a ∈ A⊥ and s ∈ S.
Proof We show that this action is continuous. Applying Lemma 1.2, let D ⊆d A⊥
and s ∈ S. First note that D ⊆ A⊥ is directed if and only if D ⊆ A is directed or
D = D ∪ {⊥} where D = ∅ or D is a directed subset of A. Therefore, two cases
may occur:
Case (1): D ⊆d A. In this case,
D .s =
D s =
x s =
x .s,
d
x∈D
d
x∈D
since the action on A is continuous.
Case (2): D = D ∪ {⊥}, where D ⊆d A or D = ∅. If D = ∅, then the result is
clear. Let D ⊆d A. Then,
d
D .s =
D
.s =
D
s =
x s
d
x∈D
d
x∈D
=
x .s
∨ (⊥.s) =
x .s.
d
x∈D
Now assume that T ⊆d S and a ∈ A⊥. If a = ⊥, then ⊥.(
If a ∈ A, then
d T ) = ⊥ =
d (⊥.s).
d
where the second equality is because A is an Sdcpo. Therefore, A⊥ is an Scpo. Now,
we show that the inclusion map ι : A → A⊥ is the universal free map. Let f : A → B
be any Sdcpo map to an Scpo B. Then, the map f : A⊥ → B defined by
is the unique cpo map with f ◦ ι = f . Now, we show that f : A⊥ → B is
actionpreserving, and so it is an Scpo map. Since the identity element of S is the top element,
the bottom element of every Scpo is a zero element (s ≤ 1 implies ⊥As ≤ ⊥A1 = ⊥A,
and so ⊥As = ⊥A), and hence f (⊥.s) = f (⊥) = ⊥B = ⊥B s = f (⊥)s, for all s ∈ S.
Also, for a = ⊥ and s ∈ S, f (a.s) = f (as) = f (as) = f (a)s = f (a)s.
Corollary 3.7 If S is a cpomonoid whose identity is the top element, then the forgetful
functor U3 : CpoS → DcpoS has a left adjoint.
Remark 3.8 The forgetful functor U3 : CpoS → DcpoS does not have a right
adjoint. Take S = {1} and apply Remark 3.5. Another way to see this, is by showing
that U3 does not preserve the initial object. Consider the example S given in Remark
3.3. Then S is the initial object in the category CpoS, but the initial object in DcpoS
is the empty poset.
4 Erratum to Adjoint Relations for PosS
In this section, we consider the bottom square of forgetful functors. Recall that the
adjoint situations related to the category of Sposets have been stated in [
3
]. The free
functor from Sacts to Sposets is described in Theorem 17 of [
3
]. There is an error in
that description which makes it true if and only if the pomonoid S has a trivial order.
In fact, it is stated there that for a given Sact A, the free Sposet is ( A, ), where
is the discrete (equality) order. But, if there are s, t ∈ S with s < t , then we may have
a in A such that as = at and so (as, at ) ∈/ . That is, ( A, ) is not necessarily an
Sposet. In the following, we correct this, and find the free adjunction to the forgetful functor
U9 : PosS → ActS.
Let A be an Sact. Consider the relation R = {(as, at ) : a ∈ A, s ≤ t } on A. Recall
the order R (see [
2
], and ≤R in [
3
]) given by
a R b if and only if there exist a1, a1, . . . , an , an ∈ A;
a = a1 Ra1 = · · · = an Ran = b,
which explicitly means that a R b if and only if there exist a1, a2, . . . , an ∈ A,
s1, . . . , sn ∈ S, t1, . . . , tn ∈ S with si ≤ ti , for all i = 1, . . . , n, and such that
a = a1s1
a1t1 = a2s2
a2t2 = a3s3 . . . antn = b
a3t3 = a4s4 . . . an−1tn−1 = an sn
Then the relation θ given by
is an Sact congruence, and the quotient Sact A/θ turns into an Sposet with the order
given by
aθ b ⇔ a R b Ra,
[a]θ ≤ [b]θ ⇔ a R b.
Theorem 4.1 For a given Sact A, the quotient Sact A/θ given above is the free
Sposet on A.
Proof First note that the Sact A/θ with the above order is an Sposet. To see this, we
show that the order is a welldefined partial order, and also the action is monotone. Let
[a]θ = [c]θ , [b]θ = [d]θ , and [a]θ ≤ [b]θ . Then a R c Ra, b R d R b, and a R b.
This gives that c R a R b R d, and so c R d, since R is transitive. Thus, the order
is welldefined. It is also a partial order, since R , and so ≤ is a preorder. Further,
it is antisymmetric. For, if [a]θ ≤ [b]θ ≤ [a]θ , then a R b Ra. Therefore, by the
definition of θ , [a]θ = [b]θ .
To see that the action is monotone, let [a] ≤ [b] and s ≤ t . Then there exist
a1, a 1, . . . , an , a n ∈ A such that a = a1 Ra 1 = a2 Ra 2 = · · · = an Ra n = b, and
by the definition of R, using the fact that S is a pomonoid, this gives
which means as R bt .
Now, we show that the natural map π : A → A/θ , a → [a], is a universal Sact
map. Let f : A → B be any Sact map to an Sposet B. Then, the map f : A/θ → B,
defined by f ([a]) = f (a), is the unique Sposet map with f ◦ π = f . To see this,
notice that θ ⊆ K er f . For, if a R b then f (a) ≤ f (b), by the definition of R ,
the hypothesis that f is an Sact map, and that B is an Sposet. Therefore, by the
Decomposition Theorem (Fundamental Homomorphism Theorem) of Sacts, there
exists the unique Sact map f as above. We further see that f is an orderpreserving
map, since [a] ≤ [b] means a R b, and so f (a) ≤ f (b).
5 Conclusion
In this final section, applying the investigations done in the above sections, we consider
the composition of forgetful functors given in all the three squares, and consider the
questions whether they have a left or a right adjoint or do not have.
Remark 5.1 Applying the compositions of some of the free functors in the above
sections, we get:
(1) If S is a cpomonoid in which the identity element is the bottom element, then
the free Scpo over a set X is X⊥ × S.
(2) If S is a cpomonoid in which the identity element is the bottom element, then
the free Scpo over a poset P is (Id( P) ∪ {∅}) × S.
(3) If S is a good cpomonoid in which the identity element is the top element, the
free Scpo over an Sact A is Id( A/θ ) ∪ {∅}, where A/θ is the Sposet given in
Theorem 4.1.
(4) If S is a good cpomonoid in which the identity element is the top element, the
free Scpo over an Sposet A is Id( A) ∪ {∅}.
(5) If S is an cpomonoid in which the identity element is the bottom element, then
the free Scpo over a dcpo A is (⊥ ⊕ A) × S. While if the identity element is
the top element, then the free Scpo over a dcpo A is ( A × S)⊥.
(6) The free Sdcpo over a set X is X × S, where X is considered as a dcpo with
the identity order.
(7) The free Sdcpo over a poset P is Id( P) × S.
(8) If S is a good Sdcpo, then the free Sdcpo over an Sact A is Id( A/θ ), where
A/θ is the Sposet given in Theorem 4.1.
(9) The free cpo over a poset P is Id( P) ∪ {∅}.
(10) The free cpo over a set X is ⊥ ⊕ X .
(11) The free dcpo over a set X is (X, =).
Remark 5.2 About the cofree functors, we have:
(1) The cofree functor Set → CpoS does not necessarily exist. This is because,
considering S = {1}, the forgetful functor U : Cpo → Set does not preserve
coproducts. Let 2 denote the two elements chain {0, 1} with 0 < 1, and let 3 denote the three
elements chain {0, a, 1} with 0 < a < 1. The coproduct of 2 and 3 in Cpo is their
coalesced sum 2 3 = ⊥ ⊕ ((2 \ {0}) ∪˙(3 \ {0})) which has four elements, whereas the
coproduct of U8(2) and U8(3) in Set is their disjoint union which has five elements.
Also, similar to Remark 3.8, it is seen that U does not preserve the initial object.
And, similarly, the cofree Scpo over a poset, a dcpo, an Sposet, and an Sact, do not
necessarily exist.
(2) The cofree Sdcpo over ActS does not necessarily exist. Take S = {1}, and see
Theorem 5.3. Similarly, the same is true for the cofree Sdcpos over a set and over a
poset.
(3) The forgetful functors from Cpo to Pos and to Set do not have right adjoints.
This is because they do not preserve initial objects. Notice that the initial object in
Cpo is the singleton poset { }, while the initial objects in Pos and Set are both the
empty set.
In the following, we see that the cofree dcpo over a set does not necessarily exist.
Theorem 5.3 The cofree dcpo over set X exists if and only if  X  = 1.
Proof If  X  = 1, then the cofree dcpo over X is ( X, =). If  X  ≥ 2, we show that the
cofree dcpo over the set X does not exist. Assume the contrary and let K ( X ) be the
cofree dcpo over X with the cofree map k : K ( X ) → X . First we see that k is oneone.
This is because, otherwise there exist x = y ∈ K ( X ) such that k(x ) = k(y) = x0.
Then, considering the map f : { } → X defined by f ( ) = x0, we see that there
exist two dcpo maps f1, f2 : { } → K ( X ), given by f1( ) = x , and f2( ) = y,
such that k ◦ f1 = f and k ◦ f2 = f . This contradicts the universal property of k, and
therefore k is oneone.
Now, take x = y ∈ X and define the map f from the threeelement chain {0, a, 1}
with 0 < a < 1, to X by f (0) = x = f (1), f (a) = y. By the universal property of
k, there exists a unique dcpo map f : 3 → K ( X ) with k ◦ f = f . Now, k( f (0)) =
f (0) = x and k( f (1)) = f (1) = x . Since k is oneone, we get f (0) = f (1).
By monotonicity of f , f (0) ≤ f (a) ≤ f (1) holds. Whence f (a) = f (0), and so
x = k( f (0)) = k( f (a)) = y, which is a contradiction.
Open Problems.
(a) Are the conditions given on S for the existence of the free objects F1, F3, and F6,
necessary?
(b) Can we give a condition on S, under which the cofree vertical functors on the side
of the diagram exist?
Acknowledgments The authors gratefully acknowledge Professor M. Mehdi Ebrahimi’s comments and
conversations during this work, specially we thank his suggestion to draw the diagram of forgetful functors
which helped us to improve the results and have a clear picture of what we were working on. Also our
sincere thanks goes to the referees for their careful reading and helpful suggestions.
1. Abramsky , S. , Jung , A. : Domain Theory, Handbook of Computer Science and Logic , vol. 3 . Clarendon Press, Oxford ( 1995 )
2. Blyth , T.S. , Janowitz , M.F.: Residuation Theory . Pergamon Press, Oxford ( 1972 )
3. BulmanFleming , S. , Mahmoudi , M.: The category of Sposets . Semigr. Forum 71 ( 3 ), 443  461 ( 2005 )
4. Crole , Roy L.: Categories for Types . Cambridge University Press, Cambridge ( 1994 )
5. Davey , B.A. , Priestly , H.A. : Introduction to Lattices and Order . Cambridge University Press, Cambridge ( 1990 )
6. Ebrahimi , M.M. , Mahmoudi , M.: The category of MSets . Ital. J. Pure Appl. Math. 9 , 123  132 ( 2001 )
7. Fiech , A. : Colimits in the category Dcpo . Math. Struct. Comput. Sci. 6 , 455  468 ( 1996 )
8. Jung , A. : Cartesian Closed Categories of Domains, Stichting Mathematisch Centrum, p. 110 . Centrum voor Wiskunde en Informatica , Amsterdam ( 1989 )
9. Jung , A. , Moshier , M. Andrew , Vickers, S. : Presenting dcpos and dcpo algebras . Electron. Notes Theor. Comput. Sci . 219 , 209  229 ( 2008 )
10. Kilp , M. , Knauer , U. , Mikhalev , A. : Monoids, Acts and Categories. Walter de Gruyter, Berlin ( 2000 )
11. Mahmoudi , M. , Moghbeli , H.: The categories of actions of a dcpomonoid on directed complete posets . ( 2014 )
12. Vickers , S. , Townsend , C. : A universal characterization of the double powerlocale . Theor. Comput. Sci . 316 , 297  321 ( 2004 )