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ADVANCES IN MATHEMATICS
Vol.39, No.6
Dec., 2010
Meet Continuity of
Posets via Lim-inf-convergence
LI Qingguo∗ ,
LI Jibo
(College of Mathematics and Econometrics, Hunan Univ., Changsha, Hunan, 410082, P. R. China)
Abstract: The relation between lim-inf-convergence and the Scott topology on a poset P is
exploited. The meet continuity of general posets is characterized in terms of lim-inf-convergence
and it turns out that for a poset there exist close relations among meet continuity, continuity
and the Scott topology.
Key words: lim-inf-convergence; Scott topology; continuous poset; meet continuous poset
MR(2000) Subject Classification: 06A06; 06B30; 54F05 / CLC number: O153.1
Document code: A
Article ID: 1000-0917(2010)06-0755-06
In the literature [1–6] et al., the class of posets and lattices in which all translations s 7−→ x∧s
preserve supremums of directed sets plays an important role. In order to depict this property, in
[1, 3, 5], the meet-continuity of semilattice was introduced in purely order theoretical terms. In
[6], the authors developed a meaningful theory of meet continuity for arbitrary dcpos in terms of
the Scott topology, stressing topological properties of the meet continuity of dcpos. For directed
complete semilattices, the definition of meet continuity in [6] is equivalent to the standard one in
[5]. Furthermore, since the meet operator is not involved in this definition, the topology properties
of meet continuity can be naturally extended to general dcpos. But these definitions are based on
the investigation of dcpos. To get rid of the restriction to directed sets, we characterize the meet
continuity of arbitrary posets via lim-inf-convergence defined in order theoretical manner in [7].
Based on the result[7] that for a poset P , the lim-inf-convergence is topological if and only if P
is continuous poset, we consider the topological properties of meet continuous posets, enriching
interplay between topological and order theoretical aspects of meet continuity of posets.
In Section 1, we exploit the relation between the lim-inf-convergence and the Scott topology
on posets. In [7], the authors showed that for an arbitrary poset P , the lim-inf-convergence is
topological if and only if P is a continuous poset by means of Kelley’s standard characterization
for a class of convergent nets to be topological[8] . We show that the topology induced by the
class of lim-inf-convergence on a poset is exactly the Scott topology. Then the previous result
can be shown directly by using the Scott topology.
Finally, in Section 2, we introduce the meet continuity of an arbitrary poset via the liminf-convergence. Since in continuous posets, the lim-inf-convergence is topological with respect
to the Scott topology, this definition of meet continuity for domains is equivalent to the one in
[6]. Furthermore, for directed complete semilattices, the definition is equivalent to the standard
one (see O-4.1, in [5]). And it turns out that for a poset, there exist close relations among meet
continuity, continuity and the Scott topology.
We recall some basic notions. Let P be a partially ordered set(or poset, for short) and A be
a subset of P . We denote the set {x ∈ P | ∃ y ∈ A : s.t. x ≤ y} by ↓ A, and the set ↑ A is defined
Received date: 2008-08-12. Revised date: 2009-07-24.
Foundation item: This research is supported by the National Natural Science Foundation of China(No. 10771056).
E-mail: ∗ [email protected]
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dually. If A = ↓ A, then A is called a downset or lower set; the dual is an upset or upper set. A
poset P is said to be directed complete and is called a dcpo if every directed subset D of P has
a supremum. The way-below relation on P can be defined in the same way as for dcpos. We
say that x is way-below y, x y, if for any directed set D ∈ P for which ∨D exists and y 6 ∨D,
then there is d ∈ D such that x 6 d. A poset P is called a continuous poset if for each a ∈ P ,
the set {x ∈ P : x a} is directed and a = ∨{x ∈ P : x a}. Throughout the paper we shall
use ∨A and ∧A to denote the supremum of A and the infimum of A respectively.
1 Lim-inf-convergence and Scott Topology
Definition 1.1[7] A net (xi )i∈I in a poset P is said to lim-inf-converge to an element
x ∈ P if there exists a directed subset D of P such that
(1) ∨D exists with ∨D > x, and
(2) ∀d ∈ D, xi > d holds eventually, that is, there exists k ∈ I such that xi > d for all i > k.
In this case we write briefly x ≡ lim-inf xi . Let L denote the class of those pairs ((xi )i∈I , x)
such that x ≡ lim-inf xi .
It is clear that any constant net (xi )i∈I in a poset P with value x lim-inf-converges to x.
If (xi )i∈I lim-inf-converges to x, then it lim-inf-converges to every y with y 6 x. Thus the
lim-inf-limits of a net are generally not unique.
Next, we recall the general relation between convergence and topology. For an arbitrary
class S of pairs ((xi )i∈I , x) consisting of a net and an element of any set P , we denote
O(S) = {U ⊆ P : whenever ((xi )i∈I , x) ∈ S and x ∈ U , then xi ∈ U holds eventually}.
Clearly, both φ and P belong to O(S), which is closed under the formation of arbitrary unions
and finite intersections, that is to say, O(S) is a topology on P .
By the very definition we know that for any ((xi )i∈I , x) ∈ L, the element x is a limit of the
net (xi )i∈I relative to the topology O(L). However, since φ and P may very well be the only
element of O(L), we need to exploit it in detail.
Lemma 1.1 Let P be a poset and U ⊆ P . Then U ∈ O(L) iff U is an upset and ∨D ∈ U
implies D ∩ U 6= φ for every directed subset D of P possessing a join.
Proof Suppose U ∈ O(L). Assume u ∈ U and u 6 x. Since the constant net (x) with
value x lim-inf-converges to x and u 6 x, we have that ((x), u) ∈ L. Because u ∈ U ∈ O(L)
holds, we conclude from the definition of O(L) that the net (x) must be eventually in U . This
means x ∈ U . Thus U is an upset. On the other hand, let D be a directed subset of P possessing
a join and ∨D ∈ U . Consider the net (xd )d∈D with xd = d. By the definition, we have that
((xd )d∈D , ∨D) ∈ L. Since ∨D ∈ U ∈ O(L), we conclude that d = xd is eventually in U , whence
D ∩ U 6= φ. Thus the necessity has been proved.
For the sufficiency, suppose that U satisfies the two conditions. We take ((xi )i∈I , x) ∈ L
with x ∈ U , and we must show that xi is eventually in U . By the definition of L, we have that
x 6 ∨D for some directed set D possessing a join and that for all d ∈ D, xi > d holds eventually.
Since U is an upset, x ∈ U implies ∨D ∈ U . So there exists some d0 ∈ D such that d0 ∈ U by
the second condition. Then for some j ∈ I, xi > d0 for all i > j, and so xi ∈ U for all i > j.
Thus U ∈ O(L).
We recall[5,9] that a subset A of a poset P is said to be Scott closed if ↓ A = A and for any
directed set D ⊆ A, ∨D ∈ A whenever ∨D exists. The complement of a Scott closed set is a Scott
open set. All the open sets of P form a topology called the Scott topology, denoted by σ(P ). By
the previous lemma, we have that the topologyO(L) is exactly the Scott topology on the poset P.
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Lemma 1.2 If P is a continuous poset, then all sets {y ∈ P : x y} for x ∈ P are Scott
open. Conversely, if y ∈ intσ (↑ x), then x y, where intσ (↑ x) means the interior of ↑ x via
Scott topology.
Proof The standard proof for continuous dcpos(see [5, Proposition II-1.6]) carries over to
continuous posets.
Proposition 1.1 Let P be a continuous poset. Then x ≡ lim-inf xi iff the net (xi )i∈I → x
with respect to σ(P ), that is to say, lim-inf-convergence is topological for the Scott topology.
Proof Since O(L) = σ(P ), x ≡ lim-inf xi implies that (xi )i∈I → x with respect to σ(P ).
Conversely, suppose that we have a convergent net (xi )i∈I → x in the Scott topology. Since
P is a continuous poset, we have the set {y ∈ P : y x} is directed and has supremum x.
Furthermore, for each y ∈ {y ∈ P : y x}, we have that {z ∈ P : y z} is a Scott open set
containing x by Lemma 1.2. Thus the net (xi )i∈I is eventually in the set {z ∈ P : y z}, and
hence xi > y holds eventually. So x ≡ lim-inf xi .
Proposition 1.2 If, in a poset P , the lim-inf-convergence is topological with respect to
σ(P ), then P is a continuous poset.
Proof Let x ∈ P . Define I = {(U, n, a) ∈ N (x) × N × P : a ∈ U }, where N (x) consists of
all Scott open sets containing x, and define an order on I to be the lexicographic order on the first
two coordinates, that is, (U, m, a) < (V, n, b) iff V is a proper subset of U or U = V and m < n.
Let xi = a for i = (U, n, a) ∈ I, then it is easy to see that (xi )i∈I is a net and converges to x
in the Scott topology. Since the lim-inf-convergence is topological with respect to σ(P ), we have
x ≡ lim-inf xi , and there exists a directed set D such that x 6 ∨D and xi > d holds eventually
for all d ∈ D. So, for all d ∈ D, there exists i = (U, m, a) ∈ I such that (V, n, b) = j > i implies
d 6 b. In particular, we have (U, m + 1, b) > (U, m, a) for all b ∈ U , and thus U ⊆↑ d, i.e.,
x ∈ intσ (↑ d). By the Lemma 1.2, d x. Since D is directed with supremum greater than or
equal to x, we conclude that x is the directed supremum of D. Thus P is a continuous poset.
The combination of Propositions 1.1 and 1.2 deduces the following theorem[7, Theorem 1] .
Theorem 1.1 For a poset P , the following statements are equivalent:
(1) The lim-inf-convergence is topological for the Scott topology σ(P );
(2) P is a continuous poset.
2 Meet Continuity of Posets and Lim-inf-convergence
We all know that a semilattice S is called meet continuous if it is a dcpo, and satisfies
x ∧ ∨D = ∨{x ∧ d : d ∈ D}, for all x ∈ S and all directed sets D ⊆ S. Since in a poset the
supremum of a directed subset and the infimum of two elements need not exist, thus we have to
define the meet continuity of an arbitrary poset in a different way.
Definition 2.1 Let P be a poset and x, y ∈ P . Define x ≺ y if for every directed subset
D of P possessing a join, the relation y 6 ∨D implies that there exists a net (xi )i∈I ⊆↓ y ∩ ↓ D,
such that x ≡ lim-inf xi .
From the definition of the relation ≺, we see the following properties easily.
Proposition 2.1 If P is a poset and a, x, y, b ∈ P , then
(1) x ≺ y implies x 6 y;
(2) x y implies x ≺ y, and
(3) a 6 x ≺ y 6 b implies a ≺ b.
Example 2.1(see Remark 1 in [7]) The relation ≺ is different from , for example, let
P = {a, b} ∪ {bi : i ∈ N}, where N denotes the set of all natural numbers. The order on P is
defined by a < b, b1 < b2 < · · · < b. By the definition of the relation ≺, we have a ≺ b. On the
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other hand, the set {bi : i ∈ N} is directed and b = ∨{bi : i ∈ N}. But for any bi ∈ {bi : i ∈ N},
a bi . So a b does not hold.
Definition 2.2 A poset P is called meet continuous if the set {y ∈ P : y ≺ x} is directed
and x = ∨{y ∈ P : y ≺ x} holds for every x ∈ P . We will say that a poset P is join continuous
iff P op is meet continuous. A complete lattice which is meet continuous as a poset will be called
a meet continuous lattice.
It follows from Proposition 2.1(1) that if x ≺ x for each x ∈ P , then the poset P is meet
continuous. Recall that for a class S of convergent nets on a set P , we say that S satisfies
the conditionIterated Limits) if ((xi )i∈I , x) ∈ S, and if ((xi,j )j∈J(i) , xi ) ∈ S for all i ∈ I, then
Q
((xi,f (i) )(i,f )∈I×M , x) ∈ S, where M = i∈I J(i) is a product of directed sets. By Lemma 4 in
[7], it is clear that if the class L satisfies the condition(Iterated Limits) in a meet continuous
poset P , then P is continuous.
Proposition 2.2 A semilattice S is meet continuous as a poset iff for every directed set
D ⊆ S possessing a join and for every x ∈ S, x ∧ ∨D = ∨{x ∧ d : d ∈ D}.
Proof First, let D be a directed set with ∨D exists, and let x ∈ S. It is clear that x ∧ ∨D
is an upper bound of the set {x ∧ d : d ∈ D}. Suppose that y is an arbitrary upper bound
of the set {x ∧ d : d ∈ D}. For any z with z ≺ x ∧ ∨D, by Definition 2.1, there exists a net
(xi )i∈I ⊆↓ (x ∧ ∨D)∩ ↓ D such that z ≡ lim-inf xi . Since ↓ (x ∧ ∨D)∩ ↓ D ⊆↓ y, we have z ≤ y
by the definition of lim-inf-convergence. Hence, x ∧ ∨D = ∨{z ∈ S : z ≺ x ∧ ∨D} ≤ y. Thus,
x ∧ ∨D = ∨{x ∧ d : d ∈ D}.
Second, if x 6 ∨D, then x = x ∧ ∨D = ∨{x ∧ d : d ∈ D}. Consider the net (xd )d∈D with
xd = x ∧ d. Now there exists a directed set {x ∧ d : d ∈ D} such that x 6 ∨{x ∧ d : d ∈ D} and
for all x ∧ d ∈ {x ∧ d : d ∈ D}, xd > x ∧ d holds eventually. Thus we have x ≺ x, and hence the
set {y ∈ S : y ≺ x} is directed by Proposition 2.1(3) and x = ∨{y ∈ S : y ≺ x}. Now we have
that S is meet continuous.
From the previous proposition, we can see that for directed complete semilattices our definition of meet continuity is equivalent to the standard one. Furthermore, if a dcpo P is meet
continuous as a poset in our sense, then for any x ∈ P and any directed set D with x 6 ∨D, x
is in the Scott closure of ↓ D∩ ↓ x. Since only in continuous posets, the lim-inf-convergence is
topological with respect to the Scott topology, the converse is not generally true.
Proposition 2.3 Every continuous poset is meet continuous.
Proof Let P be a continuous poset and x ∈ P . By the definition of the continuity the
set {y ∈ P : y x} is directed and x = ∨{y ∈ P : y x}. Let M = {y ∈ P : y x}.
Then M is directed and x = ∨M . Suppose that x 6 ∨D for any directed set D where ∨D
exists. Consider the net (xm )m∈M , where xm = m for each m ∈ M . Then x ≡ lim-inf xm and
(xm )m∈M ⊆↓ x∩ ↓ D. Thus x ≺ x, and hence P is meet continuous.
Lemma 2.1 In a meet continuous poset P , for any Scott open set U and any x ∈ P , we
have that ↑ (U ∩ ↓ x) is Scott open.
Proof Let x ∈ P and U be Scott open. If ∨D ∈↑ (U ∩ ↓ x) for some directed set D
possessing a join, then there exists z ∈ U ∩ ↓ x such that z 6 ∨D. Since P is meet continuous,
the set {t ∈ P : t ≺ z} is a directed set and z = ∨{t ∈ P : t ≺ z}. It follows from z =
∨{t ∈ P : t ≺ z} ∈ U and U Scott open that U ∩ {t ∈ P : t ≺ z} =
6 ∅. Thus, there exists
t0 ≺ z such that t0 ∈ U . Hence, there exists a net (xi )i∈I ⊆↓ z∩ ↓ D such that t0 ≡ lim-inf xi .
Thus (xi )i∈I → t0 with respect to σ(P ), and hence ↓ z∩ ↓ D ∩ U 6= φ. Now we have that
D∩ ↑ (U ∩ ↓ x) ⊇ D∩ ↑ (U ∩ ↓ z) 6= φ, which establishes that ↑ (U ∩ ↓ x) is Scott open.
Proposition 2.4 Let P be a meet continuous poset. Suppose that for any x ∈ P and any
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Scott open set U containing x, there exists y ∈ U ∩ ↓ x such that ↑ y ∩ ↓ x is a relative Scott
neighborhood of x in ↓ x. Then P is a continuous poset.
Proof Let x ∈ P . We consider the family D of all y ∈↓ x such that ↑ y ∩ ↓ x is a
relative Scott neighborhood of x in ↓ x. Let y ∈ D. Then there exists a Scott open U such that
↓ x ∩ U ⊆↑ y. By the previous proposition we have that ↑ (U ∩ ↓ x) is Scott open; note that it
is also a subset of ↑ y. Thus ↑ y is a Scott neighborhood of x in P , and hence y x. Thus
D ⊆ {a ∈ P : a x}.
Now suppose that ↑ yi ∩ ↓ x is a relative Scott neighborhood of x in ↓ x for i = 1, 2. Then
their intersection is also the neighborhood of x in ↓ x, so there exists V Scott open such that
x ∈ V ∩ ↓ x ⊆↑ y1 ∩ ↑ y2 . By hypothesis, there exists y ∈ V ∩ ↓ x such that ↑ y ∩ ↓ x is a relative
Scott neighborhood of x in ↓ x. Hence D is directed.
Finally, let z < x. Then P \ ↓ z is a Scott open set containing x, and again by hypothesis
we can pick y ∈ (P \ ↓ z)∩ ↓ x such that y ∈ D. It follows that x = ∨D. Since x is arbitrary, P
is continuous.
Let P be a poset. The topology generated by the complements P \ ↑ x of principal filters is
called the lower topology and is denoted by ω(P ). The common refinement σ(P ) ∧ ω(P ) of the
Scott topology and the lower topology is called the Lawson topology and is denoted by λ(P ). In
a poset, meet continuity enriches relationships between the Scott and Lawson topologies.
Proposition 2.5 For a meet continuous poset P , we have
(1) if U ∈ λ(P ), then ↑ U ∈ σ(P );
(2) if X is an upper set, then intσ X = intλ X;
(3) if X is a lower set, then clσ X = clλ X.
Proof (1) Suppose that y ∈↑ U and that U is a Lawson open set. Let x ∈ U such that
x 6 y. Then there exists a basic Lawson open set V \ ↑ F , where V is Scott open and F is finite,
such that x ∈ V \ ↑ F ⊆ U . Thus ↑ (V ∩ ↓ x) ⊆↑ (V \ ↑ F ) ⊆↑ U . Since the set ↑ (V ∩ ↓ x) is
Scott open by Lemma 2.1, it follows that y is in the Scott-interior of ↑ U . Since y was arbitrary
in ↑ U , the latter is Scott open.
(2) It is clear that intσ X ⊆ intλ X by the definition of the Lawson topology. By (1),
↑ intλ X ∈ σ(P ). Since ↑ intλ X ⊆↑ X = X, intλ X ⊆ intσ X.
(3) It is straightforward by (2).
Proposition 2.6 If P is a meet continuous poset, then σ(P )op is a meet continuous lattice,
and hence it is a frame.
Proof Let Γ(P ) = {A ⊆ P : P \A ∈ σ(P )} be the Scott closed set lattice(ordered by
set-theoretic inclusion), then σ(P )op ∼
= Γ(P ). Suppose that {Ai : i ∈ I} ⊆ Γ(P ) is a family
of Scott closed sets such that K ⊆ ∨i∈I Ai . Since ∨i∈I Ai = clσ (∪i∈I Ai ) holds, we have K ⊆
clσ (∪i∈I Ai ). Assume that there exists x ∈ K, such that x is not in ∨i∈I (K ∩ Ai ). Note that
∨i∈I (K ∩ Ai ) = clσ (∪i∈I (K ∩ Ai )), we can find a Scott open neighborhood U of x such that
U ∩ (∪i∈I (K ∩ Ai )) = U ∩ K ∩ (∪i∈I Ai ) = φ. And by Lemma 2.1, ↑ (U ∩ ↓ x) ∈ σ(P ). Because
x ∈↑ (U ∩ ↓ x) and x ∈ K ⊆ clσ (∪i∈I Ai ), we have that (∪i∈I Ai ) ∩ (↑ (U ∩ ↓ x)) 6= φ holds.
It follows that there exists y ∈ P and j ∈ I such that y ∈ Aj ∩ ↑ (U ∩ ↓ x). This means that
there exists a ∈ U such that a ∈↓ x ∩ ↓ y. Since both K and Aj are Scott closed sets, it follows
a ∈ U ∩ K ∩ Aj ⊆ U ∩ K ∩ (∪i∈I Ai ). This is a contradiction. Hence K = ∨i∈I (K ∩ Ai ). Thus
σ(P )op is a meet continuous lattice. Since σ(P )op is always distributive, it is a frame by Remark
O-4.3 in [5].
Corollary 2.1 Let P be a poset. Consider the following conditions:
(1) P is meet continuous;
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(2) σ(P ) is join continuous;
(3) σ(P )op is a frame.
The implications (1) ⇒ (2) ⇔ (3) are true. If P is a complete lattice, then (3) ⇒ (1).
It is well known that the Scott topology of a continuous domain L has a basis consisting
of Scott open filters. This implies that the lattice σ(L) of Scott open sets has enough coprime
elements. But, for a general poset, the Scott topology is usually short of this kind of point. Next
we investigate under what conditions the Scott topology has a basis of open filters for a poset.
Definition 2.3[5, 6] Let P be a poset. A set F ⊆ P is filtered if F 6= φ and for all x, y ∈ F ,
there is z ∈ F such that z 6 x and z 6 y. In this case, if F =↑ F , then F is called a filter.
We say that a poset with a topology has small open filtered sets if and only if each point has a
neighborhood basis of open filtered sets.
Propiosition 2.7 Let P be a meet continuous poset. Then the following are equivalent:
(1) The Scott topology σ(P ) has a basis of open filters;
(2) The Lawson topology λ(P ) has small open filtered sets.
Proof Suppose that the Scott topology σ(P ) has a basis of open filters. Let W ⊆ P be a
Lawson open neighborhood of x. Then there is a Scott open filter U and a finite set F such that
x ∈ U \ ↑ F ⊆ W . Obviously, U \ ↑ F is a Lawson open filtered set. Thus the Lawson topology
λ(P ) has small open filtered sets.
Conversely, Suppose that the Lawson topology λ(P ) has small open filtered sets. Let U
be an arbitrary Scott open neighborhood of x. Since σ(P ) ⊆ λ(P ), U is also a Lawson open
neighborhood. Thus there exists a Lawson open filtered set V such that x ∈ V ⊆ U . Note that
since P is meet continuous, ↑ V is Scott open from Proposition 2.5. Therefore, the Scott topology
σ(P ) has a basis of open filters.
References
[1] Birkhoff, G., Lattice Theory, Volume 25 of AMS Colloquium Publications, American Mathematical Society,
revised edition, 1967.
[2] Hofmann, K.H. and Stralka, A.R., The algebraic theory of compact Lawson semilattices: applications of
Galois connections to compact semilattices, Dissertationes Mathematicae, 1976, 137: 1-54.
[3] Isbell, J.R., Meet-continuous Lattices, Symposia Mathematica 1975, 16: 41-54.
[4] Isbell, J.R., Direct limits of meet-continuous lattices, Journal of Pure and Applied Algebra, 1982, 23: 33-35.
[5] Gierz, G. et al., Continuous and Domain, Cambridge: Cambridge Univ. Press, 2003.
[6] Kou Hui, Liu Yingming and Luo Maokang, On meet-continuous dcpos, GQ, Zhang et al. (eds.), Domain
Theory, Logic and Computation, Kluwer Academic Publisher, Netherlands, 2003, 117-135.
[7] Zhao Bin, Zhao Dongsheng, Lim-inf-convergence in partially ordered sets, J. Math. Anal. Appl., 2005, 309:
701-708.
[8] Kelley, J.L., General Topology, Van Nostrand, New York, 1955.
[9] Xu Luoshan, Continuity of posets via Scott topology and sobrification, Topology Appl., 2006, 153: 1886-1894.
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